Cutoff at the “entropic time” for sparse Markov chains

Bordenave, Charles; Caputo, Pietro; Salez, Justin

doi:10.1007/s00440-018-0834-0

Cited by 39 publications

(70 citation statements)

References 27 publications

(67 reference statements)

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“…After the generation of the tree T (x) a total number κ of edges has been revealed, some of which may not belong to T (x). As in [10,Lemma 7], it is not difficult to see that when exploring the out-neighborhood of x in this way the random variable κ is deterministically bounded as (1) The first h x steps of p are contained in T (x), and satisfy…”

Section: A Concentration Results For Nice Pathsmentioning

confidence: 99%

“…Indeed, this fact is a consequence of [9,10], which established that for any ε > 0, with high probability min have been revealed and then sample the τ edges generating the neighborhood B − h y (y). Let K denote the number of collisions between T (x) and B − h y (y).…”

Section: Lemma 37mentioning

confidence: 99%

“…h x as in(3.32). Following[9, Section 6.2] and[10, Section 4.1], we introduce the rooted directed tree T (x), namely the subgraph of the h x -out-neighborhood of x obtained by the following process: initially all tails and heads are unmatched and T (x) is identified with its root, x; throughout the process, we let∂ + T (x) (resp. ∂ − T (x))denote the set of unmatched tails (resp.…”

mentioning

confidence: 99%

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Stationary distribution and cover time of sparse directed configuration models

Caputo

Quattropani

2020

Probab. Theory Relat. Fields

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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.

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Section: A Concentration Results For Nice Pathsmentioning

confidence: 99%

Section: Lemma 37mentioning

confidence: 99%

mentioning

confidence: 99%

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Stationary distribution and cover time of sparse directed configuration models

Caputo

Quattropani

2020

Probab. Theory Relat. Fields

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“…We remark that a line of work by Boldi et al [11,12] analyzed the limit of PPR as α n → 0 for a fixed graph G n ; in contrast, we fix a value of α n for each G n . Finally, we will further motivate our choice of α n in Section 7.4 via recent results on the mixing time of random walks on random graphs [16,17].…”

Section: :4mentioning

confidence: 99%

A Structural Result for Personalized PageRank and its Algorithmic Consequences

Vial

Subramanian

2019

Proc. ACM Meas. Anal. Comput. Syst.

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Many systems, such as the Internet, social networks, and the power grid, can be represented as graphs. When analyzing graphs, it is often useful to compute scores describing the relative importance or distance between nodes. One example is Personalized PageRank (PPR), which assigns to each node v a vector whose i-th entry describes the importance of the i-th node from the perspective of v. PPR has proven useful in many applications, such as recommending who users should follow on social networks (if this i-th entry is large, v may be interested in following the i-th user). Unfortunately, computing n PPR vectors exactly for a graph of n nodes has complexity O (n 3), which is infeasible for many graphs of interest. In this work, we devise a scheme to estimate all n PPR vectors with bounded l 1 error and complexity O (nc), wherec < 2 depends on the degrees of the graph at hand, the desired error tolerance, and a parameter that defines PPR. This improves upon existing methods, the best of which have complexity O (n 2 log n) in our setting. Our complexity guarantee holds with high probability, for certain choices of the PPR parameter, and for a certain class of random graphs (roughly speaking, the sparse directed configuration model with heavy-tailed in-degrees); our accuracy guarantee holds with probability 1 and for arbitrary graphs and PPR parameters. The complexity result arises as a consequence of our main (structural) result, which shows that the dimensionality of the set of PPR vectors scales sublinearly in n with high probability, for the same class of random graphs and for a notion of dimensionality similar to matrix rank. It is this coupling of the PPR vectors for the nodes on a common underlying graph that allows for estimating them faster. Hence, at a high level, our scheme is analogous to (but distinct from) low-rank matrix approximation. We also note that our scheme is similar to one that was proposed in [Jeh and Widom 2003] but lacked accuracy and complexity guarantees, so another contribution of our paper is to address this gap in the literature.

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“…A somewhat related question is the long time behavior of the Markov chain with transition matrix M . This question is addressed in [10]; see also [9] for related recent progress concerning sparse random directed graphs.…”

Section: )mentioning

confidence: 99%

Spectrum of large random Markov chains: Heavy-tailed weights on the oriented complete graph

Bordenave

Caputo

Chafäı

et al. 2017

Random Matrices: Theory Appl.

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Abstract. We consider the random Markov matrix obtained by assigning i.i.d. nonnegative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disc of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disc.

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Cutoff at the “entropic time” for sparse Markov chains

Cited by 39 publications

References 27 publications

Stationary distribution and cover time of sparse directed configuration models

Stationary distribution and cover time of sparse directed configuration models

A Structural Result for Personalized PageRank and its Algorithmic Consequences

Spectrum of large random Markov chains: Heavy-tailed weights on the oriented complete graph

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