The Size of the Giant Joint Component in a Binomial Random Double Graph

Abstract

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices that supports both a red and a blue spanning tree. We show that there are critical pairs of red and blue edge densities at which a giant joint component appears. In contrast to the standard binomial graph model, the phase transition is first order: the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point. We connect this phenomenon to the properties of a certain bicoloured branching process.

Higher-order percolation processes on multiplex hypergraphs

“…where the generating functions G 0,m (x) and G 1,m, (x) are defined in Eq. (49). By considering Poisson layers with generalized degree distribution given by Eq.…”

“…Indeed both interlayer degree correlations [15] and link overlap [41][42][43] have been shown to have a very significant effect on the critical properties of interdependent percolation. This field has been growing at a very fast pace and many results related to the robustness and resilience of multiplex networks have been obtained including the formulation of interdependent percolation in network of networks [44], weak percolation [45,46], optimal percolation [47,48], combinatorial optimization problems [49], K-core multiplex percolation [50] and percolation with redundant interdependencies [51].…”