Source code for networkx.algorithms.non_randomness

r""" Computation of graph non-randomness

import math
import networkx as nx
from networkx.utils import not_implemented_for

__all__ = ["non_randomness"]

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