Abstract Given a sequence of graphs Gₙ and a fixed graph H, denote by T (H, Gₙ) the number of monochromatic copies of the graph H in a uniformly random c -coloring of the vertices of Gₙ. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs H₁, H₂, , Hd we derive the joint distribution of (T (H₁, Gₙ^ (1) ), T (H₂, Gₙ^ (2) ), , T (Hd, Gₙ^ (d) ) ), where Gₙ = (Gₙ^ (1), Gₙ^ (2), , Gₙ^ (d) ) is a collection of dense graphs on the same vertex set converging in the multiplex cut-metric. The limiting distribution is the sum of two independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.
Daros et al. (Thu,) studied this question.