Old engine for Continuous Time Bayesian Networks. Superseded by reCTBN. 🐍
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1444 lines
49 KiB
1444 lines
49 KiB
""" Functions measuring similarity using graph edit distance.
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The graph edit distance is the number of edge/node changes needed
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to make two graphs isomorphic.
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The default algorithm/implementation is sub-optimal for some graphs.
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The problem of finding the exact Graph Edit Distance (GED) is NP-hard
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so it is often slow. If the simple interface `graph_edit_distance`
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takes too long for your graph, try `optimize_graph_edit_distance`
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and/or `optimize_edit_paths`.
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At the same time, I encourage capable people to investigate
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alternative GED algorithms, in order to improve the choices available.
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"""
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import time
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from itertools import product
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import networkx as nx
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__all__ = [
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"graph_edit_distance",
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"optimal_edit_paths",
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"optimize_graph_edit_distance",
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"optimize_edit_paths",
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"simrank_similarity",
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"simrank_similarity_numpy",
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]
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def debug_print(*args, **kwargs):
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print(*args, **kwargs)
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def graph_edit_distance(
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G1,
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G2,
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node_match=None,
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edge_match=None,
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node_subst_cost=None,
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node_del_cost=None,
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node_ins_cost=None,
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edge_subst_cost=None,
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edge_del_cost=None,
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edge_ins_cost=None,
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roots=None,
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upper_bound=None,
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timeout=None,
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):
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"""Returns GED (graph edit distance) between graphs G1 and G2.
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Graph edit distance is a graph similarity measure analogous to
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Levenshtein distance for strings. It is defined as minimum cost
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of edit path (sequence of node and edge edit operations)
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transforming graph G1 to graph isomorphic to G2.
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Parameters
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----------
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G1, G2: graphs
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The two graphs G1 and G2 must be of the same type.
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node_match : callable
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A function that returns True if node n1 in G1 and n2 in G2
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should be considered equal during matching.
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The function will be called like
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node_match(G1.nodes[n1], G2.nodes[n2]).
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That is, the function will receive the node attribute
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dictionaries for n1 and n2 as inputs.
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Ignored if node_subst_cost is specified. If neither
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node_match nor node_subst_cost are specified then node
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attributes are not considered.
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edge_match : callable
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A function that returns True if the edge attribute dictionaries
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for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
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be considered equal during matching.
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The function will be called like
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edge_match(G1[u1][v1], G2[u2][v2]).
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That is, the function will receive the edge attribute
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dictionaries of the edges under consideration.
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Ignored if edge_subst_cost is specified. If neither
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edge_match nor edge_subst_cost are specified then edge
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attributes are not considered.
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node_subst_cost, node_del_cost, node_ins_cost : callable
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Functions that return the costs of node substitution, node
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deletion, and node insertion, respectively.
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The functions will be called like
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node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
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node_del_cost(G1.nodes[n1]),
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node_ins_cost(G2.nodes[n2]).
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That is, the functions will receive the node attribute
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dictionaries as inputs. The functions are expected to return
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positive numeric values.
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Function node_subst_cost overrides node_match if specified.
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If neither node_match nor node_subst_cost are specified then
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default node substitution cost of 0 is used (node attributes
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are not considered during matching).
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If node_del_cost is not specified then default node deletion
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cost of 1 is used. If node_ins_cost is not specified then
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default node insertion cost of 1 is used.
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edge_subst_cost, edge_del_cost, edge_ins_cost : callable
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Functions that return the costs of edge substitution, edge
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deletion, and edge insertion, respectively.
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The functions will be called like
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edge_subst_cost(G1[u1][v1], G2[u2][v2]),
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edge_del_cost(G1[u1][v1]),
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edge_ins_cost(G2[u2][v2]).
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That is, the functions will receive the edge attribute
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dictionaries as inputs. The functions are expected to return
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positive numeric values.
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Function edge_subst_cost overrides edge_match if specified.
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If neither edge_match nor edge_subst_cost are specified then
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default edge substitution cost of 0 is used (edge attributes
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are not considered during matching).
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If edge_del_cost is not specified then default edge deletion
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cost of 1 is used. If edge_ins_cost is not specified then
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default edge insertion cost of 1 is used.
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roots : 2-tuple
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Tuple where first element is a node in G1 and the second
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is a node in G2.
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These nodes are forced to be matched in the comparison to
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allow comparison between rooted graphs.
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upper_bound : numeric
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Maximum edit distance to consider. Return None if no edit
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distance under or equal to upper_bound exists.
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timeout : numeric
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Maximum number of seconds to execute.
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After timeout is met, the current best GED is returned.
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Examples
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--------
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>>> G1 = nx.cycle_graph(6)
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>>> G2 = nx.wheel_graph(7)
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>>> nx.graph_edit_distance(G1, G2)
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7.0
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>>> G1 = nx.star_graph(5)
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>>> G2 = nx.star_graph(5)
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>>> nx.graph_edit_distance(G1, G2, roots=(0, 0))
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0.0
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>>> nx.graph_edit_distance(G1, G2, roots=(1, 0))
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8.0
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See Also
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--------
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optimal_edit_paths, optimize_graph_edit_distance,
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is_isomorphic (test for graph edit distance of 0)
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References
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----------
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.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
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Martineau. An Exact Graph Edit Distance Algorithm for Solving
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Pattern Recognition Problems. 4th International Conference on
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Pattern Recognition Applications and Methods 2015, Jan 2015,
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Lisbon, Portugal. 2015,
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<10.5220/0005209202710278>. <hal-01168816>
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https://hal.archives-ouvertes.fr/hal-01168816
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"""
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bestcost = None
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for vertex_path, edge_path, cost in optimize_edit_paths(
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G1,
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G2,
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node_match,
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edge_match,
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node_subst_cost,
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node_del_cost,
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node_ins_cost,
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edge_subst_cost,
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edge_del_cost,
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edge_ins_cost,
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upper_bound,
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True,
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roots,
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timeout,
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):
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# assert bestcost is None or cost < bestcost
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bestcost = cost
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return bestcost
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def optimal_edit_paths(
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G1,
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G2,
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node_match=None,
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edge_match=None,
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node_subst_cost=None,
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node_del_cost=None,
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node_ins_cost=None,
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edge_subst_cost=None,
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edge_del_cost=None,
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edge_ins_cost=None,
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upper_bound=None,
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):
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"""Returns all minimum-cost edit paths transforming G1 to G2.
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Graph edit path is a sequence of node and edge edit operations
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transforming graph G1 to graph isomorphic to G2. Edit operations
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include substitutions, deletions, and insertions.
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Parameters
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----------
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G1, G2: graphs
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The two graphs G1 and G2 must be of the same type.
