Algorithms and Models for the Web Graph: 12th International by David F. Gleich, Júlia Komjáthy, Nelly Litvak

By David F. Gleich, Júlia Komjáthy, Nelly Litvak

This publication constitutes the court cases 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 awarded during this quantity have been conscientiously reviewed and chosen from 24 submissions. they're prepared in topical sections named: houses of huge graph versions, dynamic methods on huge graphs, and homes of PageRank on huge graphs.

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Extra info for Algorithms and Models for the Web Graph: 12th International Workshop, WAW 2015, Eindhoven, The Netherlands, December 10-11, 2015, Proceedings

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Suppose that n = i + 1. ˆ i+1 and G ¯ i+1 are obtained from the graph Gi by adding the Fix Gim . Graphs G m m m vertex i + 1 and m edges. These m edges can affect the number of triangles on at most m previous vertices. For example, they can be drown to at most m vertices . Such reasonings finally lead of degree d and decrease Ti (d) by at most m d (d−1) 2 i (d) ≤ M d2 for some M . to the estimate δi+1 Now let us use the induction. Consider t: i + 1 ≤ t ≤ n − 1, t > W d2 (note that the smaller values of t were already considered).

Notice that if a model of some graph H exists, so does a model with these properties. This allows us to only consider attributes with minimum degree two, since every edge in the path is generated by a different attribute. This is key to giving an upper bound for the probability of the existence of such a dense subgraph in the bipartite graph and prove the theorem. 4 Density Before turning to our main result, we need two more lemmas that establish the probability of graphs generated using G(n, m, p) have special types of dense subgraphs.

On the other hand, we showed that the metric structure of random intersection graphs is not tree-like for all values of α: the hyperbolicity (and treelength) grows at least logarithmically in n. While we only determine a lower bound for the hyperbolicity, we believe this to be the correct order of magnitude, as the diameter (a natural upper bound for the hyperbolicity) of similar model of random intersection graphs was shown to be O(log n) [25] for a similar range of parameter values. A question that naturally arises from these results is if structural sparsity should be an expected characteristic of practically relevant random graph models.

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