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Cucuringu.py
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import numba
import numpy as np
from scipy.sparse import diags
from scipy.sparse.linalg import eigs
from . import utils
from .CPAlgorithm import CPAlgorithm
class LowRankCore(CPAlgorithm):
"""LowRankCore algorithm.
M. Cucuringu, P. Rombach, S. H. Lee, and M. A. Porter Detection of core-periphery structure in networks using spectral methods and geodesic paths. Euro. J. Appl. Math., 27:846–887, 2016.
.. highlight:: python
.. code-block:: python
>>> import cpnet
>>> lrc = cpnet.LowRankCore()
>>> lrc.detect(G)
>>> pair_id = lrc.get_pair_id()
>>> coreness = lrc.get_coreness()
.. note::
- [ ] weighted
- [ ] directed
- [ ] multiple groups of core-periphery pairs
- [ ] continuous core-periphery structure
"""
def __init__(self, beta=0.1):
"""Initialize algorithm.
:param beta: parameter of the algorithm. See the original paper., defaults to 0.1
:type beta: float, optional
"""
self.beta = beta
def detect(self, G):
"""Detect a single core-periphery pair.
:param G: Graph
:type G: networkx.Graph or scipy sparse matrix
"""
A, nodelabel = utils.to_adjacency_matrix(G)
x = self._low_rank_core(A)
Q = self._score(A, None, x)
self.nodelabel = nodelabel
self.c_ = np.zeros(A.shape[0]).astype(int)
self.x_ = x.astype(int)
self.Q_ = np.sum(Q)
self.qs_ = Q
def _score(self, A, c, x):
"""Calculate the strength of core-periphery pairs.
:param A: Adjacency amtrix
:type A: scipy sparse matrix
:param c: group to which a node belongs
:type c: dict
:param x: core (x=1) or periphery (x=0)
:type x: dict
:return: strength of core-periphery
:rtype: float
"""
N = A.shape[0]
Mcc = np.dot(x.T @ A, x) / 2
Mcp = np.dot(x.T @ A, (1 - x))
Mpp = np.dot(x.T @ A, x) / 2
i = np.sum(x)
if i < 2 or i > N - 2:
return [0.0]
q = (
Mcc / float(i * (i - 1) / 2)
+ Mcp / float(i * (N - i))
- Mpp / float((N - i) * ((N - i) - 1) / 2)
)
return [q]
def _find_cut(self, A, score, b):
"""Find the best cut that maximises the objective.
:param A: adjacency matrix
:type A: scipy sparse matrix
:param score: score for each node
:type score: numpy.ndarray
:param b: prameter
:type b: float
:return: core vector
:rtype: numpy.ndarray
"""
N = A.shape[0]
qc = np.zeros(N)
qp = np.zeros(N)
od = (-score).argsort()
for i in range(b, N - b):
x = np.zeros((N, 1))
x[od[0:i]] = 1
Mcc = np.dot(x.T @ A, x)[0, 0] / 2
Mcp = np.dot(x.T @ A, (1 - x))[0, 0]
Mpp = np.dot((1 - x).T * A, (1 - x))[0, 0] / 2
qc[i] = (
Mcc / float(i * (i - 1) / 2)
+ Mcp / float(i * (N - i))
- Mpp / float((N - i) * ((N - i) - 1) / 2)
)
qp[i] = (
Mcp / float(i * (N - i))
+ Mpp / float((N - i) * ((N - i) - 1) / 2)
- Mcc / float(i * (i - 1) / 2)
)
idx_c = np.argmax(qc)
idx_p = np.argmax(qp)
if qc[idx_c] > qp[idx_p]:
Q = qc[idx_c]
x = np.zeros(N)
x[od[0:idx_c]] = 1
else:
Q = qc[idx_p]
x = np.ones(N)
x[od[0:idx_p]] = 0
Q = Q / N
return x
def _low_rank_core(self, A):
"""low rank core algorithm.
:param A: adjacency matrix
:type A: scipy sparse matrix
:return: core vector
:rtype: numpy.ndarray
"""
N = A.shape[0]
d, v = eigs(A, k=2, which="LM")
At = (np.dot(v * diags(d), v.T) > 0.5).astype(int)
score = At.sum(axis=0)
x = self._find_cut(A, score, int(np.round(N * self.beta)))
return x
class LapCore(CPAlgorithm):
"""LapCore algorithm.
M. Cucuringu, P. Rombach, S. H. Lee, and M. A. Porter Detection of core-periphery structure in networks using spectral methods and geodesic paths. Euro. J. Appl. Math., 27:846–887, 2016.
.. highlight:: python
.. code-block:: python
>>> import cpnet
>>> lc = cpnet.LapCore()
>>> lc.detect(G)
>>> pair_id = lc.get_pair_id()
>>> coreness = lc.get_coreness()
.. note::
- [ ] weighted
- [ ] directed
- [ ] multiple groups of core-periphery pairs
- [ ] continuous core-periphery structure
"""
def __init__(self, beta=0.1):
self.beta = beta
def detect(self, G):
"""Detect core-periphery structure.
:param G: Graph
:type G: networkx.Graph or scipy sparse matrix
"""
A, nodelabel = utils.to_adjacency_matrix(G)
x = self._lap_core(A)
Q = self._score(A, None, x)
self.nodelabel = nodelabel
self.c_ = np.zeros(A.shape[0]).astype(int)
self.x_ = x.astype(int)
self.Q_ = np.sum(Q)
self.qs_ = Q
def _score(self, A, c, x):
"""Calculate the strength of core-periphery pairs.
