{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [1,0]\nThe edges are: (0,5) (0,6) (0,1) (0,4) (1,5) (2,5) (2,4) (3,6) (3,4) (4,5) (4,6) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [12,9]\nnode 1: [5,5]\nnode 2: [8,6]\nnode 3: [7,6]\nnode 4: [9,11]\nnode 5: [11,11]\nnode 6: [8,9]\nnode 7: [3,2]\n", "difficulty": "easy", "doc_id": "0"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [0,0]\nnode 10: [0,0]\nThe edges are: (0,3) (0,1) (1,10) (1,9) (1,3) (1,4) (2,10) (2,3) (3,4) (5,6) (5,9) (7,8) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,6]\nnode 1: [9,11]\nnode 2: [3,5]\nnode 3: [7,8]\nnode 4: [4,6]\nnode 5: [3,3]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [1,3]\nnode 9: [3,4]\nnode 10: [4,5]\n", "difficulty": "hard", "doc_id": "1"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [0,0]\nnode 10: [0,1]\nnode 11: [1,1]\nnode 12: [0,1]\nnode 13: [1,1]\nnode 14: [0,0]\nThe edges are: (0,6) (0,3) (0,13) (0,2) (0,1) (0,9) (1,8) (1,4) (2,4) (2,10) (2,7) (4,6) (4,13) (4,12) (5,8) (5,14) (6,13) (6,12) (7,10) (8,11) (8,13) (8,10) (9,14) (10,11) (11,13) (12,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [15,9]\nnode 1: [9,12]\nnode 2: [8,9]\nnode 3: [3,3]\nnode 4: [14,11]\nnode 5: [5,7]\nnode 6: [10,12]\nnode 7: [6,3]\nnode 8: [13,7]\nnode 9: [4,5]\nnode 10: [9,10]\nnode 11: [11,9]\nnode 12: [6,9]\nnode 13: [14,17]\nnode 14: [3,2]\n", "difficulty": "hard", "doc_id": "2"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [1,1]\nThe edges are: (0,2) (0,3) (0,5) (0,4) (1,3) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,7]\nnode 1: [2,2]\nnode 2: [2,2]\nnode 3: [3,2]\nnode 4: [3,4]\nnode 5: [4,4]\n", "difficulty": "easy", "doc_id": "3"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [1,0]\nThe edges are: (0,2) (1,2) (1,4) (1,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [4,1]\nnode 3: [3,1]\nnode 4: [3,1]\n", "difficulty": "easy", "doc_id": "4"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [0,1]\nThe edges are: (0,1) (0,5) (0,3) (0,4) (1,3) (1,4) (2,3) (3,6) (3,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,14]\nnode 1: [8,12]\nnode 2: [3,5]\nnode 3: [7,12]\nnode 4: [5,10]\nnode 5: [8,12]\nnode 6: [3,5]\n", "difficulty": "easy", "doc_id": "5"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [0,1]\nnode 8: [1,1]\nnode 9: [1,0]\nThe edges are: (0,3) (1,6) (1,5) (1,3) (2,7) (2,3) (2,8) (3,6) (4,5) (5,7) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,2]\nnode 1: [6,4]\nnode 2: [7,4]\nnode 3: [3,6]\nnode 4: [1,1]\nnode 5: [2,2]\nnode 6: [5,3]\nnode 7: [2,4]\nnode 8: [2,4]\nnode 9: [2,1]\n", "difficulty": "hard", "doc_id": "6"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [0,0]\nThe edges are: (0,9) (0,4) (0,3) (0,8) (1,7) (2,9) (2,10) (3,5) (4,6) (4,8) (5,9) (5,7) (6,10) (7,8) (8,9) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,5]\nnode 1: [3,2]\nnode 2: [5,5]\nnode 3: [4,4]\nnode 4: [3,3]\nnode 5: [7,5]\nnode 6: [4,4]\nnode 7: [4,3]\nnode 8: [10,7]\nnode 9: [7,9]\nnode 10: [3,3]\n", "difficulty": "hard", "doc_id": "7"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [0,0]\nThe edges are: (0,9) (0,2) (0,8) (1,6) (1,2) (2,7) (2,3) (3,4) (3,6) (3,7) (4,9) (5,6) (6,9) (7,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,7]\nnode 1: [2,6]\nnode 2: [2,6]\nnode 3: [2,9]\nnode 4: [3,5]\nnode 5: [1,3]\nnode 6: [5,7]\nnode 7: [5,9]\nnode 8: [0,3]\nnode 9: [1,7]\n", "difficulty": "hard", "doc_id": "8"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [0,1]\nThe edges are: (0,10) (0,9) (0,7) (1,3) (2,5) (3,10) (3,8) (4,7) (5,8) (5,7) (6,10) (6,8) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,6]\nnode 1: [2,2]\nnode 2: [3,2]\nnode 3: [5,2]\nnode 4: [2,2]\nnode 5: [5,2]\nnode 6: [5,2]\nnode 7: [6,4]\nnode 8: [6,6]\nnode 9: [5,4]\nnode 10: [6,8]\n", "difficulty": "hard", "doc_id": "9"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [1,1]\nnode 10: [0,1]\nnode 11: [0,0]\nnode 12: [0,0]\nnode 13: [0,1]\nThe edges are: (0,10) (1,13) (1,4) (1,12) (2,11) (3,10) (3,9) (3,13) (3,8) (4,9) (4,5) (4,11) (5,9) (5,11) (5,12) (5,8) (5,7) (6,7) (8,11) (9,10) (10,13) (11,12) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [7,3]\nnode 2: [4,0]\nnode 3: [9,5]\nnode 4: [12,3]\nnode 5: [15,3]\nnode 6: [1,0]\nnode 7: [5,1]\nnode 8: [10,4]\nnode 9: [11,8]\nnode 10: [7,6]\nnode 11: [11,3]\nnode 12: [11,3]\nnode 13: [7,8]\n", "difficulty": "hard", "doc_id": "10"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [1,1]\nThe edges are: (0,3) (0,7) (0,1) (1,3) (1,7) (1,5) (2,3) (2,5) (3,7) (4,6) (4,5) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,9]\nnode 1: [8,11]\nnode 2: [5,5]\nnode 3: [6,10]\nnode 4: [4,2]\nnode 5: [5,5]\nnode 6: [4,3]\nnode 7: [6,9]\n", "difficulty": "easy", "doc_id": "11"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [1,1]\nThe edges are: (0,1) (1,4) (1,2) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [4,3]\nnode 2: [1,2]\nnode 3: [2,1]\nnode 4: [2,3]\n", "difficulty": "easy", "doc_id": "12"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [0,0]\nnode 11: [0,0]\nnode 12: [0,1]\nThe edges are: (0,11) (0,5) (0,7) (0,4) (1,4) (1,6) (1,9) (2,10) (2,5) (2,7) (3,8) (4,5) (4,9) (5,11) (5,9) (6,10) (6,9) (7,10) (8,9) (8,12) (10,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,3]\nnode 1: [5,4]\nnode 2: [3,6]\nnode 3: [1,2]\nnode 4: [9,5]\nnode 5: [10,5]\nnode 6: [5,7]\nnode 7: [6,6]\nnode 8: [2,2]\nnode 9: [8,6]\nnode 10: [3,3]\nnode 11: [5,2]\nnode 12: [2,6]\n", "difficulty": "hard", "doc_id": "13"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [0,0]\nnode 12: [0,1]\nnode 13: [1,1]\nThe edges are: (0,7) (0,13) (0,4) (1,10) (2,11) (2,6) (3,5) (3,10) (4,7) (4,8) (5,9) (5,12) (5,7) (5,13) (6,9) (6,12) (6,13) (7,9) (8,12) (9,11) (11,12) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,7]\nnode 1: [1,2]\nnode 2: [1,7]\nnode 3: [4,7]\nnode 4: [5,8]\nnode 5: [5,8]\nnode 6: [3,4]\nnode 7: [9,11]\nnode 8: [4,3]\nnode 9: [5,15]\nnode 10: [2,2]\nnode 11: [3,2]\nnode 12: [5,16]\nnode 13: [9,13]\n", "difficulty": "hard", "doc_id": "14"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,0]\nnode 9: [0,0]\nnode 10: [1,0]\nnode 11: [1,0]\nnode 12: [0,1]\nThe edges are: (0,12) (0,4) (1,9) (2,11) (2,4) (3,9) (3,11) (3,6) (3,5) (4,6) (6,11) (6,8) (7,9) (7,8) (8,9) (8,12) (10,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,2]\nnode 1: [1,2]\nnode 2: [5,5]\nnode 3: [4,7]\nnode 4: [2,2]\nnode 5: [3,2]\nnode 6: [9,10]\nnode 7: [2,5]\nnode 8: [4,3]\nnode 9: [4,5]\nnode 10: [2,0]\nnode 11: [5,3]\nnode 12: [1,5]\n", "difficulty": "hard", "doc_id": "15"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [1,0]\nThe edges are: (0,3) (0,5) (0,4) (1,6) (2,3) (3,5) (3,4) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,1]\nnode 1: [0,2]\nnode 2: [2,1]\nnode 3: [7,1]\nnode 4: [3,2]\nnode 5: [3,4]\nnode 6: [4,0]\n", "difficulty": "easy", "doc_id": "16"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [1,0]\nnode 10: [0,1]\nnode 11: [0,1]\nnode 12: [0,0]\nnode 13: [1,0]\nThe edges are: (0,8) (0,9) (0,6) (0,11) (1,5) (1,13) (2,3) (2,4) (2,10) (2,6) (3,8) (3,10) (3,12) (4,7) (4,11) (5,8) (5,6) (6,10) (6,9) (6,13) (7,12) (7,9) (7,13) (9,10) (11,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [12,11]\nnode 1: [6,3]\nnode 2: [12,10]\nnode 3: [11,9]\nnode 4: [9,6]\nnode 5: [7,8]\nnode 6: [19,11]\nnode 7: [10,7]\nnode 8: [8,4]\nnode 9: [14,8]\nnode 10: [12,11]\nnode 11: [8,5]\nnode 12: [5,3]\nnode 13: [11,8]\n", "difficulty": "hard", "doc_id": "17"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [1,0]\nThe edges are: (0,4) (0,6) (0,5) (1,3) (1,4) (2,4) (3,6) (3,5) (5,8) (5,10) (6,7) (6,9) (7,8) (8,10) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [8,3]\nnode 1: [1,3]\nnode 2: [1,1]\nnode 3: [8,4]\nnode 4: [1,5]\nnode 5: [3,6]\nnode 6: [1,4]\nnode 7: [6,2]\nnode 8: [4,2]\nnode 9: [5,2]\nnode 10: [6,2]\n", "difficulty": "hard", "doc_id": "18"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [0,0]\nThe edges are: (0,1) (1,3) (1,6) (2,7) (2,6) (3,7) (3,6) (3,8) (4,6) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,0]\nnode 1: [8,5]\nnode 2: [5,3]\nnode 3: [6,3]\nnode 4: [4,2]\nnode 5: [4,2]\nnode 6: [8,3]\nnode 7: [5,3]\nnode 8: [3,2]\n", "difficulty": "hard", "doc_id": "19"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,0]\nnode 8: [1,1]\nnode 9: [1,0]\nnode 10: [1,0]\nnode 11: [0,1]\nnode 12: [1,0]\nnode 13: [0,0]\nThe edges are: (0,9) (0,8) (0,4) (0,7) (1,8) (1,5) (2,12) (3,9) (3,5) (3,6) (4,7) (7,9) (7,8) (7,10) (9,11) (11,13) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,8]\nnode 1: [1,2]\nnode 2: [1,2]\nnode 3: [3,5]\nnode 4: [5,5]\nnode 5: [4,1]\nnode 6: [2,0]\nnode 7: [3,7]\nnode 8: [7,6]\nnode 9: [9,5]\nnode 10: [3,3]\nnode 11: [2,6]\nnode 12: [3,0]\nnode 13: [2,0]\n", "difficulty": "hard", "doc_id": "20"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [0,1]\nnode 10: [0,1]\nnode 11: [1,1]\nThe edges are: (0,6) (0,11) (1,4) (2,6) (2,4) (2,10) (2,7) (3,10) (3,7) (4,9) (5,11) (6,10) (7,8) (7,10) (8,11) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [0,2]\nnode 2: [2,5]\nnode 3: [1,2]\nnode 4: [2,3]\nnode 5: [2,1]\nnode 6: [4,5]\nnode 7: [3,5]\nnode 8: [2,2]\nnode 9: [2,3]\nnode 10: [2,5]\nnode 11: [5,5]\n", "difficulty": "hard", "doc_id": "21"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [0,1]\nThe edges are: (0,6) (0,2) (0,4) (1,5) (2,6) (2,7) (3,4) (4,6) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,6]\nnode 1: [2,1]\nnode 2: [3,6]\nnode 3: [2,2]\nnode 4: [6,7]\nnode 5: [2,3]\nnode 6: [5,7]\nnode 7: [1,2]\n", "difficulty": "easy", "doc_id": "22"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [0,1]\nnode 8: [1,1]\nThe edges are: (0,4) (1,8) (1,4) (1,5) (2,8) (3,4) (4,7) (6,8) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,2]\nnode 1: [2,4]\nnode 2: [2,1]\nnode 3: [0,2]\nnode 4: [2,6]\nnode 5: [1,3]\nnode 6: [3,2]\nnode 7: [1,4]\nnode 8: [3,6]\n", "difficulty": "hard", "doc_id": "23"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [1,0]\nThe edges are: (0,9) (0,7) (0,3) (0,6) (1,7) (2,4) (3,5) (4,9) (6,8) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,8]\nnode 1: [0,2]\nnode 2: [1,1]\nnode 3: [2,2]\nnode 4: [0,3]\nnode 5: [0,2]\nnode 6: [3,1]\nnode 7: [4,1]\nnode 8: [0,4]\nnode 9: [3,2]\n", "difficulty": "hard", "doc_id": "24"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,1]\nThe edges are: (0,2) (0,1) (0,4) (0,3) (1,2) (2,4) (2,5) (2,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,6]\nnode 1: [2,4]\nnode 2: [1,4]\nnode 3: [2,4]\nnode 4: [2,4]\nnode 5: [1,3]\n", "difficulty": "easy", "doc_id": "25"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [0,0]\nnode 8: [1,0]\nnode 9: [1,1]\nnode 10: [1,1]\nnode 11: [1,0]\nnode 12: [0,1]\nThe edges