The giant component of the directed configuration model revisited

Cai, Xing Shi; Perarnau, Guillem

doi:10.30757/alea.v18-55

Cited by 9 publications

(5 citation statements)

References 10 publications

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“…We will use this proposition together with ( 13), ( 14) and (15) to establish the following bounds on the survival probabilities ρ(S, x) and ζ(p). PROPOSITION 6.7.…”

Section: Applications To Preferential Attachment Graphsmentioning

confidence: 99%

“…Note that Lemma E.2, together with the fact that the position of the root is √ x for x chosen uniformly at random from [0, 1], immediately implies (15). We close this appendix with the proof of Proposition 6.7.…”

mentioning

confidence: 91%

“…Later, Cooper and Frieze [16] established the bow-tie structure of directed configuration model with maximum degree o(n 1/12 ). Later work, weakened the condition on maximum degree to o(n 1/4 ) in [24], and o( √ n) in [15]. To the best of our knowledge, this is the first work showing the bow-tie structure for oriented percolation on general expanders.…”

mentioning

confidence: 99%

“…By the same reasoning, we may bound F from below by showing that ǫ − is bounded from below by the solution of 1 = p(m − 1) log 1+(m−1)ǫ (m−1)ǫ , which inserted into (29) gives ρ(S, x) ≥ e − 1 p(m−1) , as claimed. Inserted into (15), this also gives the lower bound on ζ(p).…”

mentioning

confidence: 99%

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Locality of Random Digraphs on Expanders

Alimohammadi¹,

Borgs²,

Saberi³

2021

Preprint

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We study random digraphs on sequences of expanders with bounded average degree and weak local limit. The threshold for the existence of a giant strongly connected component, as well as the asymptotic fraction of nodes with giant fan-in or giant fan-out are local, in the sense that they are the same for two sequences with the same weak local limit. The digraph has a bow-tie structure, with all but a vanishing fraction of nodes lying either in the unique strongly connected giant and its fan-in and fan-out, or in sets with small fanin and small fan-out. All local quantities are expressed in terms of percolation on the limiting rooted graph, without any structural assumptions on the limit, allowing, in particular, for non tree-like limits.In the course of proving these results, we prove that for unoriented percolation, there is a unique giant above criticality, whose size and critical threshold are again local. An application of our methods shows that the critical threshold for bond percolation and random digraphs on preferential attachment graphs is pc = 0, with an infinite order phase transition at pc.

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“…We will use this proposition together with ( 13), ( 14) and (15) to establish the following bounds on the survival probabilities ρ(S, x) and ζ(p). PROPOSITION 6.7.…”

Section: Applications To Preferential Attachment Graphsmentioning

confidence: 99%

mentioning

confidence: 91%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Locality of Random Digraphs on Expanders

Alimohammadi¹,

Borgs²,

Saberi³

2021

Preprint

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show abstract

“…Later, Cooper and Frieze [4] provided a rigorous proof for the existence of such threshold under certain conditions of the degree sequence. This problem has been recently revisited by Cai and the second author [3], extending the range of applicability of the result.…”

Section: Introductionmentioning

confidence: 99%

Weak Components of the Directed Configuration Model

Coulson

Perarnau

2021

Trends in Mathematics

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We study the threshold for the existence of a linear order weakly connected component in the directed configuration model, confirming analytic but non-rigorous results recently obtained by Kryven [8]. We also establish convergence in probability of the fraction of vertices and edges that are contained in the largest component. As a consequence of our results, we obtain that the "separation" between the thresholds for the existence a giant weakly and strongly connected component is in some sense independent from the in-/out-degree correlation. We formalise this idea using bond percolation.

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Percolation in simple directed random graphs with a given degree distribution

Ieperen

Kryven

2023

Prob. Eng. Inf. Sci.

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We study site and bond percolation in simple directed random graphs with a given degree distribution. We derive the percolation threshold for the giant strongly connected component and the fraction of vertices in this component as a function of the percolation probability. The results are obtained for degree sequences in which the maximum degree may depend on the total number of nodes n, being asymptotically bounded by $n^{\frac{1}{9}}$ .

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The giant component of the directed configuration model revisited

Abstract: We prove a law of large numbers for the order and size of the largest strongly connected component in the directed configuration model. Our result extends previous work by Cooper and Frieze (2004).

Cited by 9 publications

References 10 publications

Locality of Random Digraphs on Expanders

Locality of Random Digraphs on Expanders

Weak Components of the Directed Configuration Model

Percolation in simple directed random graphs with a given degree distribution

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