:param A: Adjacency amtrix
:type A: scipy sparse matrix
:param c: group to which a node belongs
:type c: dict
:param x: core (x=1) or periphery (x=0)
:type x: dict
:return: strength of core-periphery
:rtype: float
"""
N = A.shape[0]
Mcc = np.dot(x.T * A, x) / 2
Mcp = np.dot(x.T * A, (1 - x))
Mpp = np.dot(x.T * A, x) / 2
i = np.sum(x)
if i < 2 or i > N - 2:
return [0.0]
q = (
Mcc / float(i * (i - 1) / 2)
+ Mcp / float(i * (N - i))
- Mpp / float((N - i) * ((N - i) - 1) / 2)
)
return [q]
def _find_cut(self, A, score, b):
"""Find the best cut that maximises the objective.
:param A: adjacency matrix
:type A: scipy sparse matrix
:param score: score for each node
:type score: numpy.ndarray
:param b: prameter
:type b: float
:return: core vector
:rtype: numpy.ndarray
"""
N = A.shape[0]
qc = np.zeros(N)
qp = np.zeros(N)
od = (-score).argsort()
for i in range(b, N - b):
x = np.zeros((N, 1))
x[od[0:i]] = 1
Mcc = np.dot(x.T * A, x)[0, 0] / 2
Mcp = np.dot(x.T * A, (1 - x))[0, 0]
Mpp = np.dot((1 - x).T * A, (1 - x))[0, 0] / 2
qc[i] = (
Mcc / float(i * (i - 1) / 2)
+ Mcp / float(i * (N - i))
- Mpp / float((N - i) * ((N - i) - 1) / 2)
)
qp[i] = (
Mcp / float(i * (N - i))
+ Mpp / float((N - i) * ((N - i) - 1) / 2)
- Mcc / float(i * (i - 1) / 2)
)
idx_c = np.argmax(qc)
idx_p = np.argmax(qp)
if qc[idx_c] > qp[idx_p]:
Q = qc[idx_c]
x = np.zeros(N)
x[od[0:idx_c]] = 1
else:
Q = qc[idx_p]
x = np.ones(N)
x[od[0:idx_p]] = 0
Q = Q / N
return x
def _lap_core(self, A):
"""low rank core algorithm.
:param A: adjacency matrix
:type A: scipy sparse matrix
:return: core vector
:rtype: numpy.ndarray
"""
N = A.shape[0]
deg = np.array(A.sum(axis=1)).reshape(-1)
denom = np.zeros(N)
denom[deg > 0] = 1.0 / (deg[deg > 0] + 1.0)
T = diags(denom) * A - diags(np.ones(N))
d, v = eigs(T, k=1, which="SR")
x = self._find_cut(A, v.T[0], int(np.round(N * self.beta)))
return x
class LapSgnCore(CPAlgorithm):
"""LowSgnCore algorithm.
M. Cucuringu, P. Rombach, S. H. Lee, and M. A. Porter Detection of core-periphery structure in networks using spectral methods and geodesic paths. Euro. J. Appl. Math., 27:846–887, 2016.
.. highlight:: python
.. code-block:: python
>>> import cpnet
>>> lsc = cpnet.LapSgnCore()
>>> lsc.detect(G)
>>> pair_id = lsc.get_pair_id()
>>> coreness = lsc.get_coreness()
.. note::
- [ ] weighted
- [ ] directed
- [ ] multiple groups of core-periphery pairs
- [ ] continuous core-periphery structure
"""
def __init__(self, beta=0.1):
"""Initialize algorithm.
:param beta: parameter of the algorithm. See the original paper., defaults to 0.1
:type beta: float, optional
"""
self.beta = beta
def detect(self, G):
"""Detect a single core-periphery pair.
:param G: Graph
:type G: networkx.Graph or scipy sparse matrix
"""
A, nodelabel = utils.to_adjacency_matrix(G)
x = self._lapsgn_core(A)
Q = self._score(A, None, x)
self.nodelabel = nodelabel
self.c_ = np.zeros(A.shape[0]).astype(int)
self.x_ = x.astype(int)
self.Q_ = np.sum(Q)
self.qs_ = Q
def _score(self, A, c, x):
"""Calculate the strength of core-periphery pairs.
:param A: Adjacency amtrix
:type A: scipy sparse matrix
:param c: group to which a node belongs
:type c: dict
:param x: core (x=1) or periphery (x=0)
:type x: dict
:return: strength of core-periphery
:rtype: float
"""
N = A.shape[0]
Mcc = np.dot(x.T @ A, x) / 2
Mcp = np.dot(x.T @ A, (1 - x))
Mpp = np.dot(x.T @ A, x) / 2
i = np.sum(x)
if i < 2 or i > N - 2:
return [0.0]
q = (
Mcc / float(i * (i - 1) / 2)
+ Mcp / float(i * (N - i))
- Mpp / float((N - i) * ((N - i) - 1) / 2)
)
return [q]
def _lapsgn_core(self, A):
"""lapsgn algorithm.
:param A: adjacency matrix
:type A: scipy sparse matrix
:return: core vector
:rtype: numpy.ndarray
"""
N = A.shape[0]
deg = np.array(A.sum(axis=0)).reshape(-1)
denom = np.zeros(N)
denom[deg > 0] = 1.0 / (deg[deg > 0] + 1.0)
T = diags(denom) * A - diags(np.ones(N))
d, v = eigs(T, k=1, which="SR")
v = np.sign(v)
x = (v.T > 0).astype(float)
x = np.array(x).reshape(-1)
if self._score(A, None, x) < self._score(A, None, 1 - x):
x = 1 - x
return x