are: (0,2) (0,1) (1,4) (2,12) (3,12) (3,7) (3,11) (4,8) (5,6) (5,9) (6,12) (6,7) (6,10) (7,8) (7,11) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,4]\nnode 1: [3,0]\nnode 2: [2,1]\nnode 3: [3,4]\nnode 4: [2,3]\nnode 5: [1,2]\nnode 6: [4,3]\nnode 7: [3,6]\nnode 8: [4,1]\nnode 9: [1,1]\nnode 10: [1,2]\nnode 11: [4,3]\nnode 12: [2,7]\n", "difficulty": "hard", "doc_id": "26"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,0]\nnode 10: [1,1]\nnode 11: [1,0]\nThe edges are: (0,9) (0,10) (0,1) (1,8) (1,5) (1,2) (2,9) (2,8) (3,9) (3,10) (4,6) (4,7) (4,11) (6,8) (9,10) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [11,8]\nnode 1: [7,8]\nnode 2: [11,10]\nnode 3: [7,4]\nnode 4: [4,4]\nnode 5: [4,4]\nnode 6: [4,4]\nnode 7: [2,1]\nnode 8: [8,8]\nnode 9: [8,7]\nnode 10: [8,6]\nnode 11: [7,4]\n", "difficulty": "hard", "doc_id": "27"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,1]\nThe edges are: (0,4) (0,2) (1,4) (1,2) (2,3) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,2]\nnode 1: [4,2]\nnode 2: [6,6]\nnode 3: [4,2]\nnode 4: [6,6]\n", "difficulty": "easy", "doc_id": "28"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [1,1]\nnode 10: [0,1]\nnode 11: [0,1]\nnode 12: [0,1]\nThe edges are: (0,1) (0,7) (0,11) (0,5) (1,9) (1,3) (1,8) (1,11) (1,5) (2,10) (2,6) (2,11) (2,5) (3,6) (3,12) (4,8) (5,9) (5,7) (5,6) (5,12) (6,12) (7,12) (8,10) (8,11) (9,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [9,9]\nnode 1: [10,14]\nnode 2: [9,9]\nnode 3: [11,7]\nnode 4: [1,3]\nnode 5: [17,14]\nnode 6: [12,13]\nnode 7: [11,10]\nnode 8: [5,4]\nnode 9: [12,10]\nnode 10: [3,6]\nnode 11: [8,10]\nnode 12: [10,11]\n", "difficulty": "hard", "doc_id": "29"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,1]\nThe edges are: (0,6) (1,7) (1,5) (2,4) (2,7) (2,6) (3,4) (3,7) (3,6) (4,7) (4,6) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,1]\nnode 1: [3,2]\nnode 2: [10,3]\nnode 3: [10,3]\nnode 4: [10,4]\nnode 5: [6,2]\nnode 6: [4,4]\nnode 7: [5,4]\n", "difficulty": "easy", "doc_id": "30"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [0,1]\nnode 12: [1,0]\nThe edges are: (0,10) (1,10) (1,9) (1,2) (2,12) (2,7) (3,11) (4,6) (4,8) (5,12) (6,10) (7,12) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [5,5]\nnode 2: [3,2]\nnode 3: [1,2]\nnode 4: [3,1]\nnode 5: [1,0]\nnode 6: [3,4]\nnode 7: [3,1]\nnode 8: [1,1]\nnode 9: [1,2]\nnode 10: [5,6]\nnode 11: [2,4]\nnode 12: [4,1]\n", "difficulty": "hard", "doc_id": "31"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [0,0]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [0,0]\nnode 13: [0,1]\nThe edges are: (0,4) (0,3) (0,12) (0,8) (0,9) (0,11) (1,13) (1,10) (1,12) (1,11) (2,3) (2,5) (2,7) (3,7) (3,6) (4,12) (4,7) (5,6) (5,11) (6,9) (7,13) (7,8) (9,13) (9,11) (10,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,3]\nnode 1: [1,4]\nnode 2: [4,2]\nnode 3: [5,5]\nnode 4: [3,5]\nnode 5: [3,2]\nnode 6: [5,1]\nnode 7: [2,1]\nnode 8: [3,4]\nnode 9: [4,5]\nnode 10: [1,2]\nnode 11: [7,4]\nnode 12: [3,3]\nnode 13: [4,5]\n", "difficulty": "hard", "doc_id": "32"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nThe edges are: (0,1) (1,3) (1,2) (2,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [3,4]\nnode 2: [3,1]\nnode 3: [2,1]\nnode 4: [1,2]\n", "difficulty": "easy", "doc_id": "33"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [1,0]\nThe edges are: (0,3) (0,1) (0,2) (1,4) (2,3) (2,4) (2,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,3]\nnode 1: [5,4]\nnode 2: [8,7]\nnode 3: [4,3]\nnode 4: [5,4]\nnode 5: [5,3]\n", "difficulty": "easy", "doc_id": "34"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [1,1]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nnode 13: [0,1]\nnode 14: [0,1]\nThe edges are: (0,2) (0,13) (1,2) (1,3) (1,10) (2,13) (2,11) (2,10) (3,8) (4,5) (4,8) (4,10) (6,10) (7,13) (9,14) (9,12) (10,11) (11,13) (13,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,3]\nnode 1: [5,5]\nnode 2: [4,11]\nnode 3: [0,4]\nnode 4: [2,6]\nnode 5: [0,1]\nnode 6: [2,3]\nnode 7: [2,2]\nnode 8: [1,2]\nnode 9: [2,3]\nnode 10: [2,6]\nnode 11: [6,6]\nnode 12: [1,1]\nnode 13: [3,8]\nnode 14: [3,3]\n", "difficulty": "hard", "doc_id": "35"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nThe edges are: (0,1) (0,6) (0,2) (0,4) (1,6) (1,3) (1,4) (1,5) (2,6) (2,5) (4,6) (4,5) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,8]\nnode 1: [7,8]\nnode 2: [4,7]\nnode 3: [1,3]\nnode 4: [5,10]\nnode 5: [6,8]\nnode 6: [6,9]\n", "difficulty": "easy", "doc_id": "36"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,1]\nnode 10: [0,1]\nnode 11: [1,1]\nThe edges are: (0,9) (0,3) (0,11) (0,7) (1,8) (1,11) (1,10) (2,6) (3,6) (4,6) (4,7) (5,8) (6,9) (7,8) (8,10) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,5]\nnode 1: [2,2]\nnode 2: [3,3]\nnode 3: [5,6]\nnode 4: [4,4]\nnode 5: [1,1]\nnode 6: [1,5]\nnode 7: [3,5]\nnode 8: [2,3]\nnode 9: [6,7]\nnode 10: [2,3]\nnode 11: [4,7]\n", "difficulty": "hard", "doc_id": "37"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [1,0]\nnode 11: [1,0]\nThe edges are: (0,3) (0,9) (0,7) (0,10) (1,9) (2,10) (3,11) (3,6) (4,8) (4,5) (4,10) (4,6) (5,8) (6,10) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,3]\nnode 1: [1,0]\nnode 2: [1,3]\nnode 3: [7,6]\nnode 4: [5,8]\nnode 5: [3,2]\nnode 6: [4,4]\nnode 7: [4,3]\nnode 8: [4,6]\nnode 9: [5,4]\nnode 10: [10,7]\nnode 11: [1,0]\n", "difficulty": "hard", "doc_id": "38"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [0,0]\nThe edges are: (0,3) (0,4) (0,6) (1,2) (1,4) (2,3) (4,5) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,5]\nnode 1: [5,3]\nnode 2: [3,2]\nnode 3: [3,3]\nnode 4: [2,4]\nnode 5: [5,4]\nnode 6: [1,3]\n", "difficulty": "easy", "doc_id": "39"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [1,1]\nThe edges are: (0,3) (0,7) (0,6) (1,2) (1,4) (3,5) (3,7) (4,5) (4,8) (4,6) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,5]\nnode 1: [4,4]\nnode 2: [2,1]\nnode 3: [5,6]\nnode 4: [7,3]\nnode 5: [5,4]\nnode 6: [7,8]\nnode 7: [7,4]\nnode 8: [3,3]\n", "difficulty": "hard", "doc_id": "40"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [1,0]\nnode 11: [0,0]\nThe edges are: (0,10) (0,5) (0,11) (0,3) (1,6) (1,2) (1,4) (1,9) (3,7) (4,8) (5,9) (5,8) (7,11) (7,8) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [6,0]\nnode 2: [1,3]\nnode 3: [3,2]\nnode 4: [3,3]\nnode 5: [6,1]\nnode 6: [1,3]\nnode 7: [2,0]\nnode 8: [5,2]\nnode 9: [2,4]\nnode 10: [4,1]\nnode 11: [3,2]\n", "difficulty": "hard", "doc_id": "41"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,0]\nThe edges are: (0,2) (0,5) (1,4) (2,7) (2,4) (3,7) (3,4) (4,6) (4,5) (5,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,2]\nnode 1: [2,2]\nnode 2: [3,5]\nnode 3: [2,3]\nnode 4: [8,5]\nnode 5: [4,5]\nnode 6: [2,2]\nnode 7: [4,2]\nnode 8: [2,1]\n", "difficulty": "hard", "doc_id": "42"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [1,1]\nnode 10: [0,0]\nnode 11: [1,1]\nnode 12: [1,1]\nnode 13: [1,1]\nThe edges are: (0,2) (0,1) (0,4) (1,7) (1,3) (2,6) (2,12) (2,4) (3,11) (3,10) (3,13) (3,4) (4,7) (4,10) (4,13) (5,6) (5,12) (6,9) (7,8) (8,9) (8,12) (9,13) (10,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,4]\nnode 1: [6,3]\nnode 2: [5,8]\nnode 3: [7,4]\nnode 4: [12,7]\nnode 5: [2,5]\nnode 6: [5,6]\nnode 7: [6,4]\nnode 8: [5,4]\nnode 9: [6,6]\nnode 10: [7,5]\nnode 11: [3,2]\nnode 12: [6,6]\nnode 13: [9,7]\n", "difficulty": "hard", "doc_id": "43"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,1]\nnode 8: [1,1]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [0,0]\nnode 12: [1,1]\nThe edges are: (0,4) (0,9) (0,12) (1,8) (2,9) (3,5) (5,6) (5,8) (5,10) (5,12) (7,8) (7,10) (8,11) (8,9) (9,11) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,5]\nnode 1: [2,2]\nnode 2: [4,3]\nnode 3: [4,5]\nnode 4: [1,2]\nnode 5: [7,5]\nnode 6: [4,5]\nnode 7: [3,4]\nnode 8: [12,13]\nnode 9: [5,9]\nnode 10: [10,10]\nnode 11: [6,5]\nnode 12: [5,7]\n", "difficulty": "hard", "doc_id": "44"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [1,0]\nThe edges are: (0,1) (0,3) (0,5) (0,4) (0,2) (1,5) (1,2) (2,3) (2,4) (3,5) (3,4) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [8,13]\nnode 1: [6,7]\nnode 2: [4,12]\nnode 3: [7,10]\nnode 4: [4,8]\nnode 5: [3,10]\nnode 6: [3,2]\n", "difficulty": "easy", "doc_id": "45"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [0,0]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [0,1]\nnode 12: [0,1]\nnode 13: [0,1]\nThe edges are: (0,3) (0,2) (0,5) (0,12) (1,11) (1,13) (1,12) (3,4) (3,5) (4,6) (5,9) (5,12) (6,7) (6,12) (7,12) (8,11) (8,10) (9,10) (11,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,10]\nnode 1: [5,6]\nnode 2: [1,3]\nnode 3: [2,8]\nnode 4: [2,4]\nnode 5: [8,10]\nnode 6: [3,5]\nnode 7: [3,5]\nnode 8: [1,4]\nnode 9: [1,6]\nnode 10: [3,4]\nnode 11: [2,6]\nnode 12: [2,13]\nnode 13: [1,5]\n", "difficulty": "hard", "doc_id": "46"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [0,1]\nThe edges are: (0,1) (1,8) (1,5) (2,7) (3,6) (3,7) (3,9) (4,8) (5,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [5,5]\nnode 2: [0,1]\nnode 3: [1,4]\nnode 4: [3,2]\nnode 5: [3,4]\nnode 6: [2,3]\nnode 7: [6,6]\nnode 8: [2,3]\nnode 9: [3,5]\n", "difficulty": "hard", "doc_id": "47"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,0]\nThe edges are: (0,1) (0,6) (0,4) (0,3) (1,6) (1,7) (2,5) (2,6) (2,7) (3,4) (4,6) (5,6) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,4]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [2,4]\nnode 5: [2,5]\nnode 6: [5,7]\nnode 7: [4,4]\n", "difficulty": "easy", "doc_id": "48"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [1,0]\nThe edges are: (0,4) (0,2) (1,9) (1,4) (1,3) (1,2) (1,8) (2,5) (4,8) (5,9) (5,8) (5,6) (6,7) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,5]\nnode 1: [3,9]\nnode 2: [5,5]\nnode 3: [2,2]\nnode 4: [4,5]\nnode 5: [3,5]\nnode 6: [2,4]\nnode 7: [2,2]\nnode 8: [5,9]\nnode 9: [4,4]\n", "difficulty": "hard", "doc_id": "49"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [1,1]\nThe edges are: (0,2) (1,4) (2,5) (2,4) (3,5) (3,6) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [2,2]\nnode 2: [5,6]\nnode 3: [2,4]\nnode 4: [3,2]\nnode 5: [3,3]\nnode 6: [3,4]\n", "difficulty": "easy", "doc_id": "50"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,0]\nThe edges are: (0,2) (0,4) (0,6) (1,4) (1,6) (3,5) (4,6) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,4]\nnode 1: [1,3]\nnode 2: [1,2]\nnode 3: [1,1]\nnode 4: [3,6]\nnode 5: [1,1]\nnode 6: [3,4]\n", "difficulty": "easy", "doc_id": "51"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,0]\nThe edges are: (0,7) (0,5) (1,7) (1,6) (2,3) (2,4) (2,6) (3,8) (3,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,3]\nnode 