Antonopoulos Chris G, Baptista Murilo S
Department of Mathematical Sciences, University of Essex, Wivenhoe Park, United Kingdom.
Institute of Complex Sciences and Mathematical Biology, University of Aberdeen, SUPA, Aberdeen, United Kingdom.
PLoS One. 2017 Apr 12;12(4):e0175389. doi: 10.1371/journal.pone.0175389. eCollection 2017.
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, HKS, 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 Nθ+1, θ > 0, extensivity can be maintained if the intra-coupling (and the inter-coupling) strength scales as N-θ, when evolution is driven by the maximisation of HKS. We then verify our analytical results by performing numerical simulations in multiplex networks formed by electrically and chemically coupled neurons.
在本文中，我们探讨了网络拓扑结构在维持熵的广延性质方面的作用。我们通过正李雅普诺夫指数之和、HKS（一个与熵相关的量），从解析和数值两方面研究了拓扑结构如何有助于维持多重网络（即子网（层）网络）中熵的广延性。我们表明，广延性不仅依赖于与内部（短程）和外部（长程）相互作用相关的动力学耦合强度之间的相互作用，还依赖于各层节点的内部度之和。对于解析处理的大小为N的网络，在其他几个结果中，我们表明，如果内部度之和（以及外部度之和）按Nθ + 1缩放，θ > 0，当演化由HKS最大化驱动时，如果内部耦合（以及外部耦合）强度按N-θ缩放，则可以维持广延性。然后，我们通过在由电耦合和化学耦合神经元组成的多重网络中进行数值模拟来验证我们的解析结果。
