Given a sequence of graphs ${G_n}_{n \geq 1}$ and fixed graph $H$, denote by $T(H, G_n)$ the number of monochromatic copies of the graph $H$ in $G_n$ in a uniformly random $c_n$-coloring of the vertices of $G_n$. In this dissertation we study the joint distribution of monochromatic subgraphs for dense multiplex networks, that is, networks with multiple layers. Specifically, given a finite collection of graphs $H_1,\ldots,H_d$, we derive the asymptotic joint distribution of $\bm T_n:= (T(H_1, G_n^{(1)}),\ldots, T(H_d, G_n^{(d)}))$, where $\bm G_n=(G_n^{(1)},\ldots, G_n^{(d)})$ is a collection of graphs on the same vertex set converging in the joint cut-metric. Under a notion of joint convergence of $\bm G_n$ in the cut metric, we show that when the number of colors $c_n=c$ is fixed, then the limiting distribution of $\bm T_n$ is the sum of two independent components, one of which is a multivariate Gaussian and the other is a sum of bivariate stochastic integrals. On the other hand, when the number of colors $c_n \rightarrow \infty$ (such that $\mathbb E[\bm T_n] \rightarrow \infty$), then the asymptotic distribution of $\bm T_n$ is a multivariate normal. This generalizes the classical birthday problem, which involves understanding the asymptotics of $T(K_s,K_n)$, the number of monochromatic $s$-cliques in a complete graph $K_n$ ($s$-matching birthdays among a group of $n$ friends), to general monochromatic subgraphs in multiplex networks. This also extends previous results on the marginal convergence of $T(H, G_n)$ and is useful in establishing the joint convergence of various subgraph counting statistics that arise from random vertex coloring of graphs. Several applications and examples are discussed.