Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

This work presents exact expressions for size distributions of weak and multilayer connected components in two generalizations of the configuration model: networks with directed edges and multiplex networks with an arbitrary number of layers. The expressions are computable in a polynomial time and, under some restrictions, are tractable from the asymptotic theory point of view. If first partial moments of the degree distribution are finite, the size distribution for two-layer connected components in multiplex networks exhibits an exponent -3/2 in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents -1/2 and -3/2.