Various real-life networks exhibit degree correlations and heterogeneous structure, with
the latter being characterized by power-law degree distribution P(k)∼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∊(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 γ has no effect on the 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∼ln N),
which is in sharp contrast to the well-known double logarithmic scaling
(d∼ln ln N)
previously obtained for uncorrelated scale-free networks with 2≤γ<3.
In addition, we present that some scaling efficiency exponents of random walks are reliant
on the degree exponent γ.

]]>