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r""" Computation of graph non-randomness
"""
import math
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["non_randomness"]
@not_implemented_for("directed")
@not_implemented_for("multigraph")
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(nx.community.label_propagation_communities(G)))
# eq. 4.4
eigenvalues = np.linalg.eigvals(nx.to_numpy_array(G, weight=weight))
nr = 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
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