Connecting complex networks to nonadditive entropies

R M De Oliveira,L R Da Silva,Constantino Tsallis,Samuraí Brito

doi:10.1038/s41598-020-80939-1

Abstract

Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space–time entanglement. Its generalization based on nonadditive q-entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a d-dimensional geographically located network with weighted links and exhibit its ‘energy’ distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann–Gibbs exponential factor is generically substituted by its q-generalisation, and is recovered in the q=1 limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.

Highlights

Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems
When the system constituents have a generically nonlocal space–time entanglement, this theory does not apply. Such is the case already pointed in 1902 by Gibbs himself, namely when the standard partition function diverges, e.g., gravitation. It is in this context that it was proposed in 1 9881 the generalisation—hereafter referred to as nonextensive statistical mechanics—of the theory based on nonadditive entropies, namely
In parallel with the above, the study of complex networks has been intensified around the w orld[3,4,5,6,7]

Summary

Introduction

Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems. When the system constituents have a generically nonlocal space–time entanglement, this theory does not apply Such is the case already pointed in 1902 by Gibbs himself, namely when the standard partition function diverges, e.g., gravitation. The brain is formed by neurones communicating through synapses All these completely different systems can be translated onto a simple set of nodes (or sites) and edges (or links) obeying some connection rule, and the tools of network science can be successfully used to study them. Typical applications of this area can be found in classical and quantum internet8,9, medicine10,11, neuroscience[12], and sociology[13,14]. It was recently argued that most real networks are not pure scale-free[15], paving the way for new possibilities to describe them

Methods

Results

Conclusion