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

TL;DR

This paper derives exact, computationally efficient formulas for the size distributions of connected components in directed and multiplex networks with arbitrary degree distributions, revealing critical exponents in their size distributions.

Contribution

It provides the first exact expressions for component size distributions in directed and multiplex networks with arbitrary degrees, including asymptotic behavior analysis.

Findings

Size distribution for two-layer multiplex components follows a -3/2 exponent at criticality.

Directed networks' weak component size distribution exhibits -1/2 and -3/2 exponents.

Formulas are computable in polynomial time and are asymptotically tractable.

Abstract

This work presents exact expressions for size distributions of weak/multilayer connected components in two generalisations of the configuration model: networks with directed edges and multiplex networks with 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 exponent $-\frac{3}{2}$ in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents, $-\frac{1}{2}$ and $-\frac{3}{2}$.