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

Zhang, Zhongzhi; Zhang, Yichao; Zhou, Shuigeng; Yin, Ming; Guan, Jihong

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

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Description

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution $P(k)\sim k^{-γ}$, where the degree exponent $γ$ 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 $γ\in (2,1+\frac{\ln 3}{\ln 2}]$, with an aim to explore the impacts of structure heterogeneity on APL and RWs. We show that the degree exponent $γ$ has no effect on APL $d$ of RSFTs: In the full range of $γ$, $d$ 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 $2 \leq γ<3$. In addition, we present that some scaling efficiency exponents of random walks are reliant on degree exponent $γ$.
The definitive verion published in Journal of Mathematical Physics