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We consider sparse digraphs generated by the configuration model with givenin-degree and out-degree sequences. We establish that with high probability thecover time is linear up to a poly-logarithmic correction. For a large class ofdegree sequences we determine the exponent $\gamma \ge 1$ of the logarithm andshow that the cover time grows as $n\log^{\gamma}(n)$, where $n$ is the numberof vertices. The results are obtained by analysing the extremal values of thestationary distribution of the digraph. In particular, we show that thestationary distribution $\pi$ is uniform up to a poly-logarithmic factor, andthat for a large class of degree sequences the minimal values of $\pi$ have theform $\frac1n\log ^{1-\gamma}(n)$, while the maximal values of $\pi$ behave as$\frac1n\log ^{1-\kappa}(n)$ for some other exponent $\kappa\in[0,1]$. Inpassing, we prove tight bounds on the diameter of the digraphs and show thatthe latter coincides with the typical distance between two vertices.

Stationary distribution and cover time of sparse directed configuration models