import pytest
np = pytest.importorskip("numpy")
pytest.importorskip("scipy")
import networkx as nx
from networkx.generators.degree_seq import havel_hakimi_graph
from networkx.generators.expanders import margulis_gabber_galil_graph
class TestLaplacian:
@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"
# fmt: off
NL = np.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]])
# fmt: on
WL = 0.5 * NL
OL = 0.3 * NL
np.testing.assert_equal(nx.laplacian_matrix(self.G).todense(), NL)
np.testing.assert_equal(nx.laplacian_matrix(self.MG).todense(), NL)
np.testing.assert_equal(
nx.laplacian_matrix(self.G, nodelist=[0, 1]).todense(),
np.array([[1, -1], [-1, 1]]),
)
np.testing.assert_equal(nx.laplacian_matrix(self.WG).todense(), WL)
np.testing.assert_equal(nx.laplacian_matrix(self.WG, weight=None).todense(), NL)
np.testing.assert_equal(
nx.laplacian_matrix(self.WG, weight="other").todense(), OL
)
def test_normalized_laplacian(self):
"Generalized Graph Laplacian"
# fmt: off
G = np.array([[ 1. , -0.408, -0.408, -0.577, 0.],
[-0.408, 1. , -0.5 , 0. , 0.],
[-0.408, -0.5 , 1. , 0. , 0.],
[-0.577, 0. , 0. , 1. , 0.],
[ 0. , 0. , 0. , 0. , 0.]])
GL = np.array([[ 1. , -0.408, -0.408, -0.577, 0. ],
[-0.408, 1. , -0.5 , 0. , 0. ],
[-0.408, -0.5 , 1. , 0. , 0. ],
[-0.577, 0. , 0. , 1. , 0. ],
[ 0. , 0. , 0. , 0. , 0. ]])
Lsl = np.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. ]])
# fmt: on
np.testing.assert_almost_equal(
nx.normalized_laplacian_matrix(self.G, nodelist=range(5)).todense(),
G,
decimal=3,
)
np.testing.assert_almost_equal(
nx.normalized_laplacian_matrix(self.G).todense(), GL, decimal=3
)
np.testing.assert_almost_equal(
nx.normalized_laplacian_matrix(self.MG).todense(), GL, decimal=3
)
np.testing.assert_almost_equal(
nx.normalized_laplacian_matrix(self.WG).todense(), GL, decimal=3
)
np.testing.assert_almost_equal(
nx.normalized_laplacian_matrix(self.WG, weight="other").todense(),
GL,
decimal=3,
)
np.testing.assert_almost_equal(
nx.normalized_laplacian_matrix(self.Gsl).todense(), Lsl, decimal=3
)
def test_directed_laplacian():
"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),
)
)
# fmt: off
GL = np.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]])
# fmt: on
L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
np.testing.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)))
# fmt: off
GL = np.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. ]])
# fmt: on
L = nx.directed_laplacian_matrix(
G, alpha=0.9, nodelist=sorted(G), walk_type="random"
)
np.testing.assert_almost_equal(L, GL, decimal=3)
# fmt: off
GL = np.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 ]])
# fmt: on
L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G), walk_type="lazy")
np.testing.assert_almost_equal(L, GL, decimal=3)
# Make a strongly connected periodic graph
G = nx.DiGraph()
G.add_edges_from(((1, 2), (2, 4), (4, 1), (1, 3), (3, 4)))
# fmt: off
GL = np.array([[ 0.5 , -0.176, -0.176, -0.25 ],
[-0.176, 0.5 , 0. , -0.176],
[-0.176, 0. , 0.5 , -0.176],
[-0.25 , -0.176, -0.176, 0.5 ]])
# fmt: on
L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
np.testing.assert_almost_equal(L, GL, decimal=3)
def test_directed_combinatorial_laplacian():
"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),
)
)
# fmt: off
GL = np.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]])
# fmt: on
L = nx.directed_combinatorial_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
np.testing.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)))
# fmt: off
GL = np.array([[ 0.1395, -0.0349, -0.0465, 0. , 0. , -0.0581],
[-0.0349, 0.093 , -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]])
# fmt: on
L = nx.directed_combinatorial_laplacian_matrix(
G, alpha=0.9, nodelist=sorted(G), walk_type="random"
)
np.testing.assert_almost_equal(L, GL, decimal=3)
# fmt: off
GL = np.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]])
# fmt: on
L = nx.directed_combinatorial_laplacian_matrix(
G, alpha=0.9, nodelist=sorted(G), walk_type="lazy"
)
np.testing.assert_almost_equal(L, GL, decimal=3)
E = nx.DiGraph(margulis_gabber_galil_graph(2))
L = nx.directed_combinatorial_laplacian_matrix(E)
# fmt: off
expected = np.array(
[[ 0.16666667, -0.08333333, -0.08333333, 0. ],
[-0.08333333, 0.16666667, 0. , -0.08333333],
[-0.08333333, 0. , 0.16666667, -0.08333333],
[ 0. , -0.08333333, -0.08333333, 0.16666667]]
)
# fmt: on
np.testing.assert_almost_equal(L, expected, decimal=6)
with pytest.raises(nx.NetworkXError):
nx.directed_combinatorial_laplacian_matrix(G, walk_type="pagerank", alpha=100)
with pytest.raises(nx.NetworkXError):
nx.directed_combinatorial_laplacian_matrix(G, walk_type="silly")