Given a digraph G, a parameter α ∈ (0, 1) and a distribution λ over the vertices of G, the generalised PageRank surf on G with parameters α and λ is the Markov chain on the vertices of G such that at each step with probability α the state is updated with an independent sample from λ, while with probability 1 − α the state is moved to a uniformly random vertex in the out-neighbourhood of the current state. The stationary distribution is the so-called PageRank vector. We analyse convergence to stationarity of this Markov chain when G is a large sparse random digraph with given degree sequences, in the limit of vanishing parameter α. We identify three scenarios: when α is much smaller than the inverse of the mixing time of G the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behaviour; when α is much larger than the inverse of the mixing time of G on the contrary one has pure exponential decay with rate α; when α is comparable to the inverse of the mixing time of G there is a mixed behaviour interpolating between cutoff and exponential decay. This trichotomy is shown to hold uniformly in the starting point and for a large class of distributions λ, including widespread as well as strongly localized measures. Our results are crucially based on recent progress in the analysis of mixing times for the simple random walk on sparse random digraphs [6,7].

Consider a graph where each of the n nodes is in one of two possible states. Herein, we analyze the synchronous k-majority dynamics, where nodes sample k neighbors uniformly at random with replacement and adopt the majority state among the nodes in the sample (potential ties are broken uniformly at random). This class of dynamics generalizes other well-known dynamics, e.g., voter and 3-majority, which have been studied in the literature as distributed algorithms for consensus.We consider a biased communication model : whenever nodes sample a neighbor they see state σ with some probability p, regardless of the state of the sampled node, and its true state with probability 1 − p. Differently from previous works where specific graph topologies-typically characterized by good expansion properties-are considered, our analysis only requires the graphs to be sufficiently dense, i.e., to have minimum degree ω(log n), without any further topological assumption.In this setting we prove two phase transition phenomena, both occurring with high probability, depending on the bias p and on the initial unbalance toward state σ. More in detail, we prove that for every k ≥ 3 there exists a p k such that if p > p k the process reaches in O(1) rounds a σ-almost-consensus, i.e., a configuration where a fraction 1 − γ of the volume is in state σ, for any arbitrarily-small positive constant γ. On the other hand, if p < p k , we look at random initial configurations in which every node is in state σ with probability 1−q independently of the others. We prove that there exists a constant q p,k such that if q < q p,k then a σ-almost-consensus is still reached in O(1) rounds, while, if q > q p,k , the process spends n ω(1) rounds in a metastable phase where the fraction of volume in state σ is around a constant value depending only on p and k.Finally we also investigate, in such a biased setting, the differences and similarities between k-majority and other closely-related dynamics, namely voter and deterministic majority.

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.

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.

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