By Kevin J. Lang (auth.), Konstantin Avrachenkov, Debora Donato, Nelly Litvak (eds.)

ISBN-10: 3540959947

ISBN-13: 9783540959946

This booklet constitutes the refereed complaints of the sixth foreign Workshop on Algorithms and types for the Web-Graph, WAW 2009, held in Barcelona, Spain, in February 2009 - co-located with WSDM 2009, the second one ACM foreign convention on net seek and information Mining.

The 14 revised complete papers offered have been rigorously reviewed and chosen from various submissions for inclusion within the ebook. The papers deal with a wide selection of themes on the topic of the examine of the Web-graph resembling theoretical and empirical research of the net graph and net 2.0 graphs, random walks on the internet and internet 2.0 graphs and their functions, and layout and function assessment of the algorithms for social networks. The workshop papers were obviously clustered in 3 topical sections on graph types for complicated networks, pagerank and internet graph, and social networks and search.

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**Additional resources for Algorithms and Models for the Web-Graph: 6th International Workshop, WAW 2009, Barcelona, Spain, February 12-13, 2009. Proceedings**

**Sample text**

Note that the average size of a component in Wn vol(G) (that is, components must grow by a factor of at has size at least Cm σ2(m+1) n 1 ′ σ 2(m+1) n. If least σ2 each iteration) and if km > 1, we must have km ≤ Cm 1 ⌉ − 1, this would imply that km = o(1) by our condition σ = o(n−κ ) , m = ⌈ 2κ 1 so after at most ⌈ 2κ ⌉ − 1 rounds, we must have km = 1 and the process will halt with a giant connected component. ) √ 0 ≤ C ′ Δ√ln n , and In the case where Δ ln n > Cσ 2 n, we note that |S0 | ≤ vol(S ǫd ǫd ′′ the average volume of components in St is at least C Δvol(G)d = ω(Δ ln n), so we ln n can form W (1) by taking just one component for n large enough, and the proof goes as above.

Av = 0. For all u ∈ N (v) go over v’s adjacency list, and for each vertex w in u’s adjacency list check if w ∈ N (v) by going over v’s adjacency list. If w ∈ N (v) then av = av + deg w − 2 + deg u − 2. Return T LG(v) + av . Theorem 4. Let G = (V, E) be an undirected graph, and let H be a triangle with a ”tail” of length one. Then, for every vertex v, the number of copies of G that are isomorphic to H and adjacent to v can be (ǫ, δ)-approximated, with time complexity O |E|2 + n · |E| log(1/δ)/ǫ2 .

Vk }. Then, for any k ≥ k∗ we have d(Hk ) ≥ (1/3)Deq (G, k). Proof. We will ﬁrst show the following: d(Hk+1 ) ≤ d(Hk ) for all k ≥ k∗ . (1) Once we show this, then for any k ≥ k∗ we have d(Hk ) = max Hj ≥ j≥k 1 1 Dal (k) ≥ Deq (k). 3 3 The middle step follows from the approximation guarantee proved in Theorem 1. To prove (1), it suﬃces to take an arbitrary value of w for which |Cw | > k∗ , and show that d(Hj−1 ) ≥ d(Hj ) for all j in the interval (|Cw+1 |, |Cw |]. We prove this by induction, ﬁrst assuming d(Hj ) ≥ d(Cw ) and then proving d(Hj−1 ) ≥ d(Cw ).

### Algorithms and Models for the Web-Graph: 6th International Workshop, WAW 2009, Barcelona, Spain, February 12-13, 2009. Proceedings by Kevin J. Lang (auth.), Konstantin Avrachenkov, Debora Donato, Nelly Litvak (eds.)

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