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 $$ , where the degree exponent $$ 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 $$ , with an aim to explore the impacts of structure heterogeneity on the APL and RWs. We show that the degree exponent $$ has no effect on the APL $$ of RSFTs: In the full range of $$ , $$ behaves as a logarithmic scaling with the number of network nodes $$ (i.e., $$ ), which is in sharp contrast to the well-known double logarithmic scaling $$ previously obtained for uncorrelated scale-free networks with $$ . In addition, we present that some scaling efficiency exponents of random walks are reliant on the degree exponent $$ .