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

TL;DR

This work develops a Tsallis-entropy framework to quantify the complexity of spatial network ensembles formed as soft random geometric graphs (SRGG) on a point process, where edges are added with distance-dependent probability $g(|x_i-x_j|)$. It derives bounds for Tsallis and Shannon entropies, and uses an Euler–Lagrange approach to obtain the maximum-entropy connection function $\tilde{g}(x)$ under general constraints, highlighting the role of pair-distance densities $f(x)$ and constraint functions $\theta_\ell(x)$. The authors validate the theory through simulation of ad hoc wireless networks with Rayleigh fading connections, comparing $\tilde{g}$ to standard connection functions (e.g., Rayleigh, MIMO) and showing close agreement; they also analyze conditional entropies and optimal parameters $x_0$ and $\eta$ that maximize entropy measures. Overall, the paper provides a principled method to assess and engineer network complexity in spatial settings, with direct implications for the design and analysis of wireless networks.

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.

Figures (5)

• Figure 1: $a)$ Random graph, $b)$ Random geometric graph and $c)$ Soft random geometric graph with $n=200$ nodes.
• Figure 2: $a)$ Shannon conditional entropy, $H^S(\mathcal{G}|\mathcal{X})/c_n$, $b)$ Tsallis conditional entropy, $H^T(\mathcal{G}|\mathcal{X})/c_n$ and $c)$ the upper bound of Tsallis conditional entropy with respect to $x_0\in (0,2]$ and $q\in (1,4]$.
• Figure 3: The pair connection probability defined in Relation \ref{['CF']} (diamond), the maximum Tsallis entropy connection probability (square) and MIMO function (triangle)
• Figure 4: The values $x_0$ of the function maximizer $H^S(\mathcal{G}|\mathcal{X})$ (dotdash) and the function maximizer $H^T(\mathcal{G}|\mathcal{X})$ (dotted) with $q=1.1$ in terms of $\eta$.
• Figure 5: $H_2(x)/2x(1-x)$ for $x \in (0,1)$(dashed line) and $H_2^T(x)$ for $q=0.2$ (dotted line), $q= 0.999$ (dotdash line) and $q=2$ (longdash line).

Theorems & Definitions (7)

• Definition 1
• Definition 2
• Theorem 3
• Remark 4
• Remark 5
• Remark 6
• Theorem 7