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

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 $1<\lambda <2$.

References in corpus (22)

• Finding and evaluating community structure in networks
• Scale-free networks are rare
• On quantum Renyi entropies: a new generalization and some properties
• The Physics of Communicability in Complex Networks
• Spectral entropies as information-theoretic tools for complex network comparison
• Vertex Intrinsic Fitness: How to Produce Arbitrary Scale-Free Networks
• Communicability Betweenness in Complex Networks
• Complex Networks from Classical to Quantum
• On the limiting behavior of parameter-dependent network centrality measures
• An information-theoretic, all-scales approach to comparing networks
• Validity of the second law in nonextensive quantum thermodynamics
• Nonadditive generalization of the quantum Kullback-Leibler divergence for measuring the degree of purification
• Wikipedia information flow analysis reveals the scale-free architecture of the Semantic Space
• Temperature of nonextensive system: Tsallis entropy as Clausius entropy
• Generalized entropy arising from a distribution of q-indices
• A Parameterized Centrality Metric for Network Analysis
• Using text analysis to quantify the similarity and evolution of scientific disciplines
• Fluctuation-dissipation relations for complex networks
• A control analysis perspective on Katz centrality
• Generalized Entropies and the Similarity of Texts
• On some properties of Tsallis hypoentropies and hypodivergences
• On the putative essential discreteness of q-generalized entropies