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 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.

Minimum stationary values of sparse random directed graphs

“…Additionally, our proof implies that whp the hitting and the cover time are both n 1+C+o(1) . Our results complement those of Caputo and Quattropani [11] who showed that if the minimum in-degree is at least 2, stationary values have logarithmic fluctuations around n −1 .…”

“…Typical distances in G n were studied by van der Hoorn and Olvera-Cravioto [26]. Caputo and Quattropani [11] showed that the diameter of G n is asymptotically equal to the typical distance, provided that δ ± ≥ 2 and ∆ ± = O(1). In our previous work [9], we showed that the diameter has different behaviour if no constraints on the minimum degree are imposed.…”

“…Caputo and Quattropani [11] studied the extremal stationary values in G n provided that δ ± ≥ 2 and ∆ ± = O(1). Let π min and π max be the smallest and largest positive values of π, respectively.…”

“…By averaging, π max ≥ 1/n. Moreover, one can check that the proof of the second inequality in (1.2) (see [11,Section 3.5]) does not use any condition on δ − . Therefore, π max = n −1+o (1) holds in the setting of Theorem 1.1.…”

“…To prove a lower bound it will suffice to lower bound the probability of reaching a particular head f ∈ E − 0 , uniformly for all starting points e ∈ E − (see Lemma 4.4 below). We use the ideas introduced in [5,6,11] to capture the weight (defined below) of typical trajectories departing from e. Our main contribution is to control the total weight of the trajectories landing at f . Define the out-entropy H + and the entropic time τ ent as…”