Griffiths phase in clique percolation in random geometric graphs

Abstract

In this study, we discuss the clique percolation in the ensembles of random geometric graphs with different kernels that quantify the geometrical constraints. For the sharp cut-off we find the wide Griffiths phase of extended criticality with the power-law behavior. One boundary of the Griffiths phase is the generalization of a percolation critical point for the ER ensemble when the percolation within the large but finite cluster emerges. The second boundary corresponds to the point in the parameter space when the percolation in the entire clustered system becomes available. For the power-law kernel, richer behavior with a clique-size-dependent boundary between the effective ER and geometric regimes has been identified. The Griffiths phase in this case exists as well. Finally, the pattern with the exponential kernel has been analyzed. We briefly discuss the possible applications of our findings.