File: //lib/python3/dist-packages/networkx/linalg/tests/test_laplacian.py
import pytest
numpy = pytest.importorskip('numpy')
npt = pytest.importorskip('numpy.testing')
scipy = pytest.importorskip('scipy')
import networkx as nx
from networkx.generators.degree_seq import havel_hakimi_graph
class TestLaplacian(object):
@classmethod
def setup_class(cls):
deg = [3, 2, 2, 1, 0]
cls.G = havel_hakimi_graph(deg)
cls.WG = nx.Graph((u, v, {'weight': 0.5, 'other': 0.3})
for (u, v) in cls.G.edges())
cls.WG.add_node(4)
cls.MG = nx.MultiGraph(cls.G)
# Graph with clsloops
cls.Gsl = cls.G.copy()
for node in cls.Gsl.nodes():
cls.Gsl.add_edge(node, node)
def test_laplacian(self):
"Graph Laplacian"
NL = numpy.array([[3, -1, -1, -1, 0],
[-1, 2, -1, 0, 0],
[-1, -1, 2, 0, 0],
[-1, 0, 0, 1, 0],
[0, 0, 0, 0, 0]])
WL = 0.5 * NL
OL = 0.3 * NL
npt.assert_equal(nx.laplacian_matrix(self.G).todense(), NL)
npt.assert_equal(nx.laplacian_matrix(self.MG).todense(), NL)
npt.assert_equal(nx.laplacian_matrix(self.G, nodelist=[0, 1]).todense(),
numpy.array([[1, -1], [-1, 1]]))
npt.assert_equal(nx.laplacian_matrix(self.WG).todense(), WL)
npt.assert_equal(nx.laplacian_matrix(self.WG, weight=None).todense(), NL)
npt.assert_equal(nx.laplacian_matrix(self.WG, weight='other').todense(), OL)
def test_normalized_laplacian(self):
"Generalized Graph Laplacian"
GL = numpy.array([[1.00, -0.408, -0.408, -0.577, 0.00],
[-0.408, 1.00, -0.50, 0.00, 0.00],
[-0.408, -0.50, 1.00, 0.00, 0.00],
[-0.577, 0.00, 0.00, 1.00, 0.00],
[0.00, 0.00, 0.00, 0.00, 0.00]])
Lsl = numpy.array([[0.75, -0.2887, -0.2887, -0.3536, 0.],
[-0.2887, 0.6667, -0.3333, 0., 0.],
[-0.2887, -0.3333, 0.6667, 0., 0.],
[-0.3536, 0., 0., 0.5, 0.],
[0., 0., 0., 0., 0.]])
npt.assert_almost_equal(nx.normalized_laplacian_matrix(self.G).todense(),
GL, decimal=3)
npt.assert_almost_equal(nx.normalized_laplacian_matrix(self.MG).todense(),
GL, decimal=3)
npt.assert_almost_equal(nx.normalized_laplacian_matrix(self.WG).todense(),
GL, decimal=3)
npt.assert_almost_equal(nx.normalized_laplacian_matrix(self.WG, weight='other').todense(),
GL, decimal=3)
npt.assert_almost_equal(nx.normalized_laplacian_matrix(self.Gsl).todense(),
Lsl, decimal=3)
def test_directed_laplacian(self):
"Directed Laplacian"
# Graph used as an example in Sec. 4.1 of Langville and Meyer,
# "Google's PageRank and Beyond". The graph contains dangling nodes, so
# the pagerank random walk is selected by directed_laplacian
G = nx.DiGraph()
G.add_edges_from(((1, 2), (1, 3), (3, 1), (3, 2), (3, 5), (4, 5), (4, 6),
(5, 4), (5, 6), (6, 4)))
GL = numpy.array([[0.9833, -0.2941, -0.3882, -0.0291, -0.0231, -0.0261],
[-0.2941, 0.8333, -0.2339, -0.0536, -0.0589, -0.0554],
[-0.3882, -0.2339, 0.9833, -0.0278, -0.0896, -0.0251],
[-0.0291, -0.0536, -0.0278, 0.9833, -0.4878, -0.6675],
[-0.0231, -0.0589, -0.0896, -0.4878, 0.9833, -0.2078],
[-0.0261, -0.0554, -0.0251, -0.6675, -0.2078, 0.9833]])
L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
npt.assert_almost_equal(L, GL, decimal=3)
# Make the graph strongly connected, so we can use a random and lazy walk
G.add_edges_from((((2, 5), (6, 1))))
GL = numpy.array([[1., -0.3062, -0.4714, 0., 0., -0.3227],
[-0.3062, 1., -0.1443, 0., -0.3162, 0.],
[-0.4714, -0.1443, 1., 0., -0.0913, 0.],
[0., 0., 0., 1., -0.5, -0.5],
[0., -0.3162, -0.0913, -0.5, 1., -0.25],
[-0.3227, 0., 0., -0.5, -0.25, 1.]])