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node_match : callable
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A function that returns True if node n1 in G1 and n2 in G2
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should be considered equal during matching.
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The function will be called like
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node_match(G1.nodes[n1], G2.nodes[n2]).
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That is, the function will receive the node attribute
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dictionaries for n1 and n2 as inputs.
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Ignored if node_subst_cost is specified. If neither
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node_match nor node_subst_cost are specified then node
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attributes are not considered.
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edge_match : callable
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A function that returns True if the edge attribute dictionaries
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for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
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be considered equal during matching.
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The function will be called like
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edge_match(G1[u1][v1], G2[u2][v2]).
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That is, the function will receive the edge attribute
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dictionaries of the edges under consideration.
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Ignored if edge_subst_cost is specified. If neither
|
|
edge_match nor edge_subst_cost are specified then edge
|
|
attributes are not considered.
|
|
|
|
node_subst_cost, node_del_cost, node_ins_cost : callable
|
|
Functions that return the costs of node substitution, node
|
|
deletion, and node insertion, respectively.
|
|
|
|
The functions will be called like
|
|
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node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
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node_del_cost(G1.nodes[n1]),
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node_ins_cost(G2.nodes[n2]).
|
|
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That is, the functions will receive the node attribute
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|
dictionaries as inputs. The functions are expected to return
|
|
positive numeric values.
|
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|
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Function node_subst_cost overrides node_match if specified.
|
|
If neither node_match nor node_subst_cost are specified then
|
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default node substitution cost of 0 is used (node attributes
|
|
are not considered during matching).
|
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If node_del_cost is not specified then default node deletion
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cost of 1 is used. If node_ins_cost is not specified then
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default node insertion cost of 1 is used.
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|
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edge_subst_cost, edge_del_cost, edge_ins_cost : callable
|
|
Functions that return the costs of edge substitution, edge
|
|
deletion, and edge insertion, respectively.
|
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|
The functions will be called like
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edge_subst_cost(G1[u1][v1], G2[u2][v2]),
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edge_del_cost(G1[u1][v1]),
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edge_ins_cost(G2[u2][v2]).
|
|
|
|
That is, the functions will receive the edge attribute
|
|
dictionaries as inputs. The functions are expected to return
|
|
positive numeric values.
|
|
|
|
Function edge_subst_cost overrides edge_match if specified.
|
|
If neither edge_match nor edge_subst_cost are specified then
|
|
default edge substitution cost of 0 is used (edge attributes
|
|
are not considered during matching).
|
|
|
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If edge_del_cost is not specified then default edge deletion
|
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cost of 1 is used. If edge_ins_cost is not specified then
|
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default edge insertion cost of 1 is used.
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upper_bound : numeric
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Maximum edit distance to consider.
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Returns
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-------
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edit_paths : list of tuples (node_edit_path, edge_edit_path)
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node_edit_path : list of tuples (u, v)
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edge_edit_path : list of tuples ((u1, v1), (u2, v2))
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cost : numeric
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Optimal edit path cost (graph edit distance).
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Examples
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--------
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>>> G1 = nx.cycle_graph(4)
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>>> G2 = nx.wheel_graph(5)
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>>> paths, cost = nx.optimal_edit_paths(G1, G2)
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>>> len(paths)
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40
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>>> cost
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5.0
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See Also
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--------
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graph_edit_distance, optimize_edit_paths
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|
|
References
|
|
----------
|
|
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
|
|
Martineau. An Exact Graph Edit Distance Algorithm for Solving
|
|
Pattern Recognition Problems. 4th International Conference on
|
|
Pattern Recognition Applications and Methods 2015, Jan 2015,
|
|
Lisbon, Portugal. 2015,
|
|
<10.5220/0005209202710278>. <hal-01168816>
|
|
https://hal.archives-ouvertes.fr/hal-01168816
|
|
|
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"""
|
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paths = list()
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bestcost = None
|
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for vertex_path, edge_path, cost in optimize_edit_paths(
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G1,
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G2,
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node_match,
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edge_match,
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node_subst_cost,
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node_del_cost,
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node_ins_cost,
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edge_subst_cost,
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edge_del_cost,
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edge_ins_cost,
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upper_bound,
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False,
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):
|
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# assert bestcost is None or cost <= bestcost
|
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if bestcost is not None and cost < bestcost:
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paths = list()
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paths.append((vertex_path, edge_path))
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bestcost = cost
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return paths, bestcost
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|
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|
def optimize_graph_edit_distance(
|
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G1,
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|
G2,
|
|
node_match=None,
|
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edge_match=None,
|
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node_subst_cost=None,
|
|
node_del_cost=None,
|
|
node_ins_cost=None,
|
|
edge_subst_cost=None,
|
|
edge_del_cost=None,
|
|
edge_ins_cost=None,
|
|
upper_bound=None,
|
|
):
|
|
"""Returns consecutive approximations of GED (graph edit distance)
|
|
between graphs G1 and G2.
|
|
|
|
Graph edit distance is a graph similarity measure analogous to
|
|
Levenshtein distance for strings. It is defined as minimum cost
|
|
of edit path (sequence of node and edge edit operations)
|
|
transforming graph G1 to graph isomorphic to G2.
|
|
|
|
Parameters
|
|
----------
|
|
G1, G2: graphs
|
|
The two graphs G1 and G2 must be of the same type.
|
|
|
|
node_match : callable
|
|
A function that returns True if node n1 in G1 and n2 in G2
|
|
should be considered equal during matching.
|
|
|
|
The function will be called like
|
|
|
|
node_match(G1.nodes[n1], G2.nodes[n2]).
|
|
|
|
That is, the function will receive the node attribute
|
|
dictionaries for n1 and n2 as inputs.
|
|
|
|
Ignored if node_subst_cost is specified. If neither
|
|
node_match nor node_subst_cost are specified then node
|
|
attributes are not considered.
|
|
|
|
edge_match : callable
|
|
A function that returns True if the edge attribute dictionaries
|
|
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
|
|
be considered equal during matching.
|
|
|
|
The function will be called like
|
|
|
|
edge_match(G1[u1][v1], G2[u2][v2]).
|
|
|
|
That is, the function will receive the edge attribute
|
|
dictionaries of the edges under consideration.
|
|
|
|
Ignored if edge_subst_cost is specified. If neither
|
|
edge_match nor edge_subst_cost are specified then edge
|
|
attributes are not considered.
|
|
|
|
node_subst_cost, node_del_cost, node_ins_cost : callable
|
|
Functions that return the costs of node substitution, node
|
|
deletion, and node insertion, respectively.
|
|
|
|
The functions will be called like
|
|
|
|
node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
|
|
node_del_cost(G1.nodes[n1]),
|
|
node_ins_cost(G2.nodes[n2]).
