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 P(k)similar to 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 is an element of(2,1+ln 3/ln 2], with an aim to explore the impacts of structure heterogeneity on the APL and RWs. We show that the degree exponent gamma has no effect on the 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 similar to ln N), which is in sharp contrast to the well-known double logarithmic scaling (d similar to ln ln N) previously obtained for uncorrelated scale-free networks with 2 <=gamma < 3. In addition, we present that some scaling efficiency exponents of random walks are reliant on the degree exponent gamma.