Growing Multiplex Networks with Arbitrary Number of Layers

TL;DR

This paper derives closed-form solutions for the joint degree distribution of heterogeneously growing multiplex networks with any number of layers, extending previous results limited to two layers and steady state.

Contribution

It provides the first analytical solutions for the joint degree distribution in multiplex networks with arbitrary layers and heterogeneous growth, including time-dependent analysis.

Findings

Closed-form steady state joint degree distribution for heterogeneously growing multiplex networks.

Time-dependent joint degree distribution for uniform growth mechanisms.

Initial network conditions influence transient dynamics and future structure.

Abstract

This paper focuses on the problem of growing multiplex networks. Currently, the results on the joint degree distribution of growing multiplex networks present in the literature pertain to the case of two layers, and are confined to the special case of homogeneous growth, and are limited to the state state (that is, the limit of infinite size). In the present paper, we obtain closed-form solutions for the joint degree distribution of heterogeneously growing multiplex networks with arbitrary number of layers in the steady state. Heterogeneous growth means that each incoming node establishes different numbers of links in different layers. We consider both uniform and preferential growth. We then extend the analysis of the uniform growth mechanism to arbitrary times. We obtain a closed-form solution for the time-dependent joint degree distribution of a growing multiplex network with arbitrary initial conditions. Throughout, theoretical findings are corroborated with Monte Carlo simulations. The results shed light on the effects of the initial network on the transient dynamics of growing multiplex networks, and takes a step towards characterizing the temporal variations of the connectivity of growing multiplex networks, as well as predicting their future structural properties.