|
|
|
|
That is, the functions will receive the node attribute
|
|
dictionaries as inputs. The functions are expected to return
|
|
positive numeric values.
|
|
|
|
Function node_subst_cost overrides node_match if specified.
|
|
If neither node_match nor node_subst_cost are specified then
|
|
default node substitution cost of 0 is used (node attributes
|
|
are not considered during matching).
|
|
|
|
If node_del_cost is not specified then default node deletion
|
|
cost of 1 is used. If node_ins_cost is not specified then
|
|
default node insertion cost of 1 is used.
|
|
|
|
edge_subst_cost, edge_del_cost, edge_ins_cost : callable
|
|
Functions that return the costs of edge substitution, edge
|
|
deletion, and edge insertion, respectively.
|
|
|
|
The functions will be called like
|
|
|
|
edge_subst_cost(G1[u1][v1], G2[u2][v2]),
|
|
edge_del_cost(G1[u1][v1]),
|
|
edge_ins_cost(G2[u2][v2]).
|
|
|
|
That is, the functions will receive the edge attribute
|
|
dictionaries as inputs. The functions are expected to return
|
|
positive numeric values.
|
|
|
|
Function edge_subst_cost overrides edge_match if specified.
|
|
If neither edge_match nor edge_subst_cost are specified then
|
|
default edge substitution cost of 0 is used (edge attributes
|
|
are not considered during matching).
|
|
|
|
If edge_del_cost is not specified then default edge deletion
|
|
cost of 1 is used. If edge_ins_cost is not specified then
|
|
default edge insertion cost of 1 is used.
|
|
|
|
upper_bound : numeric
|
|
Maximum edit distance to consider.
|
|
|
|
Returns
|
|
-------
|
|
Generator of consecutive approximations of graph edit distance.
|
|
|
|
Examples
|
|
--------
|
|
>>> G1 = nx.cycle_graph(6)
|
|
>>> G2 = nx.wheel_graph(7)
|
|
>>> for v in nx.optimize_graph_edit_distance(G1, G2):
|
|
... minv = v
|
|
>>> minv
|
|
7.0
|
|
|
|
See Also
|
|
--------
|
|
graph_edit_distance, optimize_edit_paths
|
|
|
|
References
|
|
----------
|
|
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
|
|
Martineau. An Exact Graph Edit Distance Algorithm for Solving
|
|
Pattern Recognition Problems. 4th International Conference on
|
|
Pattern Recognition Applications and Methods 2015, Jan 2015,
|
|
Lisbon, Portugal. 2015,
|
|
<10.5220/0005209202710278>. <hal-01168816>
|
|
https://hal.archives-ouvertes.fr/hal-01168816
|
|
"""
|
|
for vertex_path, edge_path, cost in optimize_edit_paths(
|
|
G1,
|
|
G2,
|
|
node_match,
|
|
edge_match,
|
|
node_subst_cost,
|
|
node_del_cost,
|
|
node_ins_cost,
|
|
edge_subst_cost,
|
|
edge_del_cost,
|
|
edge_ins_cost,
|
|
upper_bound,
|
|
True,
|
|
):
|
|
yield cost
|
|
|
|
|
|
def optimize_edit_paths(
|
|
G1,
|
|
G2,
|
|
node_match=None,
|
|
edge_match=None,
|
|
node_subst_cost=None,
|
|
node_del_cost=None,
|
|
node_ins_cost=None,
|
|
edge_subst_cost=None,
|
|
edge_del_cost=None,
|
|
edge_ins_cost=None,
|
|
upper_bound=None,
|
|
strictly_decreasing=True,
|
|
roots=None,
|
|
timeout=None,
|
|
):
|
|
"""GED (graph edit distance) calculation: advanced interface.
|
|
|
|
Graph edit path is a sequence of node and edge edit operations
|
|
transforming graph G1 to graph isomorphic to G2. Edit operations
|
|
include substitutions, deletions, and insertions.
|
|
|
|
Graph edit distance is defined as minimum cost of edit path.
|
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Parameters
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----------
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G1, G2: graphs
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The two graphs G1 and G2 must be of the same type.
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node_match : callable
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A function that returns True if node n1 in G1 and n2 in G2
|
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should be considered equal during matching.
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The function will be called like
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node_match(G1.nodes[n1], G2.nodes[n2]).
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That is, the function will receive the node attribute
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dictionaries for n1 and n2 as inputs.
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Ignored if node_subst_cost is specified. If neither
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node_match nor node_subst_cost are specified then node
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attributes are not considered.
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edge_match : callable
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A function that returns True if the edge attribute dictionaries
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for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
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be considered equal during matching.
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The function will be called like
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edge_match(G1[u1][v1], G2[u2][v2]).
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That is, the function will receive the edge attribute
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dictionaries of the edges under consideration.
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Ignored if edge_subst_cost is specified. If neither
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edge_match nor edge_subst_cost are specified then edge
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attributes are not considered.
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node_subst_cost, node_del_cost, node_ins_cost : callable
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Functions that return the costs of node substitution, node
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deletion, and node insertion, respectively.
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The functions will be called like
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node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
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node_del_cost(G1.nodes[n1]),
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node_ins_cost(G2.nodes[n2]).
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That is, the functions will receive the node attribute
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dictionaries as inputs. The functions are expected to return
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positive numeric values.
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Function node_subst_cost overrides node_match if specified.
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If neither node_match nor node_subst_cost are specified then
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default node substitution cost of 0 is used (node attributes
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are not considered during matching).
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If node_del_cost is not specified then default node deletion
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cost of 1 is used. If node_ins_cost is not specified then
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default node insertion cost of 1 is used.
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edge_subst_cost, edge_del_cost, edge_ins_cost : callable
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Functions that return the costs of edge substitution, edge
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deletion, and edge insertion, respectively.
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The functions will be called like
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edge_subst_cost(G1[u1][v1], G2[u2][v2]),
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edge_del_cost(G1[u1][v1]),
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edge_ins_cost(G2[u2][v2]).
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That is, the functions will receive the edge attribute
|
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dictionaries as inputs. The functions are expected to return
|
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positive numeric values.
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Function edge_subst_cost overrides edge_match if specified.
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If neither edge_match nor edge_subst_cost are specified then
|
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default edge substitution cost of 0 is used (edge attributes
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are not considered during matching).
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If edge_del_cost is not specified then default edge deletion
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cost of 1 is used. If edge_ins_cost is not specified then
|
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default edge insertion cost of 1 is used.
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upper_bound : numeric
|
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Maximum edit distance to consider.
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strictly_decreasing : bool
|
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If True, return consecutive approximations of strictly
|
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decreasing cost. Otherwise, return all edit paths of cost
|
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less than or equal to the previous minimum cost.
