import networkx as nx

def build_graph(candidate_pairs):
    g = nx.Graph()
    for a, b in candidate_pairs:
        g.add_edge(a, b)
    return g

def pair_graph_features(g, a, b):
    degree_map = dict(g.degree())
    degree_sum = float(degree_map.get(a, 0) + degree_map.get(b, 0))

    common = len(list(nx.common_neighbors(g, a, b))) if a in g and b in g else 0

    na = set(g.neighbors(a)) if a in g else set()
    nb = set(g.neighbors(b)) if b in g else set()

    union = len(na | nb)
    inter = len(na & nb)
    jaccard = float(inter / union) if union else 0.0

    nodes = set([a, b]) | na | nb
    sub = g.subgraph(nodes)
    possible = max(1, len(nodes) * (len(nodes) - 1) / 2)
    density = float(sub.number_of_edges() / possible)

    return {
        "graph_degree_sum": degree_sum,
        "graph_common_neighbors": float(common),
        "graph_jaccard": jaccard,
        "graph_local_density": density,
    }