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

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution [Formula: see text] , where the degree exponent [Formula: see text] 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 [Formula: see text] , with an aim to explore the impacts of structure heterogeneity on the APL and RWs. We show that the degree exponent [Formula: see text] has no effect on the APL [Formula: see text] of RSFTs: In the full range of [Formula: see text] , [Formula: see text] behaves as a logarithmic scaling with the number of network nodes [Formula: see text] (i.e., [Formula: see text]), which is in sharp contrast to the well-known double logarithmic scaling [Formula: see text] previously obtained for uncorrelated scale-free networks with [Formula: see text]. In addition, we present that some scaling efficiency exponents of random walks are reliant on the degree exponent [Formula: see text].