1: [0,3]\nnode 2: [1,1]\nnode 3: [5,2]\nnode 4: [3,1]\nnode 5: [2,1]\nnode 6: [4,3]\nnode 7: [2,3]\nnode 8: [1,0]\n", "difficulty": "hard", "doc_id": "52"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [0,0]\nnode 10: [1,1]\nnode 11: [0,0]\nnode 12: [1,0]\nnode 13: [1,1]\nThe edges are: (0,13) (0,11) (0,7) (0,5) (1,10) (1,9) (2,9) (2,3) (3,4) (3,10) (3,6) (4,13) (4,7) (4,5) (5,10) (5,13) (7,10) (7,8) (8,13) (8,11) (9,11) (9,12) (10,12) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [11,11]\nnode 1: [7,5]\nnode 2: [7,5]\nnode 3: [9,6]\nnode 4: [14,12]\nnode 5: [12,12]\nnode 6: [4,3]\nnode 7: [11,11]\nnode 8: [7,8]\nnode 9: [5,5]\nnode 10: [14,12]\nnode 11: [7,7]\nnode 12: [10,8]\nnode 13: [13,13]\n", "difficulty": "hard", "doc_id": "53"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [1,1]\nnode 9: [1,1]\nThe edges are: (0,5) (1,4) (2,7) (2,6) (3,4) (4,6) (5,7) (6,9) (7,8) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [2,5]\nnode 3: [1,1]\nnode 4: [1,5]\nnode 5: [1,2]\nnode 6: [5,3]\nnode 7: [5,4]\nnode 8: [3,3]\nnode 9: [3,5]\n", "difficulty": "hard", "doc_id": "54"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [1,0]\nThe edges are: (0,3) (0,4) (0,6) (1,5) (1,4) (2,5) (2,3) (2,6) (3,5) (3,6) (4,5) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,5]\nnode 1: [6,5]\nnode 2: [9,6]\nnode 3: [12,8]\nnode 4: [9,6]\nnode 5: [6,5]\nnode 6: [7,4]\nnode 7: [5,3]\n", "difficulty": "easy", "doc_id": "55"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,0]\nThe edges are: (0,2) (0,1) (0,3) (2,3) (2,4) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,4]\nnode 1: [1,1]\nnode 2: [5,3]\nnode 3: [4,4]\nnode 4: [5,3]\n", "difficulty": "easy", "doc_id": "56"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [0,0]\nnode 11: [0,0]\nnode 12: [0,1]\nThe edges are: (0,7) (0,2) (0,11) (1,5) (1,4) (1,7) (1,12) (1,3) (2,4) (2,3) (4,7) (4,9) (4,10) (4,6) (4,8) (4,11) (8,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,4]\nnode 1: [6,4]\nnode 2: [7,4]\nnode 3: [6,4]\nnode 4: [12,6]\nnode 5: [4,2]\nnode 6: [4,2]\nnode 7: [10,5]\nnode 8: [5,3]\nnode 9: [4,2]\nnode 10: [4,2]\nnode 11: [7,3]\nnode 12: [4,2]\n", "difficulty": "hard", "doc_id": "57"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [0,0]\nnode 10: [0,0]\nThe edges are: (0,6) (0,7) (0,2) (1,8) (1,5) (2,8) (2,5) (2,9) (3,6) (3,4) (3,10) (4,7) (4,9) (7,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,2]\nnode 1: [2,0]\nnode 2: [6,3]\nnode 3: [6,5]\nnode 4: [5,4]\nnode 5: [5,0]\nnode 6: [3,4]\nnode 7: [6,6]\nnode 8: [5,0]\nnode 9: [5,2]\nnode 10: [3,3]\n", "difficulty": "hard", "doc_id": "58"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [1,1]\nnode 11: [1,1]\nnode 12: [1,0]\nnode 13: [0,1]\nnode 14: [1,1]\nThe edges are: (0,8) (0,3) (0,9) (0,12) (1,8) (1,3) (1,5) (2,10) (2,8) (2,13) (2,7) (2,4) (2,11) (3,8) (3,13) (3,7) (3,5) (3,9) (4,6) (4,14) (5,9) (5,6) (7,13) (7,9) (9,10) (10,12) (11,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [9,12]\nnode 1: [6,9]\nnode 2: [8,9]\nnode 3: [11,18]\nnode 4: [3,9]\nnode 5: [6,14]\nnode 6: [4,3]\nnode 7: [11,18]\nnode 8: [8,16]\nnode 9: [10,11]\nnode 10: [8,11]\nnode 11: [5,9]\nnode 12: [4,2]\nnode 13: [7,14]\nnode 14: [2,1]\n", "difficulty": "hard", "doc_id": "59"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,0]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [1,0]\nnode 12: [1,1]\nThe edges are: (0,7) (0,8) (0,10) (1,6) (1,5) (2,12) (3,7) (4,6) (5,12) (6,11) (7,9) (7,11) (8,11) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,2]\nnode 1: [4,1]\nnode 2: [1,0]\nnode 3: [3,2]\nnode 4: [2,0]\nnode 5: [2,1]\nnode 6: [3,5]\nnode 7: [3,7]\nnode 8: [2,5]\nnode 9: [4,5]\nnode 10: [1,2]\nnode 11: [8,3]\nnode 12: [3,2]\n", "difficulty": "hard", "doc_id": "60"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [1,0]\nnode 11: [0,0]\nnode 12: [1,0]\nnode 13: [0,1]\nThe edges are: (0,7) (0,10) (1,8) (1,2) (2,12) (3,5) (3,12) (4,5) (4,11) (4,10) (5,13) (6,12) (7,8) (8,9) (9,11) (10,11) (10,12) (11,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [2,2]\nnode 2: [2,2]\nnode 3: [3,2]\nnode 4: [6,3]\nnode 5: [2,3]\nnode 6: [2,1]\nnode 7: [2,1]\nnode 8: [0,4]\nnode 9: [4,2]\nnode 10: [7,3]\nnode 11: [3,4]\nnode 12: [5,3]\nnode 13: [4,2]\n", "difficulty": "hard", "doc_id": "61"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [1,1]\nThe edges are: (0,2) (0,1) (0,6) (1,5) (2,4) (2,6) (3,6) (3,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,6]\nnode 1: [3,2]\nnode 2: [4,3]\nnode 3: [2,2]\nnode 4: [3,4]\nnode 5: [2,5]\nnode 6: [5,6]\n", "difficulty": "easy", "doc_id": "62"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,1]\nThe edges are: (0,6) (0,3) (0,5) (1,4) (1,7) (1,5) (2,4) (2,7) (3,4) (3,5) (4,7) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,6]\nnode 1: [7,10]\nnode 2: [5,6]\nnode 3: [6,9]\nnode 4: [8,5]\nnode 5: [9,5]\nnode 6: [2,1]\nnode 7: [7,10]\n", "difficulty": "easy", "doc_id": "63"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,1]\nThe edges are: (0,1) (1,4) (2,4) (2,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,1]\nnode 1: [2,2]\nnode 2: [1,1]\nnode 3: [2,1]\nnode 4: [3,2]\n", "difficulty": "easy", "doc_id": "64"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [0,1]\nThe edges are: (0,5) (0,4) (0,3) (1,5) (1,6) (2,5) (4,5) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [3,3]\nnode 2: [1,2]\nnode 3: [1,2]\nnode 4: [2,4]\nnode 5: [6,7]\nnode 6: [2,4]\n", "difficulty": "easy", "doc_id": "65"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [0,1]\nnode 11: [1,0]\nThe edges are: (0,3) (0,4) (0,1) (1,11) (1,5) (2,7) (3,6) (3,10) (4,11) (4,6) (4,5) (5,11) (5,7) (5,9) (5,10) (6,9) (6,10) (7,10) (8,9) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,3]\nnode 1: [4,10]\nnode 2: [2,1]\nnode 3: [6,9]\nnode 4: [5,14]\nnode 5: [15,8]\nnode 6: [10,6]\nnode 7: [7,6]\nnode 8: [2,1]\nnode 9: [3,9]\nnode 10: [7,14]\nnode 11: [12,7]\n", "difficulty": "hard", "doc_id": "66"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,0]\nnode 10: [0,0]\nnode 11: [1,1]\nnode 12: [1,1]\nThe edges are: (0,9) (1,7) (1,3) (1,12) (1,11) (2,5) (2,6) (2,8) (2,4) (2,3) (2,9) (3,9) (4,7) (4,6) (4,11) (6,10) (9,10) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [5,5]\nnode 2: [10,9]\nnode 3: [7,7]\nnode 4: [8,8]\nnode 5: [3,3]\nnode 6: [8,5]\nnode 7: [7,4]\nnode 8: [3,3]\nnode 9: [5,4]\nnode 10: [3,4]\nnode 11: [8,5]\nnode 12: [5,4]\n", "difficulty": "hard", "doc_id": "67"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [1,1]\nThe edges are: (0,4) (0,7) (0,8) (1,7) (2,7) (2,5) (3,7) (3,8) (5,7) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,4]\nnode 1: [2,3]\nnode 2: [4,4]\nnode 3: [3,4]\nnode 4: [3,3]\nnode 5: [3,5]\nnode 6: [2,3]\nnode 7: [10,10]\nnode 8: [5,5]\n", "difficulty": "hard", "doc_id": "68"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [1,1]\nnode 10: [0,0]\nThe edges are: (0,6) (0,8) (1,2) (1,3) (2,4) (2,9) (2,8) (3,10) (3,4) (3,6) (3,5) (3,8) (4,7) (4,9) (4,8) (6,9) (7,9) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [2,9]\nnode 2: [8,7]\nnode 3: [6,6]\nnode 4: [7,15]\nnode 5: [1,5]\nnode 6: [5,10]\nnode 7: [6,6]\nnode 8: [6,15]\nnode 9: [6,10]\nnode 10: [2,6]\n", "difficulty": "hard", "doc_id": "69"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [0,1]\nnode 10: [1,0]\nnode 11: [1,1]\nThe edges are: (0,9) (0,7) (0,5) (0,2) (0,10) (1,9) (2,5) (3,9) (3,10) (4,7) (4,5) (5,8) (6,10) (7,9) (7,8) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,9]\nnode 1: [3,2]\nnode 2: [2,5]\nnode 3: [5,5]\nnode 4: [0,4]\nnode 5: [4,5]\nnode 6: [2,3]\nnode 7: [7,7]\nnode 8: [0,4]\nnode 9: [3,7]\nnode 10: [5,4]\nnode 11: [2,3]\n", "difficulty": "hard", "doc_id": "70"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [1,0]\nnode 7: [0,0]\nThe edges are: (0,7) (1,3) (1,4) (1,2) (2,6) (2,4) (2,7) (3,5) (3,6) (4,5) (4,6) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [6,2]\nnode 2: [7,3]\nnode 3: [6,0]\nnode 4: [8,0]\nnode 5: [5,4]\nnode 6: [6,2]\nnode 7: [4,0]\n", "difficulty": "easy", "doc_id": "71"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [1,1]\nThe edges are: (0,1) (0,4) (2,4) (3,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,4]\nnode 1: [0,1]\nnode 2: [2,3]\nnode 3: [0,2]\nnode 4: [0,4]\nnode 5: [3,4]\n", "difficulty": "easy", "doc_id": "72"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [1,0]\nnode 12: [0,0]\nnode 13: [0,1]\nnode 14: [0,0]\nThe edges are: (0,12) (0,6) (0,10) (1,4) (1,10) (1,13) (1,2) (2,12) (2,3) (2,7) (2,11) (3,10) (4,14) (4,10) (5,9) (5,14) (8,9) (8,12) (8,14) (8,10) (9,13) (10,13) (11,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,9]\nnode 1: [6,12]\nnode 2: [3,11]\nnode 3: [3,7]\nnode 4: [3,9]\nnode 5: [2,3]\nnode 6: [2,1]\nnode 7: [2,2]\nnode 8: [4,11]\nnode 9: [3,6]\nnode 10: [8,13]\nnode 11: [4,5]\nnode 12: [5,5]\nnode 13: [3,12]\nnode 14: [2,5]\n", "difficulty": "hard", "doc_id": "73"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,1]\nThe edges are: (0,1) (0,3) (1,4) (1,3) (2,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,2]\nnode 1: [0,2]\nnode 2: [0,1]\nnode 3: [0,3]\nnode 4: [0,3]\n", "difficulty": "easy", "doc_id": "74"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [1,1]\nnode 9: [1,0]\nnode 10: [0,0]\nThe edges are: (0,3) (0,2) (1,8) (1,5) (1,6) (1,10) (2,6) (4,8) (4,7) (4,10) (5,6) (6,7) (7,8) (7,10) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,3]\nnode 1: [6,4]\nnode 2: [2,1]\nnode 3: [1,0]\nnode 4: [7,4]\nnode 5: [2,3]\nnode 6: [3,7]\nnode 7: [8,5]\nnode 8: [6,7]\nnode 9: [3,1]\nnode 10: [5,6]\n", "difficulty": "hard", "doc_id": "75"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,0]\nThe edges are: (0,6) (0,2) (0,4) (1,3) (1,5) (2,4) (2,5) (4,6) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [1,2]\nnode 2: [3,2]\nnode 3: [1,0]\nnode 4: [3,3]\nnode 5: [3,2]\nnode 6: [3,2]\n", "difficulty": "easy", "doc_id": "76"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [1,1]\nThe edges are: (0,2) (0,6) (1,9) (1,4) (3,9) (4,8) (5,6) (5,8) (5,9) (7,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,0]\nnode 1: [2,4]\nnode 2: [2,1]\nnode 3: [1,3]\nnode 4: [1,1]\nnode 5: [2,3]\nnode 6: [5,3]\nnode 7: [1,3]\nnode 8: [4,3]\nnode 9: [6,5]\n", "difficulty": "hard", "doc_id": "77"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [0,0]\nnode 7: [1,0]\nThe edges are: (0,2) (0,7) (0,6) (1,3) (1,2) (1,5) (1,6) (2,5) (2,7) (3,5) (3,7) (4,7) (4,6) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [11,3]\nnode 1: [13,3]\nnode 2: [11,6]\nnode 3: [9,5]\nnode 4: [7,2]\nnode 5: [13,6]\nnode 6: [5,3]\nnode 7: [13,4]\n", "difficulty": "easy", "doc_id": "78"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [1,1]\nnode 11: [0,0]\nnode 12: [0,1]\nnode 13: [1,1]\nnode 14: [1,0]\nThe edges are: (0,14) (0,6) (0,8) (0,5) (1,9) (1,13) (1,8) (2,7) (2,9) (2,4) (2,11) (3,7) (3,14) (3,10) (4,6) (4,13) (4,11) (5,9) (5,6) (5,11) (6,9) (6,12) (6,10) (7,13) (8,14) (9,12) (9,13) (11,14) (12,13) (13,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [10,6]\nnode 1: [7,7]\nnode 2: [9,11]\nnode 3: [7,4]\nnode 4: [10,12]\nnode 5: [8,13]\nnode 6: [10,9]\nnode 7: [8,7]\nnode 8: [6,6]\nnode 9: [14,12]\nnode 10: [6,6]\nnode 11: [9,6]\nnode 12: [8,11]\nnode 13: [14,13]\nnode 14: [11,10]\n", "difficulty": "hard", "doc_id": "79"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [1,0]\nThe edges are: (0,2) (1,2) (1,5) (2,3) (2,5) (2,4) (3,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,1]\nnode 1: [6,3]\nnode 2: [4,6]\nnode 3: [6,3]\nnode 4: [4,1]\nnode 5: [6,3]\n", "difficulty": "easy", "doc_id": "80"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,0]\nnode 9: [1,0]\nThe edges are: (0,4) (0,9) (0,2) (1,5) (1,4) (1,9) (2,8) (2,9) (3,5) (5,6) (6,7) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [2,4]\nnode 2: [5,3]\nnode 3: [1,2]\nnode 4: [6,2]\nnode 5: [5,2]\nnode 6: [2,3]\nnode 7: [2,2]\nnode 8: [2,1]\nnode 9: [7,2]\n", "difficulty": "hard", "doc_id": "81"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,0]\nThe edges are: (0,4) (0,1) (1,3) (2,4) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [4,2]\nnode 2: [1,1]\nnode 3: [1,2]\nnode 4: [5,2]\n", "difficulty": "easy", "doc_id": "82"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [0,1]\nThe edges are: (0,1) (0,7) (1,4) (2,5) (2,6) (3,5) (3,6) (4,5) (4,6) (4,7) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,4]\nnode 1: [3,2]\nnode 2: [3,2]\nnode 3: [3,2]\nnode 4: [3,6]\nnode 5: [5,2]\nnode 6: [6,2]\nnode 7: [3,2]\n", "difficulty": "easy", "doc_id": "83"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,1]\nThe edges are: (0,4) (0,3) (1,2) (2,4) (2,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,0]\nnode 1: [2,1]\nnode 2: [0,0]\nnode 3: [4,2]\nnode 4: [4,2]\n", "difficulty": "easy", "doc_id": "84"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [1,1]\nnode 9: [1,0]\nnode 10: [0,1]\nnode 11: [1,0]\nThe edges are: (0,10) (0,3) (1,11) (2,5) (2,6) (3,5) (3,10) (3,9) (4,8) (4,7) (4,6) (4,11) (6,10) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,3]\nnode 1: [2,1]\nnode 2: [2,5]\nnode 3: [7,8]\nnode 4: [6,6]\nnode 5: [3,2]\nnode 6: [7,6]\nnode 7: [3,3]\nnode 8: [5,5]\nnode 9: [4,2]\nnode 10: [4,6]\nnode 11: [4,3]\n", "difficulty": "hard", "doc_id": "85"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [1,1]\nnode 11: [0,1]\nnode 12: [1,1]\nnode 13: [1,1]\nThe edges are: (0,4) (0,11) (1,2) (1,6) (1,10) (1,12) (2,6) (2,4) (2,5) (2,12) (3,8) (3,9) (4,6) (4,11) (5,10) (5,11) (5,9) (6,13) (6,8) (6,11) (7,10) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,4]\nnode 1: [6,12]\nnode 2: [7,16]\nnode 3: [4,1]\nnode 4: [9,9]\nnode 5: [9,6]\nnode 6: [11,10]\nnode 7: [1,3]\nnode 8: [4,9]\nnode 9: [3,4]\nnode 10: [7,8]\nnode 11: [6,12]\nnode 12: [5,5]\nnode 13: [2,5]\n", "difficulty": "hard", "doc_id": "86"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [1,0]\nnode 11: [1,1]\nnode 12: [1,1]\nThe edges are: (0,4) (1,4) (1,11) (2,4) (2,8) (2,12) (3,9) (3,10) (4,7) (4,6) (4,12) (5,7) (5,8) (6,10) (7,11) (7,12) (9,11) (9,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [4,1]\nnode 2: [3,1]\nnode 3: [4,2]\nnode 4: [6,4]\nnode 5: [2,2]\nnode 6: [3,1]\nnode 7: [5,1]\nnode 8: [1,1]\nnode 9: [5,0]\nnode 10: [3,0]\nnode 11: [6,5]\nnode 12: [8,6]\n", "difficulty": "hard", "doc_id": "87"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,0]\nnode 10: [0,0]\nnode 11: [0,0]\nThe edges are: (0,5) (0,9) (1,3) (1,4) (2,11) (2,6) (2,8) (2,4) (3,6) (3,8) (3,4) (4,7) (4,8) (4,5) (5,6) (5,10) (5,8) (8,9) (9,11) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,5]\nnode 1: [6,6]\nnode 2: [12,12]\nnode 3: [12,13]\nnode 4: [14,14]\nnode 5: [11,13]\nnode 6: [7,7]\nnode 7: [4,4]\nnode 8: [13,13]\nnode 9: [7,6]\nnode 10: [5,4]\nnode 11: [4,5]\n", "difficulty": "hard", "doc_id": "88"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,1]\nnode 10: [0,1]\nnode 11: [1,0]\nThe edges are: (0,11) (1,2) (2,7) (2,5) (3,7) (3,5) (4,9) (5,11) (5,8) (6,9) (6,7) (9,10) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,2]\nnode 1: [1,1]\nnode 2: [6,5]\nnode 3: [5,4]\nnode 4: [3,2]\nnode 5: [2,3]\nnode 6: [5,4]\nnode 7: [3,2]\nnode 8: [3,2]\nnode 9: [1,5]\nnode 10: [3,2]\nnode 11: [7,4]\n", "difficulty": "hard", "doc_id": "89"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [0,0]\nThe edges are: (0,6) (0,5) (1,4) (2,6) (2,4) (2,5) (3,4) (3,5) (4,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,6]\nnode 1: [3,2]\nnode 2: [8,8]\nnode 3: [6,5]\nnode 4: [4,6]\nnode 5: [3,2]\nnode 6: [5,3]\n", "difficulty": "easy", "doc_id": "90"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [0,0]\nnode 10: [1,0]\nnode 11: [0,1]\nnode 12: [0,1]\nThe edges are: (0,5) (0,11) (1,10) (1,3) (2,8) (3,6) (3,5) (3,11) (4,12) (5,12) (5,8) (5,11) (5,9) (6,12) (7,12) (7,9) (8,11) (9,12) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,4]\nnode 1: [1,4]\nnode 2: [1,2]\nnode 3: [7,7]\nnode 4: [2,3]\nnode 5: [7,13]\nnode 6: [3,6]\nnode 7: [3,6]\nnode 8: [4,4]\nnode 9: [6,8]\nnode 10: [2,1]\nnode 11: [5,12]\nnode 12: [4,10]\n", "difficulty": "hard", "doc_id": "91"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [0,1]\nThe edges are: (0,5) (0,2) (1,5) (2,4) (3,7) (3,8) (4,7) (4,9) (4,8) (5,7) (5,6) (6,7) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,1]\nnode 1: [2,1]\nnode 2: [2,3]\nnode 3: [5,4]\nnode 4: [7,4]\nnode 5: [4,6]\nnode 6: [5,3]\nnode 7: [9,8]\nnode 8: [7,5]\nnode 9: [2,2]\n", "difficulty": "hard", "doc_id": "92"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,0]\nThe edges are: (0,6) (0,4) (1,5) (1,4) (2,3) (2,7) (3,4) (4,5) (4,6) (5,7) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,5]\nnode 1: [3,4]\nnode 2: [4,2]\nnode 3: [3,4]\nnode 4: [4,7]\nnode 5: [7,5]\nnode 6: [7,6]\nnode 7: [1,4]\n", "difficulty": "easy", "doc_id": "93"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,0]\nThe edges are: (0,5) (0,4) (1,5) (1,3) (2,3) (3,4) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [3,3]\nnode 2: [1,2]\nnode 3: [1,3]\nnode 4: [3,4]\nnode 5: [1,3]\n", "difficulty": "easy", "doc_id": "94"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [1,0]\nnode 9: [0,0]\nnode 10: [1,0]\nnode 11: [1,0]\nnode 12: [0,1]\nnode 13: [1,0]\nThe edges are: (0,13) (1,11) (1,10) (1,9) (1,5) (2,4) (2,12) (3,11) (3,7) (4,8) (5,8) (6,12) (8,11) (8,13) (9,10) (10,11) (10,13) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,0]\nnode 1: [11,5]\nnode 2: [2,1]\nnode 3: [5,2]\nnode 4: [5,1]\nnode 5: [7,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [10,3]\nnode 9: [6,1]\nnode 10: [12,3]\nnode 11: [12,2]\nnode 12: [5,4]\nnode 13: [8,1]\n", "difficulty": "hard", "doc_id": "95"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [1,1]\nThe edges are: (0,7) (1,5) (1,9) (2,4) (3,4) (3,6) (4,9) (5,8) (5,9) (5,6) (6,8) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [5,3]\nnode 2: [2,3]\nnode 3: [3,5]\nnode 4: [3,1]\nnode 5: [7,5]\nnode 6: [6,6]\nnode 7: [2,3]\nnode 8: [5,7]\nnode 9: [7,7]\n", "difficulty": "hard", "doc_id": "96"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,1]\nThe edges are: (0,6) (0,5) (0,2) (1,3) (1,4) (1,2) (3,5) (3,4) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,1]\nnode 1: [4,1]\nnode 2: [4,2]\nnode 3: [6,2]\nnode 4: [3,2]\nnode 5: [3,2]\nnode 6: [4,2]\n", "difficulty": "easy", "doc_id": "97"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [0,1]\nnode 11: [0,0]\nnode 12: [0,1]\nnode 13: [1,1]\nnode 14: [0,1]\nThe edges are: (0,6) (0,5) (0,12) (0,7) (1,6) (1,11) (1,3) (1,2) (2,8) (2,10) (3,5) (3,11) (3,12) (4,7) (5,14) (6,9) (6,12) (6,10) (7,13) (7,12) (8,13) (9,13) (10,14) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,13]\nnode 1: [2,10]\nnode 2: [7,7]\nnode 3: [8,9]\nnode 4: [2,3]\nnode 5: [1,6]\nnode 6: [11,14]\nnode 7: [5,11]\nnode 8: [1,6]\nnode 9: [2,7]\nnode 10: [1,8]\nnode 11: [3,4]\nnode 12: [5,15]\nnode 13: [9,11]\nnode 14: [3,5]\n", "difficulty": "hard", "doc_id": "98"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,1]\nThe edges are: (0,3) (0,1) (0,2) (1,4) (1,3) (2,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,1]\nnode 1: [2,0]\nnode 2: [2,0]\nnode 3: [6,1]\nnode 4: [3,1]\n", "difficulty": "easy", "doc_id": "99"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [0,1]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [0,1]\nThe edges are: (0,2) (0,10) (1,3) (1,5) (1,7) (1,8) (2,5) (2,4) (4,10) (4,8) (4,6) (5,9) (6,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,4]\nnode 1: [3,3]\nnode 2: [5,3]\nnode 3: [3,2]\nnode 4: [7,8]\nnode 5: [8,4]\nnode 6: [5,1]\nnode 7: [4,4]\nnode 8: [7,3]\nnode 9: [2,3]\nnode 10: [4,2]\n", "difficulty": "hard", "doc_id": "100"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [1,0]\nnode 7: [0,1]\nnode 8: [0,1]\nThe edges are: (0,5) (0,6) (1,5) (1,2) (2,7) (2,3) (2,5) (2,4) (3,8) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,4]\nnode 1: [6,6]\nnode 2: [9,10]\nnode 3: [3,4]\nnode 4: [3,3]\nnode 5: [9,8]\nnode 6: [2,1]\nnode 7: [6,6]\nnode 8: [1,2]\n", "difficulty": "hard", "doc_id": "101"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [0,0]\nnode 8: [1,1]\nnode 9: [1,1]\nThe edges are: (0,6) (0,2) (0,5) (1,6) (2,7) (2,8) (3,4) (3,6) (4,8) (5,7) (5,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,6]\nnode 1: [1,2]\nnode 2: [4,6]\nnode 3: [2,4]\nnode 4: [1,1]\nnode 5: [3,6]\nnode 6: [1,2]\nnode 7: [3,5]\nnode 8: [4,7]\nnode 9: [1,2]\n", "difficulty": "hard", "doc_id": "102"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [0,1]\nThe edges are: (0,3) (0,2) (0,6) (1,3) (1,2) (2,5) (2,6) (3,4) (3,5) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,6]\nnode 1: [4,5]\nnode 2: [9,3]\nnode 3: [7,2]\nnode 4: [2,2]\nnode 5: [7,6]\nnode 6: [6,5]\n", "difficulty": "easy", "doc_id": "103"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [0,1]\nnode 10: [1,0]\nThe edges are: (0,7) (0,3) (1,7) (1,8) (1,9) (1,2) (2,8) (3,7) (3,5) (3,4) (5,10) (5,6) (7,10) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [8,4]\nnode 1: [7,6]\nnode 2: [2,4]\nnode 3: [10,3]\nnode 4: [4,2]\nnode 5: [7,3]\nnode 6: [3,0]\nnode 7: [9,6]\nnode 8: [3,4]\nnode 9: [2,4]\nnode 10: [7,2]\n", "difficulty": "hard", "doc_id": "104"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [1,0]\nThe edges are: (0,4) (0,7) (0,3) (0,2) (1,8) (2,8) (2,3) (2,6) (5,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [1,2]\nnode 2: [4,7]\nnode 3: [5,4]\nnode 4: [3,3]\nnode 5: [1,2]\nnode 6: [2,1]\nnode 7: [3,3]\nnode 8: [4,1]\n", "difficulty": "hard", "doc_id": "105"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [0,1]\nThe edges are: (0,2) (0,4) (1,5) (1,3) (1,6) (1,7) (1,4) (2,7) (3,7) (4,7) (5,6) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,5]\nnode 1: [4,12]\nnode 2: [5,4]\nnode 3: [6,5]\nnode 4: [8,6]\nnode 5: [6,7]\nnode 6: [4,4]\nnode 7: [4,11]\n", "difficulty": "easy", "doc_id": "106"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [0,0]\nnode 10: [0,0]\nnode 11: [1,1]\nnode 12: [1,0]\nnode 13: [1,0]\nnode 14: [1,0]\nThe edges are: (0,12) (0,3) (0,14) (0,7) (0,11) (0,1) (1,12) (1,10) (1,7) (1,4) (1,6) (2,10) (2,14) (2,7) (2,8) (3,14) (3,7) (3,5) (4,7) (4,6) (5,9) (5,13) (5,11) (6,8) (7,8) (8,10) (9,13) (9,10) (10,12) (11,13) (11,14) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [22,18]\nnode 1: [19,12]\nnode 2: [10,12]\nnode 3: [16,13]\nnode 4: [10,9]\nnode 5: [13,8]\nnode 6: [7,9]\nnode 7: [19,16]\nnode 8: [10,7]\nnode 9: [8,5]\nnode 10: [12,10]\nnode 11: [15,11]\nnode 12: [15,10]\nnode 13: [12,6]\nnode 14: [16,9]\n", "difficulty": "hard", "doc_id": "107"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,2) (1,3) (2,4) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,2]\nnode 1: [0,2]\nnode 2: [1,3]\nnode 3: [2,3]\nnode 4: [0,4]\n", "difficulty": "easy", "doc_id": "108"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [1,1]\nThe edges are: (0,7) (0,8) (1,2) (1,8) (1,4) (2,8) (3,6) (3,7) (3,8) (3,5) (4,6) (5,6) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,6]\nnode 1: [6,5]\nnode 2: [5,5]\nnode 3: [5,10]\nnode 4: [4,4]\nnode 5: [3,5]\nnode 6: [5,6]\nnode 7: [5,7]\nnode 8: [9,10]\n", "difficulty": "hard", "doc_id": "109"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,1]\nThe edges are: (0,5) (1,3) (1,5) (1,2) (2,6) (2,5) (3,5) (4,5) (4,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [8,5]\nnode 2: [5,2]\nnode 3: [5,2]\nnode 4: [3,3]\nnode 5: [9,4]\nnode 6: [3,2]\nnode 7: [1,1]\n", "difficulty": "easy", "doc_id": "110"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,0]\nThe edges are: (0,3) (1,4) (1,2) (1,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [4,4]\nnode 2: [2,2]\nnode 3: [2,3]\nnode 4: [2,2]\n", "difficulty": "easy", "doc_id": "111"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [1,0]\nThe edges are: (0,4) (0,2) (1,2) (1,3) (2,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,0]\nnode 1: [3,0]\nnode 2: [4,0]\nnode 3: [1,0]\nnode 4: [4,0]\n", "difficulty": "easy", "doc_id": "112"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [1,0]\nnode 9: [0,0]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nnode 13: [1,0]\nnode 14: [1,1]\nThe edges are: (0,7) (0,8) (1,2) (1,10) (1,5) (2,9) (3,6) (3,9) (3,8) (4,7) (5,14) (6,8) (7,10) (7,8) (8,12) (11,14) (13,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,3]\nnode 1: [4,4]\nnode 2: [0,2]\nnode 3: [4,4]\nnode 4: [3,1]\nnode 5: [2,2]\nnode 6: [5,3]\nnode 7: [5,3]\nnode 8: [8,3]\nnode 9: [3,2]\nnode 10: [3,2]\nnode 11: [2,1]\nnode 12: [3,2]\nnode 13: [2,1]\nnode 14: [4,4]\n", "difficulty": "hard", "doc_id": "113"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [0,0]\nThe edges are: (0,3) (0,1) (0,2) (0,5) (0,6) (1,4) (1,5) (2,3) (2,6) (4,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,9]\nnode 1: [2,5]\nnode 2: [2,5]\nnode 3: [3,3]\nnode 4: [1,5]\nnode 5: [3,5]\nnode 6: [3,4]\n", "difficulty": "easy", "doc_id": "114"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [1,0]\nThe edges are: (0,3) (0,2) (0,1) (2,3) (2,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,0]\nnode 1: [1,0]\nnode 2: [3,0]\nnode 3: [2,0]\nnode 4: [1,0]\n", "difficulty": "easy", "doc_id": "115"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,0]\nnode 10: [0,1]\nnode 11: [0,0]\nThe edges are: (0,1) (0,3) (0,7) (0,6) (0,10) (0,2) (1,3) (1,11) (1,7) (1,8) (2,4) (4,5) (4,11) (4,10) (6,11) (6,7) (6,9) (8,11) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [13,4]\nnode 1: [10,6]\nnode 2: [2,5]\nnode 3: [5,5]\nnode 4: [8,6]\nnode 5: [0,2]\nnode 6: [7,6]\nnode 7: [7,5]\nnode 8: [6,5]\nnode 9: [2,0]\nnode 10: [5,8]\nnode 11: [8,5]\n", "difficulty": "hard", "doc_id": "116"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [1,1]\nnode 9: [1,0]\nnode 10: [0,0]\nThe edges are: (0,2) (0,6) (0,4) (1,2) (2,5) (2,4) (3,10) (3,8) (3,4) (4,9) (4,7) (5,7) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,7]\nnode 1: [2,3]\nnode 2: [7,6]\nnode 3: [4,5]\nnode 4: [6,10]\nnode 5: [3,4]\nnode 6: [1,2]\nnode 7: [5,4]\nnode 8: [2,4]\nnode 9: [4,4]\nnode 10: [1,2]\n", "difficulty": "hard", "doc_id": "117"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,0]\nThe edges are: (0,2) (0,3) (0,5) (1,2) (1,3) (2,4) (3,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,6]\nnode 1: [2,5]\nnode 2: [5,0]\nnode 3: [5,1]\nnode 4: [1,3]\nnode 5: [3,2]\n", "difficulty": "easy", "doc_id": "118"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [1,1]\nnode 10: [1,0]\nnode 11: [0,1]\nnode 12: [1,1]\nnode 13: [1,1]\nnode 14: [1,0]\nThe edges are: (0,12) (0,4) (1,14) (1,9) (2,6) (3,10) (3,4) (3,7) (4,5) (4,6) (5,10) (6,14) (6,8) (7,12) (7,14) (7,10) (8,12) (8,14) (8,13) (9,10) (11,14) (11,13) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,5]\nnode 1: [4,4]\nnode 2: [2,2]\nnode 3: [10,5]\nnode 4: [9,6]\nnode 5: [6,4]\nnode 6: [7,7]\nnode 7: [11,9]\nnode 8: [7,11]\nnode 9: [6,3]\nnode 10: [11,4]\nnode 11: [3,6]\nnode 12: [10,8]\nnode 13: [7,6]\nnode 14: [13,7]\n", "difficulty": "hard", "doc_id": "119"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [0,1]\nThe edges are: (0,4) (0,5) (0,6) (1,2) (2,3) (2,6) (3,4) (3,5) (3,7) (4,7) (4,6) (5,7) (5,6) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,9]\nnode 1: [1,0]\nnode 2: [1,10]\nnode 3: [6,6]\nnode 4: [3,13]\nnode 5: [3,13]\nnode 6: [7,8]\nnode 7: [5,13]\n", "difficulty": "easy", "doc_id": "120"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [1,1]\nThe edges are: (0,2) (0,1) (0,3) (1,4) (1,7) (1,5) (1,6) (4,7) (4,5) (4,6) (5,6) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,5]\nnode 1: [11,6]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [12,6]\nnode 5: [11,6]\nnode 6: [12,7]\nnode 7: [11,6]\n", "difficulty": "easy", "doc_id": "121"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [0,1]\nnode 11: [0,0]\nnode 12: [0,0]\nnode 13: [0,0]\nThe edges are: (0,8) (1,5) (1,4) (2,5) (2,3) (3,12) (3,5) (4,9) (5,8) (5,9) (6,10) (6,7) (7,13) (7,8) (8,13) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [5,6]\nnode 2: [3,5]\nnode 3: [3,4]\nnode 4: [0,0]\nnode 5: [2,2]\nnode 6: [3,1]\nnode 7: [5,3]\nnode 8: [8,7]\nnode 9: [6,7]\nnode 10: [1,1]\nnode 11: [0,0]\nnode 12: [0,1]\nnode 13: [4,2]\n", "difficulty": "hard", "doc_id": "122"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [1,0]\nnode 11: [0,0]\nThe edges are: (0,10) (0,6) (0,1) (1,10) (1,6) (1,11) (2,10) (2,3) (3,10) (3,9) (3,8) (4,11) (5,11) (5,7) (7,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,3]\nnode 1: [10,7]\nnode 2: [6,3]\nnode 3: [4,5]\nnode 4: [3,3]\nnode 5: [3,3]\nnode 6: [4,1]\nnode 7: [4,4]\nnode 8: [3,1]\nnode 9: [3,1]\nnode 10: [8,3]\nnode 11: [3,1]\n", "difficulty": "hard", "doc_id": "123"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nThe edges are: (0,1) (0,3) (1,3) (1,4) (1,2) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [3,1]\nnode 2: [2,1]\nnode 3: [4,1]\nnode 4: [3,2]\n", "difficulty": "easy", "doc_id": "124"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,1]\nThe edges are: (0,2) (0,3) (1,2) (1,5) (1,4) (2,3) (3,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,2]\nnode 1: [5,2]\nnode 2: [6,6]\nnode 3: [4,2]\nnode 4: [3,4]\nnode 5: [7,6]\n", "difficulty": "easy", "doc_id": "125"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [1,1]\nThe edges are: (0,8) (0,6) (0,5) (1,7) (1,5) (1,3) (2,8) (4,6) (5,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,4]\nnode 1: [4,2]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [3,3]\nnode 6: [1,1]\nnode 7: [1,1]\nnode 8: [4,4]\n", "difficulty": "hard", "doc_id": "126"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,0]\nnode 9: [0,0]\nnode 10: [1,0]\nThe edges are: (0,9) (0,3) (0,8) (0,2) (1,6) (2,6) (2,8) (3,10) (3,9) (3,5) (4,10) (4,9) (4,5) (5,10) (6,8) (7,8) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [9,10]\nnode 1: [1,1]\nnode 2: [5,6]\nnode 3: [4,6]\nnode 4: [3,5]\nnode 5: [6,5]\nnode 6: [6,7]\nnode 7: [3,4]\nnode 8: [5,6]\nnode 9: [9,8]\nnode 10: [6,4]\n", "difficulty": "hard", "doc_id": "127"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [0,0]\nnode 10: [1,0]\nnode 11: [0,1]\nnode 12: [0,1]\nnode 13: [0,1]\nnode 14: [0,1]\nThe edges are: (0,11) (0,6) (0,9) (1,4) (1,12) (1,14) (2,13) (2,8) (3,10) (4,6) (4,12) (4,9) (4,14) (5,7) (5,13) (5,10) (6,14) (8,9) (11,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,5]\nnode 1: [2,5]\nnode 2: [2,0]\nnode 3: [0,0]\nnode 4: [3,10]\nnode 5: [1,0]\nnode 6: [3,5]\nnode 7: [2,1]\nnode 8: [1,4]\nnode 9: [3,3]\nnode 10: [3,1]\nnode 11: [2,3]\nnode 12: [1,5]\nnode 13: [2,3]\nnode 14: [3,8]\n", "difficulty": "hard", "doc_id": "128"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [1,0]\nnode 10: [0,0]\nThe edges are: (0,4) (0,6) (0,9) (1,4) (1,9) (1,8) (2,6) (3,5) (4,9) (5,10) (6,8) (7,9) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [8,6]\nnode 1: [8,7]\nnode 2: [2,1]\nnode 3: [0,1]\nnode 4: [7,6]\nnode 5: [3,0]\nnode 6: [5,5]\nnode 7: [3,3]\nnode 8: [4,2]\nnode 9: [10,5]\nnode 10: [3,4]\n", "difficulty": "hard", "doc_id": "129"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [0,0]\nnode 7: [0,0]\nnode 8: [1,0]\nnode 9: [1,1]\nnode 10: [0,0]\nnode 11: [1,1]\nnode 12: [0,0]\nnode 13: [1,1]\nnode 14: [1,1]\nThe edges are: (0,8) (0,10) (0,11) (0,1) (1,9) (1,7) (1,5) (1,13) (2,3) (2,12) (3,10) (4,13) (5,9) (5,14) (5,11) (6,14) (6,8) (6,12) (6,13) (7,9) (7,11) (8,14) (8,10) (10,14) (10,12) (12,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [12,10]\nnode 1: [8,7]\nnode 2: [3,3]\nnode 3: [5,2]\nnode 4: [0,0]\nnode 5: [9,8]\nnode 6: [6,5]\nnode 7: [7,7]\nnode 8: [11,6]\nnode 9: [9,9]\nnode 10: [9,7]\nnode 11: [7,6]\nnode 12: [10,5]\nnode 13: [8,7]\nnode 14: [14,11]\n", "difficulty": "hard", "doc_id": "130"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [0,0]\nnode 10: [0,1]\nnode 11: [0,1]\nThe edges are: (0,8) (0,10) (0,2) (0,11) (0,6) (1,10) (1,11) (2,5) (2,9) (3,9) (4,7) (4,11) (5,9) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,3]\nnode 1: [3,3]\nnode 2: [4,4]\nnode 3: [1,0]\nnode 4: [4,4]\nnode 5: [1,0]\nnode 6: [4,6]\nnode 7: [0,1]\nnode 8: [2,4]\nnode 9: [1,0]\nnode 10: [2,6]\nnode 11: [2,7]\n", "difficulty": "hard", "doc_id": "131"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,0]\nThe edges are: (0,6) (0,1) (1,9) (1,4) (2,8) (2,3) (3,8) (3,4) (5,7) (7,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,3]\nnode 1: [2,1]\nnode 2: [3,5]\nnode 3: [2,5]\nnode 4: [5,5]\nnode 5: [2,2]\nnode 6: [1,0]\nnode 7: [3,2]\nnode 8: [5,7]\nnode 9: [5,4]\n", "difficulty": "hard", "doc_id": "132"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [0,0]\nThe edges are: (0,1) (0,3) (2,3) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,3]\nnode 1: [1,0]\nnode 2: [2,2]\nnode 3: [1,0]\nnode 4: [2,2]\n", "difficulty": "easy", "doc_id": "133"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [0,0]\nnode 8: [1,1]\nnode 9: [0,0]\nnode 10: [0,0]\nnode 11: [0,0]\nThe edges are: (0,5) (0,10) (0,1) (1,5) (1,10) (1,4) (1,3) (2,6) (3,6) (5,11) (5,7) (6,8) (7,8) (8,10) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,6]\nnode 1: [7,8]\nnode 2: [3,2]\nnode 3: [5,3]\nnode 4: [2,1]\nnode 5: [4,3]\nnode 6: [4,1]\nnode 7: [2,2]\nnode 8: [7,7]\nnode 9: [1,0]\nnode 10: [4,2]\nnode 11: [1,2]\n", "difficulty": "hard", "doc_id": "134"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [0,0]\nThe edges are: (0,3) (0,5) (0,6) (0,4) (1,5) (1,6) (1,2) (3,4) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,8]\nnode 1: [3,6]\nnode 2: [0,1]\nnode 3: [1,3]\nnode 4: [2,4]\nnode 5: [2,6]\nnode 6: [2,5]\n", "difficulty": "easy", "doc_id": "135"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nThe edges are: (0,4) (0,3) (0,2) (1,4) (1,3) (2,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,5]\nnode 1: [2,4]\nnode 2: [3,2]\nnode 3: [5,1]\nnode 4: [4,0]\n", "difficulty": "easy", "doc_id": "136"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [0,0]\nThe edges are: (0,7) (0,1) (1,2) (1,3) (2,8) (2,3) (4,5) (4,6) (5,8) (6,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,4]\nnode 1: [1,5]\nnode 2: [3,7]\nnode 3: [1,5]\nnode 4: [0,0]\nnode 5: [4,3]\nnode 6: [4,3]\nnode 7: [1,1]\nnode 8: [0,2]\n", "difficulty": "hard", "doc_id": "137"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,0]\nThe edges are: (0,3) (0,1) (0,4) (1,6) (1,5) (2,6) (2,3) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,5]\nnode 1: [2,2]\nnode 2: [2,2]\nnode 3: [4,5]\nnode 4: [2,4]\nnode 5: [1,1]\nnode 6: [3,2]\n", "difficulty": "easy", "doc_id": "138"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,1]\nnode 5: [0,1]\nnode 6: [0,1]\nThe edges are: (0,1) (1,4) (1,2) (2,5) (3,4) (4,5) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [0,5]\nnode 2: [4,6]\nnode 3: [0,2]\nnode 4: [5,7]\nnode 5: [0,5]\nnode 6: [2,3]\n", "difficulty": "easy", "doc_id": "139"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [0,0]\nnode 9: [1,1]\nThe edges are: (0,9) (0,4) (0,3) (0,5) (1,2) (2,7) (2,8) (4,6) (5,7) (5,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,6]\nnode 1: [1,1]\nnode 2: [4,1]\nnode 3: [1,4]\nnode 4: [1,5]\nnode 5: [2,7]\nnode 6: [0,1]\nnode 7: [5,3]\nnode 8: [2,2]\nnode 9: [3,6]\n", "difficulty": "hard", "doc_id": "140"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [0,1]\nnode 11: [0,1]\nnode 12: [0,1]\nThe edges are: (0,6) (0,8) (1,6) (1,8) (2,5) (2,12) (3,4) (3,5) (3,12) (4,11) (4,10) (4,7) (5,9) (6,10) (6,9) (7,10) (7,9) (9,10) (9,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,5]\nnode 1: [5,5]\nnode 2: [4,0]\nnode 3: [4,3]\nnode 4: [8,5]\nnode 5: [5,9]\nnode 6: [7,11]\nnode 7: [5,10]\nnode 8: [2,4]\nnode 9: [12,6]\nnode 10: [7,12]\nnode 11: [0,3]\nnode 12: [5,9]\n", "difficulty": "hard", "doc_id": "141"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,0]\nThe edges are: (0,9) (0,8) (1,7) (1,5) (1,2) (1,4) (2,7) (2,8) (3,5) (4,6) (5,6) (6,7) (6,9) (7,8) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [0,9]\nnode 2: [2,6]\nnode 3: [0,3]\nnode 4: [1,3]\nnode 5: [1,3]\nnode 6: [1,9]\nnode 7: [3,5]\nnode 8: [2,5]\nnode 9: [4,2]\n", "difficulty": "hard", "doc_id": "142"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [0,1]\nThe edges are: (0,4) (1,4) (1,5) (1,2) (2,4) (2,5) (3,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,2]\nnode 1: [2,3]\nnode 2: [2,3]\nnode 3: [1,0]\nnode 4: [4,2]\nnode 5: [2,5]\n", "difficulty": "easy", "doc_id": "143"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [1,1]\nnode 11: [0,1]\nnode 12: [1,1]\nnode 13: [0,0]\nThe edges are: (0,8) (0,4) (0,10) (0,9) (0,11) (1,13) (2,6) (3,7) (3,4) (3,9) (3,11) (5,9) (6,12) (6,7) (6,13) (6,8) (7,10) (8,13) (8,10) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [12,9]\nnode 1: [2,2]\nnode 2: [4,3]\nnode 3: [9,6]\nnode 4: [5,8]\nnode 5: [2,2]\nnode 6: [6,6]\nnode 7: [9,8]\nnode 8: [12,12]\nnode 9: [6,9]\nnode 10: [7,8]\nnode 11: [5,9]\nnode 12: [7,5]\nnode 13: [6,5]\n", "difficulty": "hard", "doc_id": "144"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [0,1]\nnode 10: [1,0]\nThe edges are: (0,9) (0,8) (0,7) (0,10) (1,5) (1,4) (2,7) (3,5) (3,10) (5,7) (6,10) (7,8) (8,9) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,12]\nnode 1: [2,1]\nnode 2: [3,4]\nnode 3: [2,5]\nnode 4: [2,1]\nnode 5: [7,6]\nnode 6: [1,4]\nnode 7: [4,5]\nnode 8: [6,12]\nnode 9: [3,4]\nnode 10: [6,5]\n", "difficulty": "hard", "doc_id": "145"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,1]\nThe edges are: (0,2) (0,4) (1,4) (1,6) (1,7) (2,5) (2,8) (2,3) (5,8) (6,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,5]\nnode 1: [1,5]\nnode 2: [5,2]\nnode 3: [2,3]\nnode 4: [1,1]\nnode 5: [4,4]\nnode 6: [2,2]\nnode 7: [0,1]\nnode 8: [3,6]\n", "difficulty": "hard", "doc_id": "146"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,0]\nnode 10: [1,1]\nnode 11: [1,0]\nnode 12: [0,0]\nnode 13: [1,0]\nThe edges are: (0,12) (0,10) (0,7) (0,9) (1,7) (1,2) (2,3) (2,8) (2,4) (3,12) (3,8) (4,6) (4,9) (5,8) (6,10) (6,7) (7,12) (8,13) (9,10) (9,11) (10,12) (11,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [11,5]\nnode 1: [5,3]\nnode 2: [8,2]\nnode 3: [7,3]\nnode 4: [7,6]\nnode 5: [2,0]\nnode 6: [7,3]\nnode 7: [9,4]\nnode 8: [7,5]\nnode 9: [7,3]\nnode 10: [10,6]\nnode 11: [5,3]\nnode 12: [9,4]\nnode 13: [3,0]\n", "difficulty": "hard", "doc_id": "147"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [0,0]\nThe edges are: (0,4) (0,3) (0,5) (0,1) (1,4) (2,4) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [6,1]\nnode 1: [5,2]\nnode 2: [3,1]\nnode 3: [2,1]\nnode 4: [4,1]\nnode 5: [5,2]\n", "difficulty": "easy", "doc_id": "148"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [1,0]\nnode 7: [0,1]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nnode 13: [1,0]\nnode 14: [0,1]\nThe edges are: (0,3) (0,5) (0,4) (0,14) (1,2) (1,7) (1,14) (1,11) (2,14) (3,4) (3,14) (4,8) (4,12) (4,7) (5,8) (6,13) (6,12) (6,10) (7,9) (8,10) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,5]\nnode 1: [2,6]\nnode 2: [3,4]\nnode 3: [3,7]\nnode 4: [4,6]\nnode 5: [2,4]\nnode 6: [4,0]\nnode 7: [3,5]\nnode 8: [3,2]\nnode 9: [3,1]\nnode 10: [3,2]\nnode 11: [2,2]\nnode 12: [3,2]\nnode 13: [2,0]\nnode 14: [3,8]\n", "difficulty": "hard", "doc_id": "149"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,1]\nnode 10: [0,1]\nnode 11: [0,1]\nnode 12: [0,0]\nnode 13: [0,1]\nThe edges are: (0,2) (0,12) (0,3) (1,7) (1,3) (1,13) (2,12) (3,13) (4,5) (4,10) (4,13) (6,7) (6,13) (8,12) (9,13) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,4]\nnode 1: [5,7]\nnode 2: [2,2]\nnode 3: [5,7]\nnode 4: [3,7]\nnode 5: [0,3]\nnode 6: [4,5]\nnode 7: [1,3]\nnode 8: [1,1]\nnode 9: [3,4]\nnode 10: [0,4]\nnode 11: [0,2]\nnode 12: [2,2]\nnode 13: [2,9]\n", "difficulty": "hard", "doc_id": "150"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,1]\nThe edges are: (0,4) (0,6) (0,3) (1,4) (1,6) (1,3) (2,6) (2,3) (3,7) (4,7) (5,7) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,8]\nnode 1: [7,8]\nnode 2: [5,5]\nnode 3: [1,3]\nnode 4: [1,2]\nnode 5: [4,2]\nnode 6: [0,5]\nnode 7: [4,8]\n", "difficulty": "easy", "doc_id": "151"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [0,1]\nnode 10: [0,1]\nThe edges are: (0,2) (0,10) (1,9) (2,8) (3,9) (3,7) (4,8) (4,6) (5,8) (5,7) (6,10) (7,9) (9,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,5]\nnode 1: [2,2]\nnode 2: [2,1]\nnode 3: [2,3]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [1,5]\nnode 7: [4,4]\nnode 8: [2,5]\nnode 9: [2,6]\nnode 10: [3,4]\n", "difficulty": "hard", "doc_id": "152"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [0,0]\nnode 9: [0,1]\nThe edges are: (0,4) (0,3) (0,8) (1,5) (1,4) (1,9) (1,3) (1,6) (2,7) (3,4) (3,8) (6,7) (6,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,7]\nnode 1: [8,11]\nnode 2: [1,1]\nnode 3: [3,10]\nnode 4: [2,7]\nnode 5: [0,4]\nnode 6: [2,7]\nnode 7: [3,4]\nnode 8: [2,3]\nnode 9: [2,7]\n", "difficulty": "hard", "doc_id": "153"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [0,0]\nThe edges are: (0,4) (0,5) (0,2) (1,4) (1,3) (2,3) (3,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,0]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [2,0]\nnode 4: [0,0]\nnode 5: [0,0]\n", "difficulty": "easy", "doc_id": "154"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [1,1]\nnode 10: [1,1]\nnode 11: [1,0]\nThe edges are: (0,6) (0,7) (1,9) (1,2) (1,4) (1,8) (1,3) (1,7) (2,9) (2,5) (2,10) (3,5) (3,8) (6,9) (6,10) (8,9) (9,10) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [8,6]\nnode 2: [9,7]\nnode 3: [6,5]\nnode 4: [3,4]\nnode 5: [4,3]\nnode 6: [7,4]\nnode 7: [5,5]\nnode 8: [6,7]\nnode 9: [13,10]\nnode 10: [8,7]\nnode 11: [3,1]\n", "difficulty": "hard", "doc_id": "155"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [1,1]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [0,0]\nnode 10: [1,1]\nnode 11: [1,0]\nnode 12: [1,1]\nnode 13: [0,0]\nThe edges are: (0,3) (1,7) (1,3) (1,11) (2,8) (2,7) (2,9) (3,5) (3,11) (4,9) (5,10) (6,8) (8,12) (9,11) (9,13) (9,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [3,7]\nnode 2: [4,6]\nnode 3: [7,7]\nnode 4: [2,2]\nnode 5: [3,4]\nnode 6: [2,2]\nnode 7: [5,3]\nnode 8: [4,1]\nnode 9: [4,3]\nnode 10: [2,2]\nnode 11: [7,7]\nnode 12: [4,4]\nnode 13: [2,2]\n", "difficulty": "hard", "doc_id": "156"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [1,0]\nThe edges are: (0,1) (1,5) (2,5) (2,4) (3,5) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,1]\nnode 1: [2,3]\nnode 2: [3,2]\nnode 3: [2,2]\nnode 4: [4,2]\nnode 5: [5,1]\n", "difficulty": "easy", "doc_id": "157"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,0]\nThe edges are: (0,7) (0,3) (1,4) (1,8) (1,5) (2,4) (2,6) (3,7) (3,5) (4,5) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,1]\nnode 1: [3,4]\nnode 2: [3,1]\nnode 3: [4,4]\nnode 4: [2,3]\nnode 5: [5,1]\nnode 6: [0,1]\nnode 7: [3,2]\nnode 8: [3,1]\n", "difficulty": "hard", "doc_id": "158"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [0,1]\nThe edges are: (0,3) (0,1) (1,2) (1,6) (2,4) (2,6) (2,3) (2,5) (3,6) (4,6) (4,7) (5,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [1,5]\nnode 