Entropy of spatial network with applications to non-extensive statistical mechanics

TL;DR

This paper introduces a novel method using Tsallis entropy to analyze the complexity of random geometric networks, with applications to wireless networks and insights into their maximal complexity.

Contribution

It develops a new approach for calculating Tsallis entropy in spatial networks and identifies connection functions that maximize entropy under constraints.

Findings

Wireless networks modeled are nearly maximally complex.

The proposed method effectively identifies optimal connection functions.

Tsallis entropy provides a useful measure for network complexity analysis.

Abstract

A new method is proposed for analyzing complexity and studying the information in random geometric networks using Tsallis entropy tool. Tsallis entropy of the ensemble of random geometric networks is calculated based on the components of the random connection model on the point process which is obtained by connecting the points with a probability that depends on their relative positions (10.1016/j.indag.2022.05.002, 2022). According to information theory and conditional discussion, the bounds for Shannon and Tsallis entropies of the ensemble of this random graph are presented. Using this function and Lagrange's formula, the connection function that provides the maximum Tsallis entropy based on general constraints is obtained. Then, a simulation-based example is presented to clarify the application of the proposed method in the study of ad hoc wireless networks. By observing the obtained results, it can be stated that the wireless networks that adhere to the model studied here are almost maximally complex. Also, Tsallis conditional entropy maximizing function is compared with other connection functions using numerical calculations and the optimal value for the maximization of conditional entropies is obtained.