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roots : 2-tuple
|
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Tuple where first element is a node in G1 and the second
|
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is a node in G2.
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These nodes are forced to be matched in the comparison to
|
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allow comparison between rooted graphs.
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timeout : numeric
|
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Maximum number of seconds to execute.
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After timeout is met, the current best GED is returned.
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Returns
|
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-------
|
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Generator of tuples (node_edit_path, edge_edit_path, cost)
|
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node_edit_path : list of tuples (u, v)
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edge_edit_path : list of tuples ((u1, v1), (u2, v2))
|
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cost : numeric
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See Also
|
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--------
|
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graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths
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|
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References
|
|
----------
|
|
.. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
|
|
Martineau. An Exact Graph Edit Distance Algorithm for Solving
|
|
Pattern Recognition Problems. 4th International Conference on
|
|
Pattern Recognition Applications and Methods 2015, Jan 2015,
|
|
Lisbon, Portugal. 2015,
|
|
<10.5220/0005209202710278>. <hal-01168816>
|
|
https://hal.archives-ouvertes.fr/hal-01168816
|
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|
|
"""
|
|
# TODO: support DiGraph
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|
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import numpy as np
|
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from scipy.optimize import linear_sum_assignment
|
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|
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class CostMatrix:
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def __init__(self, C, lsa_row_ind, lsa_col_ind, ls):
|
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# assert C.shape[0] == len(lsa_row_ind)
|
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# assert C.shape[1] == len(lsa_col_ind)
|
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# assert len(lsa_row_ind) == len(lsa_col_ind)
|
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# assert set(lsa_row_ind) == set(range(len(lsa_row_ind)))
|
|
# assert set(lsa_col_ind) == set(range(len(lsa_col_ind)))
|
|
# assert ls == C[lsa_row_ind, lsa_col_ind].sum()
|
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self.C = C
|
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self.lsa_row_ind = lsa_row_ind
|
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self.lsa_col_ind = lsa_col_ind
|
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self.ls = ls
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|
|
def make_CostMatrix(C, m, n):
|
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# assert(C.shape == (m + n, m + n))
|
|
lsa_row_ind, lsa_col_ind = linear_sum_assignment(C)
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|
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# Fixup dummy assignments:
|
|
# each substitution i<->j should have dummy assignment m+j<->n+i
|
|
# NOTE: fast reduce of Cv relies on it
|
|
# assert len(lsa_row_ind) == len(lsa_col_ind)
|
|
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
|
|
subst_ind = list(k for k, i, j in indexes if i < m and j < n)
|
|
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
|
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dummy_ind = list(k for k, i, j in indexes if i >= m and j >= n)
|
|
# assert len(subst_ind) == len(dummy_ind)
|
|
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
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lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n
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|
|
|
return CostMatrix(
|
|
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
|
|
)
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|
|
def extract_C(C, i, j, m, n):
|
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# assert(C.shape == (m + n, m + n))
|
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row_ind = [k in i or k - m in j for k in range(m + n)]
|
|
col_ind = [k in j or k - n in i for k in range(m + n)]
|
|
return C[row_ind, :][:, col_ind]
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|
|
def reduce_C(C, i, j, m, n):
|
|
# assert(C.shape == (m + n, m + n))
|
|
row_ind = [k not in i and k - m not in j for k in range(m + n)]
|
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col_ind = [k not in j and k - n not in i for k in range(m + n)]
|
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return C[row_ind, :][:, col_ind]
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|
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def reduce_ind(ind, i):
|
|
# assert set(ind) == set(range(len(ind)))
|
|
rind = ind[[k not in i for k in ind]]
|
|
for k in set(i):
|
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rind[rind >= k] -= 1
|
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return rind
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|
|
def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=[]):
|
|
"""
|
|
Parameters:
|
|
u, v: matched vertices, u=None or v=None for
|
|
deletion/insertion
|
|
pending_g, pending_h: lists of edges not yet mapped
|
|
Ce: CostMatrix of pending edge mappings
|
|
matched_uv: partial vertex edit path
|
|
list of tuples (u, v) of previously matched vertex
|
|
mappings u<->v, u=None or v=None for
|
|
deletion/insertion
|
|
|
|
Returns:
|
|
list of (i, j): indices of edge mappings g<->h
|
|
localCe: local CostMatrix of edge mappings
|
|
(basically submatrix of Ce at cross of rows i, cols j)
|
|
"""
|
|
M = len(pending_g)
|
|
N = len(pending_h)
|
|
# assert Ce.C.shape == (M + N, M + N)
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|
|
|
g_ind = [
|
|
i
|
|
for i in range(M)
|
|
if pending_g[i][:2] == (u, u)
|
|
or any(pending_g[i][:2] in ((p, u), (u, p)) for p, q in matched_uv)
|