2: [5,6]\nnode 3: [1,5]\nnode 4: [1,5]\nnode 5: [0,3]\nnode 6: [3,5]\nnode 7: [2,2]\n", "difficulty": "easy", "doc_id": "159"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [1,1]\nnode 9: [0,1]\nnode 10: [1,1]\nnode 11: [0,1]\nThe edges are: (0,11) (1,10) (1,11) (1,9) (1,3) (2,4) (2,6) (2,7) (2,8) (2,9) (3,7) (3,8) (4,9) (5,9) (8,10) (10,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [5,6]\nnode 2: [4,4]\nnode 3: [5,7]\nnode 4: [4,4]\nnode 5: [1,0]\nnode 6: [3,4]\nnode 7: [5,6]\nnode 8: [6,8]\nnode 9: [5,10]\nnode 10: [5,8]\nnode 11: [3,7]\n", "difficulty": "hard", "doc_id": "160"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [1,0]\nnode 7: [0,0]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [1,1]\nnode 11: [1,1]\nThe edges are: (0,8) (0,2) (0,11) (0,3) (0,9) (0,7) (1,8) (1,2) (3,7) (3,10) (4,11) (4,9) (4,7) (5,11) (5,7) (5,6) (7,11) (7,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [10,6]\nnode 1: [2,0]\nnode 2: [7,7]\nnode 3: [10,10]\nnode 4: [6,5]\nnode 5: [5,6]\nnode 6: [2,1]\nnode 7: [13,10]\nnode 8: [7,7]\nnode 9: [11,11]\nnode 10: [2,1]\nnode 11: [13,12]\n", "difficulty": "hard", "doc_id": "161"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [0,0]\nThe edges are: (0,2) (0,1) (0,5) (0,4) (2,5) (2,3) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,2]\nnode 1: [3,1]\nnode 2: [7,3]\nnode 3: [2,1]\nnode 4: [6,2]\nnode 5: [6,2]\n", "difficulty": "easy", "doc_id": "162"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,0]\nThe edges are: (0,1) (0,5) (0,2) (1,4) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,1]\nnode 1: [2,4]\nnode 2: [1,2]\nnode 3: [1,2]\nnode 4: [3,2]\nnode 5: [1,2]\n", "difficulty": "easy", "doc_id": "163"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,1]\nnode 8: [1,1]\nnode 9: [0,1]\nnode 10: [1,0]\nThe edges are: (0,6) (0,2) (0,3) (0,4) (1,2) (1,7) (2,10) (3,10) (3,9) (3,7) (4,10) (4,5) (5,10) (5,8) (5,9) (6,8) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [9,4]\nnode 1: [3,3]\nnode 2: [9,9]\nnode 3: [12,11]\nnode 4: [11,10]\nnode 5: [12,7]\nnode 6: [6,6]\nnode 7: [3,4]\nnode 8: [8,6]\nnode 9: [7,7]\nnode 10: [10,6]\n", "difficulty": "hard", "doc_id": "164"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [0,1]\nnode 9: [1,1]\nThe edges are: (0,2) (0,6) (0,9) (1,4) (1,3) (2,5) (2,9) (3,6) (3,8) (4,7) (5,7) (6,9) (8,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [8,6]\nnode 1: [4,4]\nnode 2: [5,7]\nnode 3: [7,2]\nnode 4: [4,1]\nnode 5: [5,3]\nnode 6: [6,9]\nnode 7: [3,2]\nnode 8: [4,6]\nnode 9: [10,7]\n", "difficulty": "hard", "doc_id": "165"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [0,1]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [0,0]\nnode 10: [0,1]\nnode 11: [0,1]\nnode 12: [1,1]\nnode 13: [0,0]\nThe edges are: (0,8) (0,4) (0,2) (1,4) (2,4) (3,6) (4,8) (4,13) (4,12) (5,7) (6,8) (7,11) (7,9) (8,10) (9,13) (10,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,10]\nnode 1: [1,5]\nnode 2: [1,8]\nnode 3: [1,1]\nnode 4: [0,12]\nnode 5: [0,2]\nnode 6: [0,3]\nnode 7: [0,3]\nnode 8: [3,11]\nnode 9: [0,3]\nnode 10: [0,5]\nnode 11: [0,2]\nnode 12: [2,7]\nnode 13: [1,6]\n", "difficulty": "hard", "doc_id": "166"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [1,1]\nThe edges are: (0,4) (1,5) (1,3) (1,4) (2,3) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [6,3]\nnode 2: [2,0]\nnode 3: [2,4]\nnode 4: [5,5]\nnode 5: [4,5]\n", "difficulty": "easy", "doc_id": "167"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nThe edges are: (0,1) (0,5) (1,3) (1,4) (1,2) (3,4) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,3]\nnode 1: [5,2]\nnode 2: [2,2]\nnode 3: [3,3]\nnode 4: [6,4]\nnode 5: [2,1]\n", "difficulty": "easy", "doc_id": "168"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [0,1]\nThe edges are: (0,5) (0,6) (0,3) (1,4) (2,5) (2,4) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,4]\nnode 1: [0,1]\nnode 2: [0,3]\nnode 3: [1,2]\nnode 4: [0,0]\nnode 5: [1,3]\nnode 6: [1,4]\n", "difficulty": "easy", "doc_id": "169"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [1,0]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,1]\nnode 10: [0,0]\nnode 11: [0,0]\nnode 12: [0,1]\nnode 13: [1,0]\nnode 14: [0,0]\nThe edges are: (0,9) (0,5) (1,14) (1,11) (1,8) (2,6) (2,12) (3,11) (3,6) (3,12) (4,13) (4,10) (5,6) (7,14) (7,13) (8,14) (8,10) (11,14) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,2]\nnode 1: [4,2]\nnode 2: [5,5]\nnode 3: [7,7]\nnode 4: [2,2]\nnode 5: [5,5]\nnode 6: [3,3]\nnode 7: [2,1]\nnode 8: [2,1]\nnode 9: [2,2]\nnode 10: [2,0]\nnode 11: [4,3]\nnode 12: [4,4]\nnode 13: [2,0]\nnode 14: [4,2]\n", "difficulty": "hard", "doc_id": "170"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [0,1]\nThe edges are: (0,7) (0,6) (0,3) (1,7) (1,2) (2,7) (2,3) (3,7) (3,6) (3,4) (4,7) (4,6) (5,7) (5,6)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [14,9]\nnode 1: [8,5]\nnode 2: [11,7]\nnode 3: [16,14]\nnode 4: [14,9]\nnode 5: [10,6]\nnode 6: [9,11]\nnode 7: [12,14]\n", "difficulty": "easy", "doc_id": "171"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,1]\nnode 8: [1,1]\nnode 9: [0,1]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nThe edges are: (0,4) (1,2) (1,9) (1,7) (1,4) (1,8) (2,8) (2,10) (3,11) (3,6) (3,5) (3,8) (4,9) (4,11) (4,5) (5,10) (6,7) (6,10) (7,9) (7,11) (9,10) (10,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [11,12]\nnode 2: [6,8]\nnode 3: [9,9]\nnode 4: [12,13]\nnode 5: [5,6]\nnode 6: [5,7]\nnode 7: [11,12]\nnode 8: [7,8]\nnode 9: [8,11]\nnode 10: [9,9]\nnode 11: [6,7]\nnode 12: [1,2]\n", "difficulty": "hard", "doc_id": "172"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [1,0]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,1]\nnode 8: [0,1]\nThe edges are: (0,5) (0,3) (0,4) (1,8) (1,7) (1,2) (2,6) (2,4) (3,7) (5,6) (6,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,1]\nnode 1: [5,3]\nnode 2: [2,3]\nnode 3: [5,1]\nnode 4: [5,1]\nnode 5: [4,1]\nnode 6: [4,2]\nnode 7: [1,3]\nnode 8: [1,3]\n", "difficulty": "hard", "doc_id": "173"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [1,0]\nnode 9: [1,0]\nThe edges are: (0,6) (0,1) (0,2) (0,7) (1,7) (1,9) (2,9) (3,5) (3,9) (4,7) (5,8) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,6]\nnode 1: [1,7]\nnode 2: [1,4]\nnode 3: [1,2]\nnode 4: [0,3]\nnode 5: [3,0]\nnode 6: [1,5]\nnode 7: [5,4]\nnode 8: [1,0]\nnode 9: [5,2]\n", "difficulty": "hard", "doc_id": "174"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [1,0]\nnode 11: [1,0]\nnode 12: [0,0]\nThe edges are: (0,3) (0,1) (0,11) (0,10) (0,9) (1,8) (1,5) (2,11) (2,6) (3,7) (3,9) (4,11) (4,12) (4,6) (5,7) (5,11) (5,12) (6,7) (6,10) (6,9) (8,11) (8,10) (10,11) (10,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [15,7]\nnode 1: [7,1]\nnode 2: [9,4]\nnode 3: [5,3]\nnode 4: [11,4]\nnode 5: [11,4]\nnode 6: [7,6]\nnode 7: [9,3]\nnode 8: [11,5]\nnode 9: [10,4]\nnode 10: [16,5]\nnode 11: [12,5]\nnode 12: [6,3]\n", "difficulty": "hard", "doc_id": "175"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [1,0]\nnode 4: [0,0]\nThe edges are: (0,4) (1,2) (1,4) (2,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [4,1]\nnode 2: [2,2]\nnode 3: [2,0]\nnode 4: [1,1]\n", "difficulty": "easy", "doc_id": "176"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,0]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [1,1]\nThe edges are: (0,7) (0,6) (0,1) (0,4) (0,8) (0,3) (1,3) (2,7) (2,8) (3,5) (3,6) (5,8) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,4]\nnode 1: [6,4]\nnode 2: [2,1]\nnode 3: [4,7]\nnode 4: [3,3]\nnode 5: [4,1]\nnode 6: [7,5]\nnode 7: [4,5]\nnode 8: [5,6]\n", "difficulty": "hard", "doc_id": "177"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [0,0]\nnode 5: [0,0]\nThe edges are: (0,2) (0,5) (0,1) (1,4) (2,5) (2,3) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,2]\nnode 1: [0,2]\nnode 2: [0,3]\nnode 3: [0,2]\nnode 4: [0,1]\nnode 5: [0,2]\n", "difficulty": "easy", "doc_id": "178"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [0,1]\nnode 4: [1,0]\nThe edges are: (0,3) (1,2) (1,3) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,1]\nnode 1: [4,2]\nnode 2: [0,2]\nnode 3: [0,4]\nnode 4: [3,1]\n", "difficulty": "easy", "doc_id": "179"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [1,1]\nnode 7: [1,0]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [1,0]\nnode 11: [1,0]\nnode 12: [0,1]\nnode 13: [1,0]\nThe edges are: (0,5) (0,11) (0,2) (1,9) (1,11) (1,12) (1,10) (2,7) (2,9) (2,3) (4,9) (6,7) (6,12) (6,10) (6,13) (7,9) (8,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [4,7]\nnode 2: [9,3]\nnode 3: [2,0]\nnode 4: [2,2]\nnode 5: [3,0]\nnode 6: [6,6]\nnode 7: [7,3]\nnode 8: [1,1]\nnode 9: [9,2]\nnode 10: [6,2]\nnode 11: [6,1]\nnode 12: [6,2]\nnode 13: [4,1]\n", "difficulty": "hard", "doc_id": "180"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,0]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [1,1]\nnode 11: [0,1]\nThe edges are: (0,7) (1,8) (1,3) (1,7) (1,2) (1,10) (3,5) (3,6) (3,11) (4,5) (6,7) (6,10) (7,9)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,1]\nnode 1: [1,2]\nnode 2: [3,3]\nnode 3: [6,8]\nnode 4: [1,2]\nnode 5: [0,1]\nnode 6: [1,2]\nnode 7: [7,5]\nnode 8: [3,3]\nnode 9: [1,1]\nnode 10: [5,5]\nnode 11: [0,1]\n", "difficulty": "hard", "doc_id": "181"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 5, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,1]\nThe edges are: (0,3) (1,5) (2,3) (2,4) (3,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [0,1]\nnode 2: [4,3]\nnode 3: [0,2]\nnode 4: [0,1]\nnode 5: [4,3]\n", "difficulty": "easy", "doc_id": "182"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [0,0]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,1]\nnode 10: [1,0]\nnode 11: [1,1]\nnode 12: [0,0]\nnode 13: [1,0]\nnode 14: [0,1]\nThe edges are: (0,13) (0,2) (1,10) (1,12) (1,4) (2,5) (3,10) (3,7) (3,14) (4,8) (6,7) (6,14) (7,12) (8,11) (9,10) (9,11) (9,14) (10,13) (13,14)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,2]\nnode 1: [3,2]\nnode 2: [3,0]\nnode 3: [7,5]\nnode 4: [2,1]\nnode 5: [1,1]\nnode 6: [4,3]\nnode 7: [1,2]\nnode 8: [1,1]\nnode 9: [7,5]\nnode 10: [6,4]\nnode 11: [3,3]\nnode 12: [2,1]\nnode 13: [8,4]\nnode 14: [5,5]\n", "difficulty": "hard", "doc_id": "183"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [1,0]\nnode 10: [1,1]\nnode 11: [0,1]\nnode 12: [1,0]\nnode 13: [0,0]\nnode 14: [0,1]\nThe edges are: (0,8) (0,6) (1,11) (1,8) (1,9) (1,13) (2,5) (2,4) (3,8) (3,10) (3,12) (3,4) (4,8) (5,8) (5,6) (5,10) (6,14) (6,9) (6,12) (6,13) (7,9) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [8,5]\nnode 1: [6,6]\nnode 2: [4,7]\nnode 3: [9,10]\nnode 4: [10,8]\nnode 5: [12,9]\nnode 6: [5,12]\nnode 