Clique percolation in random graphs

Abstract

As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two k cliques means that they share at least l<k vertices. In this paper we develop a theoretical approach to study clique percolation in Erdős-Rényi graphs, which gives not only the exact solutions of the critical point, but also the corresponding order parameter. Based on this, we prove theoretically that the fraction ψ of cliques in the giant clique cluster always makes a continuous phase transition as the classical percolation. However, the fraction ϕ of vertices in the giant clique cluster for l>1 makes a step-function-like discontinuous phase transition in the thermodynamic limit and a continuous phase transition for l=1. More interesting, our analysis shows that at the critical point, the order parameter ϕ(c) for l>1 is neither 0 nor 1, but a constant depending on k and l. All these theoretical findings are in agreement with the simulation results, which give theoretical support and clarification for previous simulation studies of clique percolation.

Finite-size scaling of clique percolation on two-dimensional Moore lattices

“…It was found that the value of the critical exponent β depends on the definition of the order parameter [11,18]. There are two ways to define the order parameter.…”

“…Therefore, for a system with the bond density p approaching the critical point p c , the corresponding mapped site percolation has the same order ofp approaching towardp c , which infers that the critical exponents of the clique percolation are identical with those of the site percolation, thus belonging to the same universality class. It is known that clique network on an ER graph can also be mapped onto a new ER graph [17,18]. However, the clique percolation on an ER graph does not belong to the same universality class as the bond percolation, as the critical exponents of clique percolation on an ER graph also depend on the value of l. These differences between the ER graph and the Moore neighborhood can be attributed to the following two factors.…”

“…When l is larger than 1, the critical exponents are different, indicating new universal classes. Moreover, the values of the critical exponents, e.g., the most general exponent β, decay as the clique connection parameter l increases, which may lead to a transition from continuous (β > 0) to discontinuous (β = 0) phase transition at large l. Li et al [18] also proposed a theoretical analysis for the k-clique percolation and observed the dependence of the critical exponents on l. Therefore, the parameter l serves as a new crucial factor to the critical behavior of the system and induces new universality classes different from the normal bond percolation model. Clique percolation is thus a general model very succinct and appropriate for understanding various critical behaviors, universality classes, and discontinuous transition in percolation.…”

“…The above works associated with k-clique percolation are mainly on ER random graphs [11,15,[17][18][19], a typical infinite dimensional system which has solid analytical results. However, the conclusion obtained from such an infinite dimension system may not be applicable to low-dimensional systems [20][21][22][23].…”

Natural emergence of a core structure in networks via clique percolation

“…Their work has been broadened to less restrictive forms of percolation. For example, two k-cliques are considered to be adjacent if they share l vertices with 1 l k − 1 [16], [17].…”