Extensivity in infinitely large multiplex networks

Abstract

In this paper, we investigate the relationship between the coupling strengths and the extensive behaviour of the sum of the positive Lyapunov exponents of multiplex networks formed by coupled dynamical units. Considering networks where the dynamics of the nodes is given by the shift map, we do not only demonstrate which are the relevant parameters leading to extensivity, but also provide exact formulas how they are related. A distinct result was to show that it is always possible to construct infinitely large extensive networks by attaching, with rescaled inter-connections, infinitely many smaller networks. These smaller networks are effectively the building blocks of the large network. This is because these building blocks can have arbitrary topology and the strength of connections among nodes only depends on the block size, and not on the size of the whole network.