By David F. Gleich, Júlia Komjáthy, Nelly Litvak
This ebook constitutes the complaints of the twelfth overseas Workshop on Algorithms and versions for the net Graph, WAW 2015, held in Eindhoven, The Netherlands, in December 2015.
The 15 complete papers offered during this quantity have been rigorously reviewed and chosen from 24 submissions. they're equipped in topical sections named: homes of enormous graph versions, dynamic methods on huge graphs, and homes of PageRank on huge graphs.
Read or Download Algorithms and Models for the Web Graph: 12th International Workshop, WAW 2015, Eindhoven, The Netherlands, December 10-11, 2015, Proceedings PDF
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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
3) where we establish a negative result by giving a logarithmic lower bound, for all values of α. The details of the proof can be found in the full version of the paper . Our approach is based on a special type of path, which gives natural lower bounds on the hyperbolicity. Definition 8. Let G = G(n, m, p) be a random intersection graph. The k-path P = v1 , v2 , . . , 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.
A graph G = (V, E) is δ-hyperbolic if for all x, y, z ∈ V , for every choice of geodesic (shortest) paths between them —denoted P [x, y], P [x, z], P [y, z] —we have ∀v ∈ P [x, y], ∃w ∈ P [x, z] ∪ P [z, y] : dG (v, w) ≤ δ, where dG (u, v) is shortest-path distance in G. That is, if G is δ-hyperbolic, then for each triple of vertices x, y, z, and every choice of three shortest paths connecting them pairwise, each point on the shortest path from x to y must be within distance δ of a point on one of the other 34 M.
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 → +∞. Suppose that EX12 < ∞ and EY1 < ∞. Then d(v1 ) converges in distribution to a random variable Λ3 having the probability distribution λr r = 0, 1, . . (4) P(Λ3 = r) = Ee−λ3 3 , r! Here λ3 = Y1 a2 b1 . (iii) Assume that m/n → 0. Suppose that EX1 < ∞. Then P(d(v1 ) = 0) = 1 − o(1). Using the fact that a Poisson random variable is highly concentrated around its mean one can show that for a power law distribution P(λi > r) ∼ ci r−κi , with some ci , κi > 0, we have P(Λi > r) ∼ ci r−κi , for i = 0, 1, 3.