Maintaining extensivity in evolutionary multiplex networks

In this paper, we explore the role of network topology on maintaining the extensive property of entropy. We study analytically and numerically how the topology contributes to maintaining extensivity of entropy in multiplex networks, i.e. networks of subnetworks (layers), by means of the sum of the positive Lyapunov exponents, $H_{KS}$, a quantity related to entropy. We show that extensivity relies not only on the interplay between the coupling strengths of the dynamics associated to the intra (short-range) and inter (long-range) interactions, but also on the sum of the intra-degrees of the nodes of the layers. For the analytically treated networks of size $N$, among several other results, we show that if the sum of the intra-degrees (and the sum of inter-degrees) scales as $N^{\theta+1},\;\theta>0$, extensivity can be maintained if the intra-coupling (and the inter-coupling) strength scales as $N^{-\theta}$, when evolution is driven by the maximisation of $H_{KS}$. We then verify our analytical results by performing numerical simulations in multiplex networks formed by electrically and chemically coupled neurons.