By Ravi Kumar, D Sivakumar

This publication constitutes the refereed complaints of the seventh overseas Workshop on Algorithms and types for the Web-Graph, WAW 2010, held in Stanford, CA, united states, in December 2010, which used to be co-located with the sixth overseas Workshop on net and community Economics (WINE 2010). The thirteen revised complete papers and the invited paper awarded have been rigorously reviewed and chosen from 19 submissions.

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Extra info for Algorithms and Models for the Web-Graph: 7th International Workshop, WAW 2010, Stanford, CA, USA, December 13-14, 2010, Proceedings

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Rocklin and A. Pinar Formally, we work on the following problem. Given a graph G = (V, E) with multiple similarity measurements for each edge wi1 , wi2 , . . , wiK ∈ RK , and a groundtruth clustering for this graph C ∗ . Our goal is to find a weighting vector α ∈ RK , such that the C ∗ is an optimal clustering for the graph G, whose edges are weighted K as wi = j=1 αj wij . Note that this is only a semi-formal definition, as we have not formally defined what we mean by an optimal clustering. In addition to the well-known difficulty of defining what a good clustering means, matching to the ground-truth data has specific challenges, which we discuss in the subsequent section.

As we have already mentioned, the major challenge comes from the underlying structure of RIGs, which involves both a set of nodes and a set of attributes, as well as a set of different probabilities p = {pw | w ∈ W }. Moreover, the edges in RIG are not independent. Hence, a RIG cannot be treated as an Erd˝os-R´enyi random graph Gn,pˆ, with the edge probability pˆ = 1 − w∈W (1 − p2w ). However, in [12], the authors provide the comparison among Gn,pˆ and G(n, m, p), showing that for m = nα and α > 6, these two classes of graphs have asymptotically the same properties.

Given any sequence of nodes v0 , v1 , v2 , . , the probability that a given attribute w is first discovered at time t < n is P[Γw = t] = P[Ivt ,w = 1, Ivt−1 ,w = 0, . . , Iv0 ,w = 0] = pw (1 − pw )t . If an attribute w is not discovered by time n − 1, we set Γw = ∞ and note that P[Γw = ∞] = (1 − pw )n . From the independence of the random variables Iv,w , it follows that the discovery times {Γw : w ∈ W } are mutually independent. We now focus on describing the distribution of φt = α∈Wt qα . For t ≥ 0, we have t φt = d qα = α∈Wt t I(Γw =j) qw = qα = j=0 α∈Wj \Wj−1 j=0 w∈W I(Γw ≤t) qw .

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