|
]
|
|
h_ind = [
|
|
j
|
|
for j in range(N)
|
|
if pending_h[j][:2] == (v, v)
|
|
or any(pending_h[j][:2] in ((q, v), (v, q)) for p, q in matched_uv)
|
|
]
|
|
m = len(g_ind)
|
|
n = len(h_ind)
|
|
|
|
if m or n:
|
|
C = extract_C(Ce.C, g_ind, h_ind, M, N)
|
|
# assert C.shape == (m + n, m + n)
|
|
|
|
# Forbid structurally invalid matches
|
|
# NOTE: inf remembered from Ce construction
|
|
for k, i in zip(range(m), g_ind):
|
|
g = pending_g[i][:2]
|
|
for l, j in zip(range(n), h_ind):
|
|
h = pending_h[j][:2]
|
|
if nx.is_directed(G1) or nx.is_directed(G2):
|
|
if any(
|
|
g == (p, u) and h == (q, v) or g == (u, p) and h == (v, q)
|
|
for p, q in matched_uv
|
|
):
|
|
continue
|
|
else:
|
|
if any(
|
|
g in ((p, u), (u, p)) and h in ((q, v), (v, q))
|
|
for p, q in matched_uv
|
|
):
|
|
continue
|
|
if g == (u, u):
|
|
continue
|
|
if h == (v, v):
|
|
continue
|
|
C[k, l] = inf
|
|
|
|
localCe = make_CostMatrix(C, m, n)
|
|
ij = list(
|
|
(
|
|
g_ind[k] if k < m else M + h_ind[l],
|
|
h_ind[l] if l < n else N + g_ind[k],
|
|
)
|
|
for k, l in zip(localCe.lsa_row_ind, localCe.lsa_col_ind)
|
|
if k < m or l < n
|
|
)
|
|
|
|
else:
|
|
ij = []
|
|
localCe = CostMatrix(np.empty((0, 0)), [], [], 0)
|
|
|
|
return ij, localCe
|
|
|
|
def reduce_Ce(Ce, ij, m, n):
|
|
if len(ij):
|
|
i, j = zip(*ij)
|
|
m_i = m - sum(1 for t in i if t < m)
|
|
n_j = n - sum(1 for t in j if t < n)
|
|
return make_CostMatrix(reduce_C(Ce.C, i, j, m, n), m_i, n_j)
|
|
else:
|
|
return Ce
|
|
|
|
def get_edit_ops(
|
|
matched_uv, pending_u, pending_v, Cv, pending_g, pending_h, Ce, matched_cost
|
|
):
|
|
"""
|
|
Parameters:
|
|
matched_uv: partial vertex edit path
|
|
list of tuples (u, v) of vertex mappings u<->v,
|
|
u=None or v=None for deletion/insertion
|
|
pending_u, pending_v: lists of vertices not yet mapped
|
|
Cv: CostMatrix of pending vertex mappings
|
|
pending_g, pending_h: lists of edges not yet mapped
|
|
Ce: CostMatrix of pending edge mappings
|
|
matched_cost: cost of partial edit path
|
|
|
|
Returns:
|
|
sequence of
|
|
(i, j): indices of vertex mapping u<->v
|
|
Cv_ij: reduced CostMatrix of pending vertex mappings
|
|
(basically Cv with row i, col j removed)
|
|
list of (x, y): indices of edge mappings g<->h
|
|
Ce_xy: reduced CostMatrix of pending edge mappings
|
|
(basically Ce with rows x, cols y removed)
|
|
cost: total cost of edit operation
|
|
NOTE: most promising ops first
|
|
"""
|
|
m = len(pending_u)
|
|
n = len(pending_v)
|
|
# assert Cv.C.shape == (m + n, m + n)
|
|
|
|
# 1) a vertex mapping from optimal linear sum assignment
|
|
i, j = min(
|
|
(k, l) for k, l in zip(Cv.lsa_row_ind, Cv.lsa_col_ind) if k < m or l < n
|
|
)
|
|
xy, localCe = match_edges(
|
|
pending_u[i] if i < m else None,
|
|
pending_v[j] if j < n else None,
|
|
pending_g,
|
|
pending_h,
|
|
Ce,
|
|
matched_uv,
|
|
)
|
|
Ce_xy = reduce_Ce(Ce, xy, len(pending_g), len(pending_h))
|
|
# assert Ce.ls <= localCe.ls + Ce_xy.ls
|
|
if prune(matched_cost + Cv.ls + localCe.ls + Ce_xy.ls):
|
|
pass
|
|
else:
|
|
# get reduced Cv efficiently
|
|
Cv_ij = CostMatrix(
|
|
reduce_C(Cv.C, (i,), (j,), m, n),
|
|
reduce_ind(Cv.lsa_row_ind, (i, m + j)),
|
|
reduce_ind(Cv.lsa_col_ind, (j, n + i)),
|
|
Cv.ls - Cv.C[i, j],
|
|
)
|
|
yield (i, j), Cv_ij, xy, Ce_xy, Cv.C[i, j] + localCe.ls
|
|
|
|
# 2) other candidates, sorted by lower-bound cost estimate
|
|
other = list()
|
|
fixed_i, fixed_j = i, j
|
|
if m <= n:
|
|
candidates = (
|
|
(t, fixed_j)
|
|
for t in range(m + n)
|
|
if t != fixed_i and (t < m or t == m + fixed_j)
|
|
)
|
|
else:
|
|
candidates = (
|
|
(fixed_i, t)
|
|
for t in range(m + n)
|
|
if t != fixed_j and (t < n or t == n + fixed_i)
|
|
)
|
|
for i, j in candidates:
|
|
if prune(matched_cost + Cv.C[i, j] + Ce.ls):
|
|
continue
|
|
Cv_ij = make_CostMatrix(
|
|
reduce_C(Cv.C, (i,), (j,), m, n),
|
|
m - 1 if i < m else m,
|
|
n - 1 if j < n else n,
|
|
)
|
|
# assert Cv.ls <= Cv.C[i, j] + Cv_ij.ls
|
|
if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + Ce.ls):
|
|
continue
|
|
xy, localCe = match_edges(
|
|
pending_u[i] if i < m else None,
|
|
pending_v[j] if j < n else None,
|
|
pending_g,
|
|
pending_h,
|
|
Ce,
|
|
matched_uv,
|
|
)
|
|
if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + localCe.ls):
|
|
continue
|
|
Ce_xy = reduce_Ce(Ce, xy, len(pending_g), len(pending_h))
|
|
# assert Ce.ls <= localCe.ls + Ce_xy.ls
|
|
if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + localCe.ls + Ce_xy.ls):
|
|
continue
|
|
other.append(((i, j), Cv_ij, xy, Ce_xy, Cv.C[i, j] + localCe.ls))
|
|
|
|
yield from sorted(other, key=lambda t: t[4] + t[1].ls + t[3].ls)
|
|
|
|
def get_edit_paths(
|
|
matched_uv,
|
|
pending_u,
|
|
pending_v,
|
|
Cv,
|
|
matched_gh,
|
|
pending_g,
|
|
pending_h,
|
|
Ce,
|
|
matched_cost,
|
|
):
|
|
"""
|
|
Parameters:
|
|
matched_uv: partial vertex edit path
|
|
list of tuples (u, v) of vertex mappings u<->v,
|
|
u=None or v=None for deletion/insertion
|
|
pending_u, pending_v: lists of vertices not yet mapped
|
|
Cv: CostMatrix of pending vertex mappings
|
|
matched_gh: partial edge edit path
|
|
list of tuples (g, h) of edge mappings g<->h,
|
|
g=None or h=None for deletion/insertion
|
|
pending_g, pending_h: lists of edges not yet mapped
|
|
Ce: CostMatrix of pending edge mappings
|
|
matched_cost: cost of partial edit path
|
|
|
|
Returns:
|
|
sequence of (vertex_path, edge_path, cost)
|
|
vertex_path: complete vertex edit path
|
|
list of tuples (u, v) of vertex mappings u<->v,
|
|
u=None or v=None for deletion/insertion
|
|
edge_path: complete edge edit path
|
|
list of tuples (g, h) of edge mappings g<->h,
|
|
g=None or h=None for deletion/insertion
|
|
cost: total cost of edit path
|
|
NOTE: path costs are non-increasing
|
|
"""
|
|
# debug_print('matched-uv:', matched_uv)
|
|
# debug_print('matched-gh:', matched_gh)
|
|
# debug_print('matched-cost:', matched_cost)
|
|
# debug_print('pending-u:', pending_u)
|
|
# debug_print('pending-v:', pending_v)
|
|
# debug_print(Cv.C)
|
|
# assert list(sorted(G1.nodes)) == list(sorted(list(u for u, v in matched_uv if u is not None) + pending_u))
|
|
# assert list(sorted(G2.nodes)) == list(sorted(list(v for u, v in matched_uv if v is not None) + pending_v))
|
|
# debug_print('pending-g:', pending_g)
|
|
# debug_print('pending-h:', pending_h)
|
|
# debug_print(Ce.C)
|
|
# assert list(sorted(G1.edges)) == list(sorted(list(g for g, h in matched_gh if g is not None) + pending_g))
|
|
# assert list(sorted(G2.edges)) == list(sorted(list(h for g, h in matched_gh if h is not None) + pending_h))
|
|
# debug_print()
|
|
|
|
if prune(matched_cost + Cv.ls + Ce.ls):
|
|
return
|
|
|
|
if not max(len(pending_u), len(pending_v)):