7: [1,2]\nnode 8: [11,14]\nnode 9: [8,4]\nnode 10: [6,7]\nnode 11: [3,4]\nnode 12: [8,5]\nnode 13: [6,4]\nnode 14: [4,2]\n", "difficulty": "hard", "doc_id": "184"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,0]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,0]\nnode 9: [1,0]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nnode 13: [0,0]\nnode 14: [0,1]\nThe edges are: (0,8) (0,9) (1,14) (1,11) (1,4) (2,8) (3,4) (4,5) (4,10) (5,12) (5,10) (6,7) (6,8) (6,11) (7,12) (8,13) (9,13) (9,12) (11,13) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,2]\nnode 1: [4,1]\nnode 2: [1,2]\nnode 3: [1,0]\nnode 4: [6,1]\nnode 5: [4,0]\nnode 6: [4,4]\nnode 7: [3,0]\nnode 8: [4,0]\nnode 9: [5,0]\nnode 10: [3,0]\nnode 11: [7,1]\nnode 12: [6,2]\nnode 13: [4,3]\nnode 14: [2,1]\n", "difficulty": "hard", "doc_id": "185"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 14, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [0,1]\nnode 6: [0,1]\nnode 7: [1,0]\nnode 8: [1,1]\nnode 9: [0,0]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nnode 13: [0,0]\nnode 14: [1,0]\nThe edges are: (0,10) (0,4) (1,10) (1,5) (1,7) (1,12) (2,14) (2,4) (2,6) (3,10) (3,5) (3,7) (4,8) (4,7) (4,12) (6,10) (6,12) (7,14) (8,11) (8,12) (9,11) (12,13)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,4]\nnode 1: [5,4]\nnode 2: [5,2]\nnode 3: [4,2]\nnode 4: [6,3]\nnode 5: [3,2]\nnode 6: [3,5]\nnode 7: [7,4]\nnode 8: [5,5]\nnode 9: [1,1]\nnode 10: [4,2]\nnode 11: [3,0]\nnode 12: [9,3]\nnode 13: [1,2]\nnode 14: [3,1]\n", "difficulty": "hard", "doc_id": "186"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,1]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [0,1]\nnode 7: [1,0]\nThe edges are: (0,1) (0,4) (0,3) (0,5) (0,7) (1,6) (1,5) (2,4) (2,5) (3,7) (4,5) (5,6) (5,7) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [4,6]\nnode 1: [7,5]\nnode 2: [2,4]\nnode 3: [4,2]\nnode 4: [5,4]\nnode 5: [6,5]\nnode 6: [3,5]\nnode 7: [8,5]\n", "difficulty": "easy", "doc_id": "187"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [1,1]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [0,0]\nnode 10: [1,1]\nnode 11: [1,0]\nnode 12: [0,0]\nThe edges are: (0,2) (0,10) (0,8) (0,3) (1,2) (1,10) (2,7) (2,3) (2,4) (2,12) (2,5) (3,6) (4,8) (4,11) (5,11) (5,12) (6,8) (6,9) (7,10) (8,10) (8,12) (9,12) (10,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [9,13]\nnode 1: [4,6]\nnode 2: [15,18]\nnode 3: [5,10]\nnode 4: [8,10]\nnode 5: [8,10]\nnode 6: [6,8]\nnode 7: [4,6]\nnode 8: [8,14]\nnode 9: [3,6]\nnode 10: [12,16]\nnode 11: [4,3]\nnode 12: [10,12]\n", "difficulty": "hard", "doc_id": "188"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [1,0]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,1]\nnode 8: [1,1]\nnode 9: [1,1]\nnode 10: [0,0]\nnode 11: [0,1]\nnode 12: [1,0]\nThe edges are: (0,7) (0,2) (0,10) (1,9) (1,10) (2,3) (4,5) (4,7) (5,7) (6,7) (6,10) (7,12) (8,11) (8,9) (9,10) (10,11) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [7,2]\nnode 1: [3,3]\nnode 2: [2,1]\nnode 3: [2,0]\nnode 4: [3,1]\nnode 5: [4,1]\nnode 6: [5,2]\nnode 7: [2,6]\nnode 8: [3,2]\nnode 9: [4,5]\nnode 10: [5,5]\nnode 11: [3,6]\nnode 12: [5,1]\n", "difficulty": "hard", "doc_id": "189"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 12, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [0,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,0]\nnode 6: [0,0]\nnode 7: [0,1]\nnode 8: [1,0]\nnode 9: [1,0]\nnode 10: [1,0]\nnode 11: [0,1]\nnode 12: [1,0]\nThe edges are: (0,5) (1,3) (1,2) (1,4) (2,5) (2,7) (2,12) (2,8) (2,11) (3,12) (3,10) (4,11) (5,6) (6,10) (7,8) (7,9) (7,11) (8,12) (9,12) (11,12)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,1]\nnode 1: [5,3]\nnode 2: [7,7]\nnode 3: [2,2]\nnode 4: [1,3]\nnode 5: [6,2]\nnode 6: [1,1]\nnode 7: [6,6]\nnode 8: [7,4]\nnode 9: [4,2]\nnode 10: [4,0]\nnode 11: [7,5]\nnode 12: [8,6]\n", "difficulty": "hard", "doc_id": "190"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,0]\nnode 3: [0,0]\nnode 4: [0,1]\nnode 5: [1,1]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [0,1]\nnode 9: [1,0]\nnode 10: [0,1]\nThe edges are: (0,2) (1,8) (1,9) (1,4) (2,7) (2,5) (2,4) (3,8) (3,9) (4,6) (5,8) (5,10) (6,7) (6,8) (8,10)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,3]\nnode 1: [5,2]\nnode 2: [5,2]\nnode 3: [3,2]\nnode 4: [5,8]\nnode 5: [6,7]\nnode 6: [5,2]\nnode 7: [4,6]\nnode 8: [5,10]\nnode 9: [2,3]\nnode 10: [3,4]\n", "difficulty": "hard", "doc_id": "191"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,0]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [1,0]\nnode 5: [0,0]\nnode 6: [0,0]\nnode 7: [1,1]\nnode 8: [0,1]\nThe edges are: (0,2) (0,1) (0,3) (1,6) (1,7) (2,6) (2,4) (4,7) (5,6) (5,7) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,4]\nnode 1: [4,4]\nnode 2: [5,5]\nnode 3: [2,2]\nnode 4: [3,2]\nnode 5: [2,2]\nnode 6: [5,4]\nnode 7: [6,6]\nnode 8: [1,1]\n", "difficulty": "hard", "doc_id": "192"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 10, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [0,0]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [1,0]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,1]\nnode 9: [1,1]\nnode 10: [0,1]\nThe edges are: (0,4) (0,1) (1,10) (1,6) (2,3) (2,10) (2,9) (3,7) (3,6) (4,5) (4,10) (5,10) (6,10) (6,9) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,3]\nnode 1: [3,5]\nnode 2: [1,6]\nnode 3: [5,8]\nnode 4: [1,3]\nnode 5: [2,3]\nnode 6: [1,8]\nnode 7: [0,2]\nnode 8: [1,2]\nnode 9: [4,6]\nnode 10: [5,10]\n", "difficulty": "hard", "doc_id": "193"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,1]\nThe edges are: (0,2) (1,2) (2,3) (2,4) (3,4)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,3]\nnode 1: [2,3]\nnode 2: [5,2]\nnode 3: [3,4]\nnode 4: [4,4]\n", "difficulty": "easy", "doc_id": "194"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 8, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,0]\nnode 2: [1,1]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [0,0]\nnode 8: [0,1]\nThe edges are: (0,3) (1,8) (1,6) (2,4) (2,3) (2,6) (3,4) (5,7) (7,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [3,1]\nnode 1: [3,1]\nnode 2: [6,4]\nnode 3: [2,5]\nnode 4: [4,3]\nnode 5: [1,2]\nnode 6: [1,4]\nnode 7: [1,0]\nnode 8: [1,4]\n", "difficulty": "hard", "doc_id": "195"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 7, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,0]\nnode 5: [0,1]\nnode 6: [1,1]\nnode 7: [1,1]\nThe edges are: (0,7) (0,3) (1,5) (1,4) (1,3) (2,5) (3,5) (3,4) (4,5) (5,6) (5,7) (6,7)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [5,4]\nnode 1: [9,9]\nnode 2: [5,4]\nnode 3: [8,11]\nnode 4: [9,8]\nnode 5: [8,12]\nnode 6: [7,6]\nnode 7: [7,8]\n", "difficulty": "easy", "doc_id": "196"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 11, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,0]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [1,1]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [1,1]\nnode 8: [1,0]\nnode 9: [0,0]\nnode 10: [1,0]\nnode 11: [0,0]\nThe edges are: (0,8) (1,4) (1,2) (1,7) (2,11) (2,7) (2,6) (3,4) (3,9) (4,10) (5,8) (5,11) (5,9) (6,10) (8,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [0,1]\nnode 1: [4,2]\nnode 2: [8,2]\nnode 3: [1,2]\nnode 4: [6,4]\nnode 5: [2,2]\nnode 6: [4,2]\nnode 7: [5,3]\nnode 8: [4,0]\nnode 9: [2,1]\nnode 10: [3,1]\nnode 11: [3,2]\n", "difficulty": "hard", "doc_id": "197"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 6, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,1]\nnode 1: [0,1]\nnode 2: [1,0]\nnode 3: [0,1]\nnode 4: [0,1]\nnode 5: [0,0]\nnode 6: [1,1]\nThe edges are: (0,6) (0,1) (1,6) (1,5) (2,5) (3,6) (3,4) (4,6) (4,5)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [1,6]\nnode 1: [2,8]\nnode 2: [1,2]\nnode 3: [1,6]\nnode 4: [2,8]\nnode 5: [2,4]\nnode 6: [4,8]\n", "difficulty": "easy", "doc_id": "198"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 9, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [0,0]\nnode 1: [1,0]\nnode 2: [0,1]\nnode 3: [1,0]\nnode 4: [1,1]\nnode 5: [1,1]\nnode 6: [0,1]\nnode 7: [0,1]\nnode 8: [0,1]\nnode 9: [1,1]\nThe edges are: (0,9) (1,6) (1,2) (2,3) (2,5) (2,7) (3,9) (3,4) (3,5) (4,9) (5,7) (6,8)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [2,1]\nnode 1: [4,3]\nnode 2: [5,10]\nnode 3: [8,6]\nnode 4: [5,5]\nnode 5: [7,8]\nnode 6: [0,3]\nnode 7: [4,4]\nnode 8: [1,1]\nnode 9: [6,6]\n", "difficulty": "hard", "doc_id": "199"}
{"question": "In an undirected graph, the nodes are numbered from 0 to 4, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,1]\nnode 1: [0,1]\nnode 2: [0,1]\nnode 3: [1,1]\nnode 4: [0,1]\nThe edges are: (0,1) (1,4) (2,1) (2,4) (4,3)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:\nLayer 1:\nnode 0: sum of embeddings of node 1, [0,1] = [0,1].\nnode 1: sum of embeddings of node 0, node 2, node 4, [1,1] + [0,1] + [0,1] = [1,3].\nnode 2: sum of embeddings of node 1, node 4, [0,1] + [0,1] = [0,2].\nnode 3: sum of embeddings of node 4, [0,1] = [0,1].\nnode 4: sum of embeddings of node 1, node 2, node 3, [0,1] + [0,1] + [1,1] = [1,3].\n\nAfter layer 1, the node embeddings are updated to:\nnode 0: [0,1]\nnode 1: [1,3]\nnode 2: [0,2]\nnode 3: [0,1]\nnode 4: [1,3]\n\nLayer 2:\nnode 0: sum of embeddings of node 1, [1,3] = [1,3].\nnode 1: sum of embeddings of node 0, node 2, node 4, [0,1] + [0,2] + [1,3] = [1,6].\nnode 2: sum of embeddings of node 1, node 4, [1,3] + [1,3] = [2,6].\nnode 3: sum of embeddings of node 4, [1,3] = [1,3].\nnode 4: sum of embeddings of node 1, node 2, node 3, [1,3] + [0,2] + [0,1] = [1,6].\n\nSo the answer is:\nnode 0: [1,3]\nnode 1: [1,6]\nnode 2: [2,6]\nnode 3: [1,3]\nnode 4: [1,6]\n\nIn an undirected graph, the nodes are numbered from 0 to 13, and every node has an embedding. (i,j) means that node i and node j are connected with an undirected edge.\nEmbeddings:\nnode 0: [1,0]\nnode 1: [1,1]\nnode 2: [1,1]\nnode 3: [1,1]\nnode 4: [0,0]\nnode 5: [0,0]\nnode 6: [1,0]\nnode 7: [0,1]\nnode 8: [1,0]\nnode 9: [0,1]\nnode 10: [0,0]\nnode 11: [1,0]\nnode 12: [1,0]\nnode 13: [1,1]\nThe edges are: (0,7) (0,9) (0,6) (0,3) (1,8) (1,6) (1,12) (1,5) (1,13) (1,10) (2,7) (2,9) (2,6) (3,9) (3,11) (4,11) (5,6) (6,11) (7,8) (8,9) (8,13) (9,11)\nIn a simple graph convolution layer, each node's embedding is updated by the sum of its neighbors' embeddings.\nQ: What's the embedding of each node after two layers of simple graph convolution layer?\nA:", "answer": "The answer is:\nnode 0: [14,6]\nnode 1: [12,10]\nnode 2: [12,5]\nnode 3: [9,7]\nnode 4: [2,2]\nnode 5: [8,3]\nnode 6: [11,9]\nnode 7: [5,9]\nnode 8: [14,5]\nnode 9: [9,12]\nnode 10: [4,1]\nnode 11: [12,5]\nnode 12: [4,1]\nnode 13: [6,5]\n", "difficulty": "hard", "doc_id": "200"}