243 lines
9.7 KiB
Python
243 lines
9.7 KiB
Python
import pytest
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np = pytest.importorskip("numpy")
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pytest.importorskip("scipy")
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import networkx as nx
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from networkx.generators.degree_seq import havel_hakimi_graph
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from networkx.generators.expanders import margulis_gabber_galil_graph
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class TestLaplacian:
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@classmethod
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def setup_class(cls):
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deg = [3, 2, 2, 1, 0]
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cls.G = havel_hakimi_graph(deg)
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cls.WG = nx.Graph(
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(u, v, {"weight": 0.5, "other": 0.3}) for (u, v) in cls.G.edges()
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)
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cls.WG.add_node(4)
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cls.MG = nx.MultiGraph(cls.G)
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# Graph with clsloops
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cls.Gsl = cls.G.copy()
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for node in cls.Gsl.nodes():
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cls.Gsl.add_edge(node, node)
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def test_laplacian(self):
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"Graph Laplacian"
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# fmt: off
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NL = np.array([[ 3, -1, -1, -1, 0],
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[-1, 2, -1, 0, 0],
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[-1, -1, 2, 0, 0],
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[-1, 0, 0, 1, 0],
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[ 0, 0, 0, 0, 0]])
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# fmt: on
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WL = 0.5 * NL
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OL = 0.3 * NL
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np.testing.assert_equal(nx.laplacian_matrix(self.G).todense(), NL)
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np.testing.assert_equal(nx.laplacian_matrix(self.MG).todense(), NL)
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np.testing.assert_equal(
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nx.laplacian_matrix(self.G, nodelist=[0, 1]).todense(),
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np.array([[1, -1], [-1, 1]]),
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)
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np.testing.assert_equal(nx.laplacian_matrix(self.WG).todense(), WL)
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np.testing.assert_equal(nx.laplacian_matrix(self.WG, weight=None).todense(), NL)
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np.testing.assert_equal(
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nx.laplacian_matrix(self.WG, weight="other").todense(), OL
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)
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def test_normalized_laplacian(self):
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"Generalized Graph Laplacian"
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# fmt: off
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G = np.array([[ 1. , -0.408, -0.408, -0.577, 0.],
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[-0.408, 1. , -0.5 , 0. , 0.],
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[-0.408, -0.5 , 1. , 0. , 0.],
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[-0.577, 0. , 0. , 1. , 0.],
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[ 0. , 0. , 0. , 0. , 0.]])
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GL = np.array([[ 1. , -0.408, -0.408, -0.577, 0. ],
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[-0.408, 1. , -0.5 , 0. , 0. ],
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[-0.408, -0.5 , 1. , 0. , 0. ],
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[-0.577, 0. , 0. , 1. , 0. ],
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[ 0. , 0. , 0. , 0. , 0. ]])
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Lsl = np.array([[ 0.75 , -0.2887, -0.2887, -0.3536, 0. ],
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[-0.2887, 0.6667, -0.3333, 0. , 0. ],
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[-0.2887, -0.3333, 0.6667, 0. , 0. ],
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[-0.3536, 0. , 0. , 0.5 , 0. ],
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[ 0. , 0. , 0. , 0. , 0. ]])
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# fmt: on
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np.testing.assert_almost_equal(
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nx.normalized_laplacian_matrix(self.G, nodelist=range(5)).todense(),
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G,
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decimal=3,
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)
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np.testing.assert_almost_equal(
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nx.normalized_laplacian_matrix(self.G).todense(), GL, decimal=3
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)
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np.testing.assert_almost_equal(
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nx.normalized_laplacian_matrix(self.MG).todense(), GL, decimal=3
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)
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np.testing.assert_almost_equal(
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nx.normalized_laplacian_matrix(self.WG).todense(), GL, decimal=3
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)
