160 lines
4.8 KiB
Python
160 lines
4.8 KiB
Python
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"""Functions for computing large cliques."""
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import networkx as nx
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from networkx.utils import not_implemented_for
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from networkx.algorithms.approximation import ramsey
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__all__ = ["clique_removal", "max_clique", "large_clique_size"]
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def max_clique(G):
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r"""Find the Maximum Clique
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Finds the $O(|V|/(log|V|)^2)$ apx of maximum clique/independent set
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in the worst case.
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Parameters
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----------
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G : NetworkX graph
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Undirected graph
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Returns
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-------
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clique : set
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The apx-maximum clique of the graph
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Notes
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------
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A clique in an undirected graph G = (V, E) is a subset of the vertex set
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`C \subseteq V` such that for every two vertices in C there exists an edge
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connecting the two. This is equivalent to saying that the subgraph
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induced by C is complete (in some cases, the term clique may also refer
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to the subgraph).
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A maximum clique is a clique of the largest possible size in a given graph.
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The clique number `\omega(G)` of a graph G is the number of
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vertices in a maximum clique in G. The intersection number of
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G is the smallest number of cliques that together cover all edges of G.
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https://en.wikipedia.org/wiki/Maximum_clique
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References
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----------
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.. [1] Boppana, R., & Halldórsson, M. M. (1992).
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Approximating maximum independent sets by excluding subgraphs.
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BIT Numerical Mathematics, 32(2), 180–196. Springer.
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doi:10.1007/BF01994876
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"""
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if G is None:
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raise ValueError("Expected NetworkX graph!")
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# finding the maximum clique in a graph is equivalent to finding
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# the independent set in the complementary graph
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cgraph = nx.complement(G)
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iset, _ = clique_removal(cgraph)
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return iset
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def clique_removal(G):
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r""" Repeatedly remove cliques from the graph.
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Results in a $O(|V|/(\log |V|)^2)$ approximation of maximum clique
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and independent set. Returns the largest independent set found, along
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with found maximal cliques.
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Parameters
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----------
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G : NetworkX graph
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Undirected graph
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Returns
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-------
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max_ind_cliques : (set, list) tuple
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2-tuple of Maximal Independent Set and list of maximal cliques (sets).
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References
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----------
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.. [1] Boppana, R., & Halldórsson, M. M. (1992).
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Approximating maximum independent sets by excluding subgraphs.
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BIT Numerical Mathematics, 32(2), 180–196. Springer.
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"""
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graph = G.copy()
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c_i, i_i = ramsey.ramsey_R2(graph)
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cliques = [c_i]
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isets = [i_i]
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while graph:
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graph.remove_nodes_from(c_i)
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c_i, i_i = ramsey.ramsey_R2(graph)
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if c_i:
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cliques.append(c_i)
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if i_i:
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isets.append(i_i)
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# Determine the largest independent set as measured by cardinality.
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maxiset = max(isets, key=len)
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return maxiset, cliques
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@not_implemented_for("directed")
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@not_implemented_for("multigraph")
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def large_clique_size(G):
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"""Find the size of a large clique in a graph.
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A *clique* is a subset of nodes in which each pair of nodes is
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adjacent. This function is a heuristic for finding the size of a
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large clique in the graph.
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Parameters
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----------
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G : NetworkX graph
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Returns
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-------
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int
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The size of a large clique in the graph.
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Notes
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-----
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This implementation is from [1]_. Its worst case time complexity is
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:math:`O(n d^2)`, where *n* is the number of nodes in the graph and
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*d* is the maximum degree.
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This function is a heuristic, which means it may work well in
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practice, but there is no rigorous mathematical guarantee on the
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ratio between the returned number and the actual largest clique size
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in the graph.
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References
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----------
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.. [1] Pattabiraman, Bharath, et al.
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"Fast Algorithms for the Maximum Clique Problem on Massive Graphs
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with Applications to Overlapping Community Detection."
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*Internet Mathematics* 11.4-5 (2015): 421--448.
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<https://doi.org/10.1080/15427951.2014.986778>
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See also
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--------
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:func:`networkx.algorithms.approximation.clique.max_clique`
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A function that returns an approximate maximum clique with a
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guarantee on the approximation ratio.
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:mod:`networkx.algorithms.clique`
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Functions for finding the exact maximum clique in a graph.
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"""
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degrees = G.degree
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def _clique_heuristic(G, U, size, best_size):
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if not U:
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return max(best_size, size)
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u = max(U, key=degrees)
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U.remove(u)
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N_prime = {v for v in G[u] if degrees[v] >= best_size}
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return _clique_heuristic(G, U & N_prime, size + 1, best_size)
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best_size = 0
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nodes = (u for u in G if degrees[u] >= best_size)
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for u in nodes:
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neighbors = {v for v in G[u] if degrees[v] >= best_size}
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best_size = _clique_heuristic(G, neighbors, 1, best_size)
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return best_size
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