96 lines
2.8 KiB
Python
96 lines
2.8 KiB
Python
|
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
|