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np.testing.assert_almost_equal(
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nx.normalized_laplacian_matrix(self.WG, weight="other").todense(),
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GL,
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decimal=3,
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)
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np.testing.assert_almost_equal(
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nx.normalized_laplacian_matrix(self.Gsl).todense(), Lsl, decimal=3
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)
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def test_directed_laplacian():
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"Directed Laplacian"
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# Graph used as an example in Sec. 4.1 of Langville and Meyer,
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# "Google's PageRank and Beyond". The graph contains dangling nodes, so
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# the pagerank random walk is selected by directed_laplacian
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G = nx.DiGraph()
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G.add_edges_from(
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(
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(1, 2),
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(1, 3),
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(3, 1),
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(3, 2),
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(3, 5),
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(4, 5),
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(4, 6),
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(5, 4),
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(5, 6),
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(6, 4),
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)
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)
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# fmt: off
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GL = np.array([[ 0.9833, -0.2941, -0.3882, -0.0291, -0.0231, -0.0261],
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[-0.2941, 0.8333, -0.2339, -0.0536, -0.0589, -0.0554],
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[-0.3882, -0.2339, 0.9833, -0.0278, -0.0896, -0.0251],
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[-0.0291, -0.0536, -0.0278, 0.9833, -0.4878, -0.6675],
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[-0.0231, -0.0589, -0.0896, -0.4878, 0.9833, -0.2078],
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[-0.0261, -0.0554, -0.0251, -0.6675, -0.2078, 0.9833]])
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# fmt: on
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L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
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np.testing.assert_almost_equal(L, GL, decimal=3)
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# Make the graph strongly connected, so we can use a random and lazy walk
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G.add_edges_from(((2, 5), (6, 1)))
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# fmt: off
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GL = np.array([[ 1. , -0.3062, -0.4714, 0. , 0. , -0.3227],
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[-0.3062, 1. , -0.1443, 0. , -0.3162, 0. ],
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[-0.4714, -0.1443, 1. , 0. , -0.0913, 0. ],
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[ 0. , 0. , 0. , 1. , -0.5 , -0.5 ],
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[ 0. , -0.3162, -0.0913, -0.5 , 1. , -0.25 ],
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[-0.3227, 0. , 0. , -0.5 , -0.25 , 1. ]])
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# fmt: on
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L = nx.directed_laplacian_matrix(
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G, alpha=0.9, nodelist=sorted(G), walk_type="random"
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)
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np.testing.assert_almost_equal(L, GL, decimal=3)
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# fmt: off
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GL = np.array([[ 0.5 , -0.1531, -0.2357, 0. , 0. , -0.1614],
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[-0.1531, 0.5 , -0.0722, 0. , -0.1581, 0. ],
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[-0.2357, -0.0722, 0.5 , 0. , -0.0456, 0. ],
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[ 0. , 0. , 0. , 0.5 , -0.25 , -0.25 ],
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[ 0. , -0.1581, -0.0456, -0.25 , 0.5 , -0.125 ],
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[-0.1614, 0. , 0. , -0.25 , -0.125 , 0.5 ]])
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# fmt: on
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L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G), walk_type="lazy")
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np.testing.assert_almost_equal(L, GL, decimal=3)
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# Make a strongly connected periodic graph
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G = nx.DiGraph()
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G.add_edges_from(((1, 2), (2, 4), (4, 1), (1, 3), (3, 4)))
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# fmt: off
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GL = np.array([[ 0.5 , -0.176, -0.176, -0.25 ],
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[-0.176, 0.5 , 0. , -0.176],
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[-0.176, 0. , 0.5 , -0.176],
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[-0.25 , -0.176, -0.176, 0.5 ]])
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# fmt: on
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L = nx.directed_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
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np.testing.assert_almost_equal(L, GL, decimal=3)
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def test_directed_combinatorial_laplacian():
