Thresholds and Fluctuations of Submultiplexes in Random Multiplex Networks

In a multiplex network a common set of nodes is connected through different types of interactions, each represented as a separate graph (layer) within the network. In this paper, we study the asymptotic properties of submultiplexes, the counterparts of subgraphs (motifs) in single-layer networks, in the correlated Erd\H{o}s-R\'{e}nyi multiplex model. This is a random multiplex model with two layers, where the graphs in each layer marginally follow the classical (single-layer) Erd\H{o}s-R\'{e}nyi model, while the edges across layers are correlated. We derive the precise threshold condition for the emergence of a fixed submultiplex $\boldsymbol{H}$ in a random multiplex sampled from the correlated Erd\H{o}s-R\'{e}nyi model. Specifically, we show that the satisfiability region, the regime where the random multiplex contains infinitely many copies of $\boldsymbol{H}$, forms a polyhedral subset of $\mathbb{R}^3$. Furthermore, within this region the count of $\boldsymbol{H}$ is asymptotically normal, with an explicit convergence rate in the Wasserstein distance. We also establish various Poisson approximation results for the count of $\boldsymbol{H}$ on the boundary of the threshold, which depends on a notion of balance of submultiplexes. Collectively, these results provide an asymptotic theory for small submultiplexes in the correlated multiplex model, analogous to the classical theory of small subgraphs in random graphs.