Stationary distribution and cover time of sparse directed configuration models

Abstract

AbstractWe consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent $$\gamma \ge 1$$ γ ≥ 1 of the logarithm and show that the cover time grows as $$n\log ^{\gamma }(n)$$ n log γ ( n ) , where n is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution $$\pi $$ π is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of $$\pi $$ π have the form $$\frac{1}{n}\log ^{1-\gamma }(n)$$ 1 n log 1 - γ ( n ) , while the maximal values of $$\pi $$ π behave as $$\frac{1}{n}\log ^{1-\kappa }(n)$$ 1 n log 1 - κ ( n ) for some other exponent $$\kappa \in [0,1]$$ κ ∈ [ 0 , 1 ] . In passing, we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices.