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") @nx._dispatchable(edge_attrs="weight") def non_randomness(G, k=None, weight="weight"): """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 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. weight : string or None, optional (default=None) The name of an edge attribute that holds the numerical value used as a weight. If None, then each edge has weight 1, i.e., the graph is binary. 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) >>> nr, nr_rd = nx.non_randomness(G, 2, "weight") Notes ----- This computes Eq. (4.4) and (4.5) in Ref. [1]_. If a weight field is passed, this algorithm will use the eigenvalues of the weighted adjacency matrix to compute Eq. (4.4) and (4.5). 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 eigenvalues = np.linalg.eigvals(nx.to_numpy_array(G, weight=weight)) nr = float(np.real(np.sum(eigenvalues[: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