83 lines
2.3 KiB
Python
83 lines
2.3 KiB
Python
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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):
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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 binary, 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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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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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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Notes
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-----
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This computes Eq. (4.4) and (4.5) in Ref. [1]_.
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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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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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try:
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import numpy as np
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except ImportError as e:
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msg = "non_randomness requires NumPy: http://numpy.org/"
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raise ImportError(msg) from e
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# eq. 4.4
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nr = np.real(np.sum(np.linalg.eigvals(nx.to_numpy_array(G))[: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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# eq. 4.5
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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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