|
|
# assert not len(pending_g)
|
|
# assert not len(pending_h)
|
|
# path completed!
|
|
# assert matched_cost <= maxcost.value
|
|
maxcost.value = min(maxcost.value, matched_cost)
|
|
yield matched_uv, matched_gh, matched_cost
|
|
|
|
else:
|
|
edit_ops = get_edit_ops(
|
|
matched_uv,
|
|
pending_u,
|
|
pending_v,
|
|
Cv,
|
|
pending_g,
|
|
pending_h,
|
|
Ce,
|
|
matched_cost,
|
|
)
|
|
for ij, Cv_ij, xy, Ce_xy, edit_cost in edit_ops:
|
|
i, j = ij
|
|
# assert Cv.C[i, j] + sum(Ce.C[t] for t in xy) == edit_cost
|
|
if prune(matched_cost + edit_cost + Cv_ij.ls + Ce_xy.ls):
|
|
continue
|
|
|
|
# dive deeper
|
|
u = pending_u.pop(i) if i < len(pending_u) else None
|
|
v = pending_v.pop(j) if j < len(pending_v) else None
|
|
matched_uv.append((u, v))
|
|
for x, y in xy:
|
|
len_g = len(pending_g)
|
|
len_h = len(pending_h)
|
|
matched_gh.append(
|
|
(
|
|
pending_g[x] if x < len_g else None,
|
|
pending_h[y] if y < len_h else None,
|
|
)
|
|
)
|
|
sortedx = list(sorted(x for x, y in xy))
|
|
sortedy = list(sorted(y for x, y in xy))
|
|
G = list(
|
|
(pending_g.pop(x) if x < len(pending_g) else None)
|
|
for x in reversed(sortedx)
|
|
)
|
|
H = list(
|
|
(pending_h.pop(y) if y < len(pending_h) else None)
|
|
for y in reversed(sortedy)
|
|
)
|
|
|
|
yield from get_edit_paths(
|
|
matched_uv,
|
|
pending_u,
|
|
pending_v,
|
|
Cv_ij,
|
|
matched_gh,
|
|
pending_g,
|
|
pending_h,
|
|
Ce_xy,
|
|
matched_cost + edit_cost,
|
|
)
|
|
|
|
# backtrack
|
|
if u is not None:
|
|
pending_u.insert(i, u)
|
|
if v is not None:
|
|
pending_v.insert(j, v)
|
|
matched_uv.pop()
|
|
for x, g in zip(sortedx, reversed(G)):
|
|
if g is not None:
|
|
pending_g.insert(x, g)
|
|
for y, h in zip(sortedy, reversed(H)):
|
|
if h is not None:
|
|
pending_h.insert(y, h)
|
|
for t in xy:
|
|
matched_gh.pop()
|
|
|
|
# Initialization
|
|
|
|
pending_u = list(G1.nodes)
|
|
pending_v = list(G2.nodes)
|
|
|
|
initial_cost = 0
|
|
if roots:
|
|
root_u, root_v = roots
|
|
if root_u not in pending_u or root_v not in pending_v:
|
|
raise nx.NodeNotFound("Root node not in graph.")
|
|
|
|
# remove roots from pending
|
|
pending_u.remove(root_u)
|
|
pending_v.remove(root_v)
|
|
|
|
# cost matrix of vertex mappings
|
|
m = len(pending_u)
|
|
n = len(pending_v)
|
|
C = np.zeros((m + n, m + n))
|
|
if node_subst_cost:
|
|
C[0:m, 0:n] = np.array(
|
|
[
|
|
node_subst_cost(G1.nodes[u], G2.nodes[v])
|
|
for u in pending_u
|
|
for v in pending_v
|
|
]
|
|
).reshape(m, n)
|
|
if roots:
|
|
initial_cost = node_subst_cost(G1.nodes[root_u], G2.nodes[root_v])
|
|
elif node_match:
|
|
C[0:m, 0:n] = np.array(
|
|
[
|
|
1 - int(node_match(G1.nodes[u], G2.nodes[v]))
|
|
for u in pending_u
|
|
for v in pending_v
|
|
]
|
|
).reshape(m, n)
|
|
if roots:
|
|
initial_cost = 1 - node_match(G1.nodes[root_u], G2.nodes[root_v])
|
|
else:
|
|
# all zeroes
|
|
pass
|
|
# assert not min(m, n) or C[0:m, 0:n].min() >= 0
|
|
if node_del_cost:
|
|
del_costs = [node_del_cost(G1.nodes[u]) for u in pending_u]
|
|
else:
|
|
del_costs = [1] * len(pending_u)
|
|
# assert not m or min(del_costs) >= 0
|
|
if node_ins_cost:
|
|
ins_costs = [node_ins_cost(G2.nodes[v]) for v in pending_v]
|
|
else:
|
|
ins_costs = [1] * len(pending_v)
|
|
# assert not n or min(ins_costs) >= 0
|
|
inf = C[0:m, 0:n].sum() + sum(del_costs) + sum(ins_costs) + 1
|
|
C[0:m, n : n + m] = np.array(
|
|
[del_costs[i] if i == j else inf for i in range(m) for j in range(m)]
|
|
).reshape(m, m)
|
|
C[m : m + n, 0:n] = np.array(
|
|
[ins_costs[i] if i == j else inf for i in range(n) for j in range(n)]
|
|
).reshape(n, n)
|
|
Cv = make_CostMatrix(C, m, n)
|
|
# debug_print(f"Cv: {m} x {n}")
|
|
# debug_print(Cv.C)
|
|
|
|
pending_g = list(G1.edges)
|
|
pending_h = list(G2.edges)
|
|
|
|
# cost matrix of edge mappings
|
|
m = len(pending_g)
|
|
n = len(pending_h)
|
|
C = np.zeros((m + n, m + n))
|
|
if edge_subst_cost:
|
|
C[0:m, 0:n] = np.array(
|
|
[
|
|
edge_subst_cost(G1.edges[g], G2.edges[h])
|
|
for g in pending_g
|
|
for h in pending_h
|
|
]
|
|
).reshape(m, n)
|
|
elif edge_match:
|
|
C[0:m, 0:n] = np.array(
|
|
[
|
|
1 - int(edge_match(G1.edges[g], G2.edges[h]))
|
|
for g in pending_g
|
|
for h in pending_h
|
|
]
|
|
).reshape(m, n)
|
|
else:
|
|
# all zeroes
|
|
pass
|
|
# assert not min(m, n) or C[0:m, 0:n].min() >= 0
|
|
if edge_del_cost:
|
|
del_costs = [edge_del_cost(G1.edges[g]) for g in pending_g]
|
|
else:
|
|
del_costs = [1] * len(pending_g)
|
|
# assert not m or min(del_costs) >= 0
|
|
