Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks

TL;DR

This paper analytically investigates how degree heterogeneity affects average path length and random walks in deterministic, negatively correlated scale-free networks, revealing that APL scales logarithmically with network size regardless of heterogeneity.

Contribution

It demonstrates that in recursive scale-free trees, the average path length is unaffected by degree exponent, contrasting with uncorrelated networks, and explores how random walk properties depend on heterogeneity.

Findings

APL scales as log N, independent of gamma

Random walk efficiency depends on degree heterogeneity

Contrasts with uncorrelated scale-free networks

Abstract

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution $P(k)\sim k^{-\gamma}$, where the degree exponent $\gamma$ describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various $\gamma \in (2,1+\frac{\ln 3}{\ln 2}]$, with an aim to explore the impacts of structure heterogeneity on APL and RWs. We show that the degree exponent $\gamma$ has no effect on APL $d$ of RSFTs: In the full range of $\gamma$, $d$ behaves as a logarithmic scaling with the number of network nodes $N$ (i.e. $d \sim \ln N$), which is in sharp contrast to the well-known double logarithmic scaling ($d \sim \ln \ln N$) previously obtained for uncorrelated scale-free networks with $2 \leq \gamma <3$. In addition, we present that some scaling efficiency exponents of random walks are reliant on degree exponent $\gamma$.