Stationary distribution and cover time of sparse directed configuration models

Abstract

We 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 polylogarithmic correction. For a large class of degree sequences we determine the exponent γ ≥ 1 of the logarithm and show that the cover time grows as 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 π is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of π have the form 1 n log 1−γ (n), while the maximal values of π behave as 1 n log 1−κ (n) for some other exponent 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.

The diameter of the directed configuration model

“…Recently, there has been some progress on the diameter of the supercritical directed configuration model. Caputo and Quattropani [11] determined the asymptotic behaviour of the diameter of…”

“…One of the motivations to study the diameter of directed random graphs is its close connection to the properties of a random walk on it. For instance, in [11] the authors used their results on neighbourhood expansion to determine the extremal values for the stationary distribution of a random walk in G n , with implications on its cover time. Typical values of the stationary distribution were previously obtained by Bordenave, Caputo and Salez [8,9] as an intermediate step to bound the mixing time of the random walk.…”

“…We make no assumption on the rate of convergence in Condition 1.1, thus, we cannot determine the second order term of diam( G n ). Under explicit convergence rate assumptions, it might be possible to find the second order term, as in [11,24].…”

Mixing time trichotomy in regenerating dynamic digraphs

“…any path of length s has weight between ∆ −s and 2δ −s . We refer to [12,Lemma 3.1] for a proof of this fact. Since ∆ −s and 2δ −s are e −o(t) in this case, the result follows again by the case s = 0.…”

“…For an explicit construction, we can generate recursively the walks and the environment, letting the trajectories reveal the configuration η, the -th trajectory living in the environment discovered by the previous − 1 trajectories; see [12,Lemma 3.11] for more details. Therefore, it is sufficient to show that it is possible to find a constant C > 0 such that for every sufficiently large n…”