## Mercurial > repos > shellac > sam_consensus_v3

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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |

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