L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G), walk_type='random')
npt.assert_almost_equal(L, GL, decimal=3)
GL = numpy.array([[0.5, -0.1531, -0.2357, 0., 0., -0.1614],
[-0.1531, 0.5, -0.0722, 0., -0.1581, 0.],
[-0.2357, -0.0722, 0.5, 0., -0.0456, 0.],
[0., 0., 0., 0.5, -0.25, -0.25],
[0., -0.1581, -0.0456, -0.25, 0.5, -0.125],
[-0.1614, 0., 0., -0.25, -0.125, 0.5]])
L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G), walk_type='lazy')
npt.assert_almost_equal(L, GL, decimal=3)
def test_directed_combinatorial_laplacian(self):
"Directed combinatorial Laplacian"
# Graph used as an example in Sec. 4.1 of Langville and Meyer,
# "Google's PageRank and Beyond". The graph contains dangling nodes, so
# the pagerank random walk is selected by directed_laplacian
G = nx.DiGraph()
G.add_edges_from(((1, 2), (1, 3), (3, 1), (3, 2), (3, 5), (4, 5), (4, 6),
(5, 4), (5, 6), (6, 4)))
GL = numpy.array([[0.0366, -0.0132, -0.0153, -0.0034, -0.0020, -0.0027],
[-0.0132, 0.0450, -0.0111, -0.0076, -0.0062, -0.0069],
[-0.0153, -0.0111, 0.0408, -0.0035, -0.0083, -0.0027],
[-0.0034, -0.0076, -0.0035, 0.3688, -0.1356, -0.2187],
[-0.0020, -0.0062, -0.0083, -0.1356, 0.2026, -0.0505],
[-0.0027, -0.0069, -0.0027, -0.2187, -0.0505, 0.2815]])
L = nx.directed_combinatorial_laplacian_matrix(G, alpha=0.9,
nodelist=sorted(G))
npt.assert_almost_equal(L, GL, decimal=3)
# Make the graph strongly connected, so we can use a random and lazy walk
G.add_edges_from((((2, 5), (6, 1))))
GL = numpy.array([[0.1395, -0.0349, -0.0465, 0, 0, -0.0581],
[-0.0349, 0.0930, -0.0116, 0, -0.0465, 0],
[-0.0465, -0.0116, 0.0698, 0, -0.0116, 0],
[0, 0, 0, 0.2326, -0.1163, -0.1163],
[0, -0.0465, -0.0116, -0.1163, 0.2326, -0.0581],
[-0.0581, 0, 0, -0.1163, -0.0581, 0.2326]])
L = nx.directed_combinatorial_laplacian_matrix(G, alpha=0.9,
nodelist=sorted(G),
walk_type='random')
npt.assert_almost_equal(L, GL, decimal=3)
GL = numpy.array([[0.0698, -0.0174, -0.0233, 0, 0, -0.0291],
[-0.0174, 0.0465, -0.0058, 0, -0.0233, 0],
[-0.0233, -0.0058, 0.0349, 0, -0.0058, 0],
[0, 0, 0, 0.1163, -0.0581, -0.0581],
[0, -0.0233, -0.0058, -0.0581, 0.1163, -0.0291],
[-0.0291, 0, 0, -0.0581, -0.0291, 0.1163]])
L = nx.directed_combinatorial_laplacian_matrix(G, alpha=0.9,
nodelist=sorted(G),
walk_type='lazy')
npt.assert_almost_equal(L, GL, decimal=3)