By David F. Gleich, Júlia Komjáthy, Nelly Litvak
This e-book constitutes the court cases of the twelfth overseas Workshop on Algorithms and types for the internet Graph, WAW 2015, held in Eindhoven, The Netherlands, in December 2015.
The 15 complete papers provided during this quantity have been conscientiously reviewed and chosen from 24 submissions. they're equipped in topical sections named: homes of huge graph versions, dynamic methods on huge graphs, and houses of PageRank on huge graphs.
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Extra resources for Algorithms and Models for the Web Graph: 12th International Workshop, WAW 2015, Eindhoven, The Netherlands, December 10-11, 2015, Proceedings
Vk+1 in G is called a k-special path if all the internal vertices of P have degree two in G and there exists another disjoint path connecting v1 and vk+1 in G. We allow for the second path to have length 0: this occurs if P is a k-cycle such that all but one vertex of P has degree two in G. Lemma 10. Let k be a positive integer and G = G(n, m, p). If G contains a k-special path, then G has hyperbolicity at least k4 . Proof. Let P be a k-special path in G. By deﬁnition, P is part of some cycle C of length at least k.
In particular, we let Sk denote the number of k-special bipartite paths, and condition on the fact that the exposed graph has a giant component of size at least δn. We prove that E[Sξ log n ] = ω(1), and that Sk is tightly concentrated about its mean, using second moment methods. 5 Conclusion and Open Problems In this paper we have determined the conditions under which random intersection graphs exhibit two types of algorithmically useful structure. We proved that graphs in G(n, m, p) are structurally sparse (have bounded expansion) precisely when the number of attributes in the associated bipartite graph grows faster than the number of nodes (α > 1).
Let m, n → ∞. (i) Assume that m/n → β for some β ∈ (0, +∞). Suppose that EX12 < ∞ and EY1 < ∞. Then d(v1 ) converges in distribution to the random variable Λ1 d∗ = τj , (2) j=1 where τ1 , τ2 , . . are independent and identically distributed random variables independent of the random variable Λ1 . They are distributed as follows. For r = 0, 1, 2, . . , we have P(τ1 = r) = r+1 P(Λ0 = r + 1) EΛ0 and P(Λi = r) = E e−λi λri , r! i = 0, 1. (3) Here λ0 = X1 b1 β −1/2 and λ1 = Y1 a1 β 1/2 . (ii) Assume that m/n → +∞.