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"""Generates graphs with a given eigenvector structure"""
import networkx as nx
from networkx.utils import np_random_state
__all__ = ["spectral_graph_forge"]
@np_random_state(3)
def spectral_graph_forge(G, alpha, transformation="identity", seed=None):
"""Returns a random simple graph with spectrum resembling that of `G`
This algorithm, called Spectral Graph Forge (SGF), computes the
eigenvectors of a given graph adjacency matrix, filters them and
builds a random graph with a similar eigenstructure.
SGF has been proved to be particularly useful for synthesizing
realistic social networks and it can also be used to anonymize
graph sensitive data.
Parameters
----------
G : Graph
alpha : float
Ratio representing the percentage of eigenvectors of G to consider,
values in [0,1].
transformation : string, optional
Represents the intended matrix linear transformation, possible values
are 'identity' and 'modularity'
seed : integer, random_state, or None (default)
Indicator of numpy random number generation state.
See :ref:`Randomness<randomness>`.
Returns
-------
H : Graph
A graph with a similar eigenvector structure of the input one.
Raises
------
NetworkXError
If transformation has a value different from 'identity' or 'modularity'
Notes
-----
Spectral Graph Forge (SGF) generates a random simple graph resembling the
global properties of the given one.
It leverages the low-rank approximation of the associated adjacency matrix
driven by the *alpha* precision parameter.
SGF preserves the number of nodes of the input graph and their ordering.
This way, nodes of output graphs resemble the properties of the input one
and attributes can be directly mapped.
It considers the graph adjacency matrices which can optionally be
transformed to other symmetric real matrices (currently transformation
options include *identity* and *modularity*).
The *modularity* transformation, in the sense of Newman's modularity matrix
allows the focusing on community structure related properties of the graph.
SGF applies a low-rank approximation whose fixed rank is computed from the
ratio *alpha* of the input graph adjacency matrix dimension.
This step performs a filtering on the input eigenvectors similar to the low
pass filtering common in telecommunications.
The filtered values (after truncation) are used as input to a Bernoulli
sampling for constructing a random adjacency matrix.
References
----------
.. [1] L. Baldesi, C. T. Butts, A. Markopoulou, "Spectral Graph Forge:
Graph Generation Targeting Modularity", IEEE Infocom, '18.
https://arxiv.org/abs/1801.01715
.. [2] M. Newman, "Networks: an introduction", Oxford university press,
2010
Examples
--------
>>> G = nx.karate_club_graph()
>>> H = nx.spectral_graph_forge(G, 0.3)
>>>
"""
import numpy as np
import scipy as sp
import scipy.stats # call as sp.stats
available_transformations = ["identity", "modularity"]
alpha = np.clip(alpha, 0, 1)
A = nx.to_numpy_array(G)
n = A.shape[1]
level = int(round(n * alpha))
if transformation not in available_transformations:
msg = f"{transformation!r} is not a valid transformation. "
msg += f"Transformations: {available_transformations}"
raise nx.NetworkXError(msg)
K = np.ones((1, n)) @ A
B = A
if transformation == "modularity":
B -= K.T @ K / K.sum()
# Compute low-rank approximation of B
evals, evecs = np.linalg.eigh(B)
k = np.argsort(np.abs(evals))[::-1] # indices of evals in descending order
evecs[:, k[np.arange(level, n)]] = 0 # set smallest eigenvectors to 0
B = evecs @ np.diag(evals) @ evecs.T
if transformation == "modularity":
B += K.T @ K / K.sum()
B = np.clip(B, 0, 1)
np.fill_diagonal(B, 0)
for i in range(n - 1):
B[i, i + 1 :] = sp.stats.bernoulli.rvs(B[i, i + 1 :], random_state=seed)
B[i + 1 :, i] = np.transpose(B[i, i + 1 :])
H = nx.from_numpy_array(B)
return H
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