Characterising heavy-tailed networks using q-generalised entropy and q-adjacency kernels

TL;DR

This paper introduces a novel method for characterising heavy-tailed networks using q-generalised entropy and kernels, enabling robust comparison and thermodynamic-like analysis of complex networks with inverse power law degree distributions.

Contribution

It presents a new approach employing q-generalised adjacency kernels and divergence measures for analyzing and comparing heavy-tailed networks, with a thermodynamic interpretation.

Findings

The method effectively distinguishes between heavy-tailed networks.

Q-generalised divergence provides a robust comparison metric.

Thermodynamic-like description offers new insights into network structure.

Abstract

Heavy-tailed networks, which have degree distributions characterised by slower than exponentially bounded tails, are common in many different situations. Some interesting cases, where heavy tails are characterised by inverse powers $\lambda$ in the range $1<\lambda<2,$ arise for associative knowledge networks, and semantic and linguistic networks. In these cases, the differences between the networks are often delicate, calling for robust methods to characterise the differences. Here, we introduce a method for comparing networks using a density matrix based on q-generalised adjacency matrix kernels. It is shown that comparison of networks can then be performed using the q-generalised Kullback-Leibler divergence. In addition, the q-generalised divergence can be interpreted as a q-generalised free energy, which enables the thermodynamic-like macroscopic description of the heavy-tailed networks. The viability of the q-generalised adjacency kernels and the thermodynamic-like description in characterisation of complex networks is demonstrated using a simulated set of networks, which are modular and heavy-tailed with a degree distribution of inverse power law in the range $1<\lambda<2$.