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"Directed combinatorial Laplacian"
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# Graph used as an example in Sec. 4.1 of Langville and Meyer,
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# "Google's PageRank and Beyond". The graph contains dangling nodes, so
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# the pagerank random walk is selected by directed_laplacian
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G = nx.DiGraph()
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G.add_edges_from(
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(
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(1, 2),
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(1, 3),
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(3, 1),
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(3, 2),
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(3, 5),
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(4, 5),
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(4, 6),
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(5, 4),
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(5, 6),
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(6, 4),
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)
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)
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# fmt: off
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GL = np.array([[ 0.0366, -0.0132, -0.0153, -0.0034, -0.0020, -0.0027],
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[-0.0132, 0.0450, -0.0111, -0.0076, -0.0062, -0.0069],
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[-0.0153, -0.0111, 0.0408, -0.0035, -0.0083, -0.0027],
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[-0.0034, -0.0076, -0.0035, 0.3688, -0.1356, -0.2187],
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[-0.0020, -0.0062, -0.0083, -0.1356, 0.2026, -0.0505],
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[-0.0027, -0.0069, -0.0027, -0.2187, -0.0505, 0.2815]])
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# fmt: on
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L = nx.directed_combinatorial_laplacian_matrix(G, alpha=0.9, nodelist=sorted(G))
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np.testing.assert_almost_equal(L, GL, decimal=3)
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# Make the graph strongly connected, so we can use a random and lazy walk
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G.add_edges_from(((2, 5), (6, 1)))
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# fmt: off
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GL = np.array([[ 0.1395, -0.0349, -0.0465, 0. , 0. , -0.0581],
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[-0.0349, 0.093 , -0.0116, 0. , -0.0465, 0. ],
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[-0.0465, -0.0116, 0.0698, 0. , -0.0116, 0. ],
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[ 0. , 0. , 0. , 0.2326, -0.1163, -0.1163],
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[ 0. , -0.0465, -0.0116, -0.1163, 0.2326, -0.0581],
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[-0.0581, 0. , 0. , -0.1163, -0.0581, 0.2326]])
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# fmt: on
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L = nx.directed_combinatorial_laplacian_matrix(
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G, alpha=0.9, nodelist=sorted(G), walk_type="random"
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)
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np.testing.assert_almost_equal(L, GL, decimal=3)
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# fmt: off
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GL = np.array([[ 0.0698, -0.0174, -0.0233, 0. , 0. , -0.0291],
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[-0.0174, 0.0465, -0.0058, 0. , -0.0233, 0. ],
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[-0.0233, -0.0058, 0.0349, 0. , -0.0058, 0. ],
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[ 0. , 0. , 0. , 0.1163, -0.0581, -0.0581],
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[ 0. , -0.0233, -0.0058, -0.0581, 0.1163, -0.0291],
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[-0.0291, 0. , 0. , -0.0581, -0.0291, 0.1163]])
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# fmt: on
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L = nx.directed_combinatorial_laplacian_matrix(
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G, alpha=0.9, nodelist=sorted(G), walk_type="lazy"
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)
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np.testing.assert_almost_equal(L, GL, decimal=3)
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E = nx.DiGraph(margulis_gabber_galil_graph(2))
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L = nx.directed_combinatorial_laplacian_matrix(E)
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# fmt: off
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expected = np.array(
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[[ 0.16666667, -0.08333333, -0.08333333, 0. ],
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[-0.08333333, 0.16666667, 0. , -0.08333333],
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[-0.08333333, 0. , 0.16666667, -0.08333333],
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[ 0. , -0.08333333, -0.08333333, 0.16666667]]
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)
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# fmt: on
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np.testing.assert_almost_equal(L, expected, decimal=6)
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with pytest.raises(nx.NetworkXError):
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nx.directed_combinatorial_laplacian_matrix(G, walk_type="pagerank", alpha=100)
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with pytest.raises(nx.NetworkXError):
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nx.directed_combinatorial_laplacian_matrix(G, walk_type="silly")
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