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r""" Computation of graph non-randomness |
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""" |
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import math |
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import networkx as nx |
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from networkx.utils import not_implemented_for |
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__all__ = ["non_randomness"] |
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@not_implemented_for("directed") |
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@not_implemented_for("multigraph") |
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def non_randomness(G, k=None, weight="weight"): |
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"""Compute the non-randomness of graph G. |
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The first returned value nr is the sum of non-randomness values of all |
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edges within the graph (where the non-randomness of an edge tends to be |
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small when the two nodes linked by that edge are from two different |
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communities). |
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The second computed value nr_rd is a relative measure that indicates |
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to what extent graph G is different from random graphs in terms |
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of probability. When it is close to 0, the graph tends to be more |
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likely generated by an Erdos Renyi model. |
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Parameters |
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---------- |
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G : NetworkX graph |
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Graph must be symmetric, connected, and without self-loops. |
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k : int |
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The number of communities in G. |
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If k is not set, the function will use a default community |
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detection algorithm to set it. |
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weight : string or None, optional (default=None) |
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The name of an edge attribute that holds the numerical value used |
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as a weight. If None, then each edge has weight 1, i.e., the graph is |
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binary. |
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Returns |
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------- |
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non-randomness : (float, float) tuple |
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Non-randomness, Relative non-randomness w.r.t. |
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Erdos Renyi random graphs. |
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Raises |
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------ |
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NetworkXException |
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if the input graph is not connected. |
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NetworkXError |
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if the input graph contains self-loops. |
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Examples |
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-------- |
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>>> G = nx.karate_club_graph() |
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>>> nr, nr_rd = nx.non_randomness(G, 2) |
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>>> nr, nr_rd = nx.non_randomness(G, 2, 'weight') |
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Notes |
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----- |
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This computes Eq. (4.4) and (4.5) in Ref. [1]_. |
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If a weight field is passed, this algorithm will use the eigenvalues |
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of the weighted adjacency matrix to compute Eq. (4.4) and (4.5). |
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References |
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---------- |
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.. [1] Xiaowei Ying and Xintao Wu, |
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On Randomness Measures for Social Networks, |
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SIAM International Conference on Data Mining. 2009 |
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""" |
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import numpy as np |
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if not nx.is_connected(G): |
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raise nx.NetworkXException("Non connected graph.") |
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if len(list(nx.selfloop_edges(G))) > 0: |
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raise nx.NetworkXError("Graph must not contain self-loops") |
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if k is None: |
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k = len(tuple(nx.community.label_propagation_communities(G))) |
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eigenvalues = np.linalg.eigvals(nx.to_numpy_array(G, weight=weight)) |
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nr = np.real(np.sum(eigenvalues[:k])) |
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n = G.number_of_nodes() |
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m = G.number_of_edges() |
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p = (2 * k * m) / (n * (n - k)) |
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nr_rd = (nr - ((n - 2 * k) * p + k)) / math.sqrt(2 * k * p * (1 - p)) |
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return nr, nr_rd |
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