if edge_ins_cost:
|
|
ins_costs = [edge_ins_cost(G2.edges[h]) for h in pending_h]
|
|
else:
|
|
ins_costs = [1] * len(pending_h)
|
|
# assert not n or min(ins_costs) >= 0
|
|
inf = C[0:m, 0:n].sum() + sum(del_costs) + sum(ins_costs) + 1
|
|
C[0:m, n : n + m] = np.array(
|
|
[del_costs[i] if i == j else inf for i in range(m) for j in range(m)]
|
|
).reshape(m, m)
|
|
C[m : m + n, 0:n] = np.array(
|
|
[ins_costs[i] if i == j else inf for i in range(n) for j in range(n)]
|
|
).reshape(n, n)
|
|
Ce = make_CostMatrix(C, m, n)
|
|
# debug_print(f'Ce: {m} x {n}')
|
|
# debug_print(Ce.C)
|
|
# debug_print()
|
|
|
|
class MaxCost:
|
|
def __init__(self):
|
|
# initial upper-bound estimate
|
|
# NOTE: should work for empty graph
|
|
self.value = Cv.C.sum() + Ce.C.sum() + 1
|
|
|
|
maxcost = MaxCost()
|
|
|
|
if timeout is not None:
|
|
if timeout <= 0:
|
|
raise nx.NetworkXError("Timeout value must be greater than 0")
|
|
start = time.perf_counter()
|
|
|
|
def prune(cost):
|
|
if timeout is not None:
|
|
if time.perf_counter() - start > timeout:
|
|
return True
|
|
if upper_bound is not None:
|
|
if cost > upper_bound:
|
|
return True
|
|
if cost > maxcost.value:
|
|
return True
|
|
elif strictly_decreasing and cost >= maxcost.value:
|
|
return True
|
|
|
|
# Now go!
|
|
|
|
done_uv = [] if roots is None else [roots]
|
|
|
|
for vertex_path, edge_path, cost in get_edit_paths(
|
|
done_uv, pending_u, pending_v, Cv, [], pending_g, pending_h, Ce, initial_cost
|
|
):
|
|
# assert sorted(G1.nodes) == sorted(u for u, v in vertex_path if u is not None)
|
|
# assert sorted(G2.nodes) == sorted(v for u, v in vertex_path if v is not None)
|
|
# assert sorted(G1.edges) == sorted(g for g, h in edge_path if g is not None)
|
|
# assert sorted(G2.edges) == sorted(h for g, h in edge_path if h is not None)
|
|
# print(vertex_path, edge_path, cost, file = sys.stderr)
|
|
# assert cost == maxcost.value
|
|
yield list(vertex_path), list(edge_path), cost
|
|
|
|
|
|
def _is_close(d1, d2, atolerance=0, rtolerance=0):
|
|
"""Determines whether two adjacency matrices are within
|
|
a provided tolerance.
|
|
|
|
Parameters
|
|
----------
|
|
d1 : dict
|
|
Adjacency dictionary
|
|
|
|
d2 : dict
|
|
Adjacency dictionary
|
|
|
|
atolerance : float
|
|
Some scalar tolerance value to determine closeness
|
|
|
|
rtolerance : float
|
|
A scalar tolerance value that will be some proportion
|
|
of ``d2``'s value
|
|
|
|
Returns
|
|
-------
|
|
closeness : bool
|
|
If all of the nodes within ``d1`` and ``d2`` are within
|
|
a predefined tolerance, they are considered "close" and
|
|
this method will return True. Otherwise, this method will
|
|
return False.
|
|
|
|
"""
|
|
# Pre-condition: d1 and d2 have the same keys at each level if they
|
|
# are dictionaries.
|
|
if not isinstance(d1, dict) and not isinstance(d2, dict):
|
|
return abs(d1 - d2) <= atolerance + rtolerance * abs(d2)
|
|
return all(all(_is_close(d1[u][v], d2[u][v]) for v in d1[u]) for u in d1)
|
|
|
|
|
|
def simrank_similarity(
|
|
G,
|
|
source=None,
|
|
target=None,
|
|
importance_factor=0.9,
|
|
max_iterations=100,
|
|
tolerance=1e-4,
|
|
):
|
|
"""Returns the SimRank similarity of nodes in the graph ``G``.
|
|
|
|
SimRank is a similarity metric that says "two objects are considered
|
|
to be similar if they are referenced by similar objects." [1]_.
|
|
|
|
The pseudo-code definition from the paper is::
|
|
|
|
def simrank(G, u, v):
|
|
in_neighbors_u = G.predecessors(u)
|
|
in_neighbors_v = G.predecessors(v)
|
|
scale = C / (len(in_neighbors_u) * len(in_neighbors_v))
|
|
return scale * sum(simrank(G, w, x)
|
|
for w, x in product(in_neighbors_u,
|
|
in_neighbors_v))
|
|
|
|
where ``G`` is the graph, ``u`` is the source, ``v`` is the target,
|
|
and ``C`` is a float decay or importance factor between 0 and 1.
|
|
|
|
The SimRank algorithm for determining node similarity is defined in
|
|
[2]_.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
A NetworkX graph
|
|
|
|
source : node
|
|
If this is specified, the returned dictionary maps each node
|
|
``v`` in the graph to the similarity between ``source`` and
|
|
``v``.
|
|
|
|
target : node
|
|
If both ``source`` and ``target`` are specified, the similarity
|
|
value between ``source`` and ``target`` is returned. If
|
|
``target`` is specified but ``source`` is not, this argument is
|
|
ignored.
|
|
|
|
importance_factor : float
|
|
The relative importance of indirect neighbors with respect to
|
|
direct neighbors.
