Dynamically Weighted Clique Evolution Model in Clique Networks

On weighted clique graphs

Let K(G) be the clique graph of a graph G. A m-weighting of K(G) consists on giving to each m-size subset of its vertices a weight equal to the size of the intersection of the m corresponding cliques of G. The 2weighted clique graph was previously considered by McKee. In this work we obtain a characterization of weighted clique graphs similar to Roberts and Spencer’s characterization for clique...

The maximum weighted clique problem and Hopfield networks

Clique percolation in random networks.

The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdos-Rényi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p(c) (k) = [...

Evolution towards the Maximum Clique

As is well known, the problem of finding a maximum clique in a graph is NP-hard. Nevertheless, NP-hard problems may have easy instances. This paper proposes a new, global optimization algorithm which tries to exploit favourable data constellations, focussing on the continuous problem formulation: maximize a quadratic form over the standard simplex. Some general connections of the latter problem...

Clique partitions and clique coverings

Several new tools are presented for determining the number of cliques needed to (edge-)partition a graph . For a graph on n vertices, the clique partition number can grow cn z times as fast as the clique covering number, where c is at least 1/64. If in a clique on n vertices, the edges between en° vertices are deleted, Z--a < 1, then the number of cliques needed to partition what is left is asy...