|
|
|
|
max_iterations : integer
|
|
Maximum number of iterations.
|
|
|
|
tolerance : float
|
|
Error tolerance used to check convergence. When an iteration of
|
|
the algorithm finds that no similarity value changes more than
|
|
this amount, the algorithm halts.
|
|
|
|
Returns
|
|
-------
|
|
similarity : dictionary or float
|
|
If ``source`` and ``target`` are both ``None``, this returns a
|
|
dictionary of dictionaries, where keys are node pairs and value
|
|
are similarity of the pair of nodes.
|
|
|
|
If ``source`` is not ``None`` but ``target`` is, this returns a
|
|
dictionary mapping node to the similarity of ``source`` and that
|
|
node.
|
|
|
|
If neither ``source`` nor ``target`` is ``None``, this returns
|
|
the similarity value for the given pair of nodes.
|
|
|
|
Examples
|
|
--------
|
|
If the nodes of the graph are numbered from zero to *n - 1*, where *n*
|
|
is the number of nodes in the graph, you can create a SimRank matrix
|
|
from the return value of this function where the node numbers are
|
|
the row and column indices of the matrix::
|
|
|
|
>>> from numpy import array
|
|
>>> G = nx.cycle_graph(4)
|
|
>>> sim = nx.simrank_similarity(G)
|
|
>>> lol = [[sim[u][v] for v in sorted(sim[u])] for u in sorted(sim)]
|
|
>>> sim_array = array(lol)
|
|
|
|
References
|
|
----------
|
|
.. [1] https://en.wikipedia.org/wiki/SimRank
|
|
.. [2] G. Jeh and J. Widom.
|
|
"SimRank: a measure of structural-context similarity",
|
|
In KDD'02: Proceedings of the Eighth ACM SIGKDD
|
|
International Conference on Knowledge Discovery and Data Mining,
|
|
pp. 538--543. ACM Press, 2002.
|
|
"""
|
|
prevsim = None
|
|
|
|
# build up our similarity adjacency dictionary output
|
|
newsim = {u: {v: 1 if u == v else 0 for v in G} for u in G}
|
|
|
|
# These functions compute the update to the similarity value of the nodes
|
|
# `u` and `v` with respect to the previous similarity values.
|
|
def avg_sim(s):
|
|
return sum(newsim[w][x] for (w, x) in s) / len(s) if s else 0.0
|
|
|
|
def sim(u, v):
|
|
Gadj = G.pred if G.is_directed() else G.adj
|
|
return importance_factor * avg_sim(list(product(Gadj[u], Gadj[v])))
|
|
|
|
for _ in range(max_iterations):
|
|
if prevsim and _is_close(prevsim, newsim, tolerance):
|
|
break
|
|
prevsim = newsim
|
|
newsim = {
|
|
u: {v: sim(u, v) if u is not v else 1 for v in newsim[u]} for u in newsim
|
|
}
|
|
|
|
if source is not None and target is not None:
|
|
return newsim[source][target]
|
|
if source is not None:
|
|
return newsim[source]
|
|
return newsim
|
|
|
|
|
|
def simrank_similarity_numpy(
|
|
G,
|
|
source=None,
|
|
target=None,
|
|
importance_factor=0.9,
|
|
max_iterations=100,
|
|
tolerance=1e-4,
|
|
):
|
|
"""Calculate SimRank of nodes in ``G`` using matrices with ``numpy``.
|
|
|
|
The SimRank algorithm for determining node similarity is defined in
|
|
[1]_.
|
|
|
|
Parameters
|
|
----------
|
|
G : NetworkX graph
|
|
A NetworkX graph
|
|
|
|
source : node
|
|
If this is specified, the returned dictionary maps each node
|
|
``v`` in the graph to the similarity between ``source`` and
|
|
``v``.
|
|
|
|
target : node
|
|
If both ``source`` and ``target`` are specified, the similarity
|
|
value between ``source`` and ``target`` is returned. If
|
|
``target`` is specified but ``source`` is not, this argument is
|
|
ignored.
|
|
|
|
importance_factor : float
|
|
The relative importance of indirect neighbors with respect to
|
|
direct neighbors.
|
|
|
|
max_iterations : integer
|
|
Maximum number of iterations.
|
|
|
|
tolerance : float
|
|
Error tolerance used to check convergence. When an iteration of
|
|
the algorithm finds that no similarity value changes more than
|
|
this amount, the algorithm halts.
|
|
|
|
Returns
|
|
-------
|
|
similarity : numpy matrix, numpy array or float
|
|
If ``source`` and ``target`` are both ``None``, this returns a
|
|
Matrix containing SimRank scores of the nodes.
|
|
|
|
If ``source`` is not ``None`` but ``target`` is, this returns an
|
|
Array containing SimRank scores of ``source`` and that
|
|
node.
|
|
|
|
If neither ``source`` nor ``target`` is ``None``, this returns
|
|
the similarity value for the given pair of nodes.
|
|
|
|
Examples
|
|
--------
|
|
>>> from numpy import array
|
|
>>> G = nx.cycle_graph(4)
|
|
>>> sim = nx.simrank_similarity_numpy(G)
|
|
|
|
References
|
|
----------
|
|
.. [1] G. Jeh and J. Widom.
|
|
"SimRank: a measure of structural-context similarity",
|
|
In KDD'02: Proceedings of the Eighth ACM SIGKDD
|
|
International Conference on Knowledge Discovery and Data Mining,
|
|
pp. 538--543. ACM Press, 2002.
|
|
"""
|
|
# This algorithm follows roughly
|
|
#
|
|
# S = max{C * (A.T * S * A), I}
|
|
#
|
|
# where C is the importance factor, A is the column normalized
|
|
# adjacency matrix, and I is the identity matrix.
|
|
import numpy as np
|
|
|
|
adjacency_matrix = nx.to_numpy_array(G)
|
|
|
|
# column-normalize the ``adjacency_matrix``
|
|
adjacency_matrix /= adjacency_matrix.sum(axis=0)
|
|
|
|
newsim = np.eye(adjacency_matrix.shape[0], dtype=np.float64)
|
|
for _ in range(max_iterations):
|
|
prevsim = np.copy(newsim)
|
|
newsim = importance_factor * np.matmul(
|
|
np.matmul(adjacency_matrix.T, prevsim), adjacency_matrix
|
|
)
|
|
np.fill_diagonal(newsim, 1.0)
|
|
|
|
if np.allclose(prevsim, newsim, atol=tolerance):
|
|
break
|
|
|
|
if source is not None and target is not None:
|
|
return newsim[source, target]
|
|
if source is not None:
|
|
return newsim[source]
|
|
return newsim
|
|
|