By Andrei Broder (auth.), Ravi Kumar, Dandapani Sivakumar (eds.)
This booklet constitutes the refereed court cases 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 offered have been rigorously reviewed and chosen from 19 submissions.
Read or Download Algorithms and Models for the Web-Graph: 7th International Workshop, WAW 2010, Stanford, CA, USA, December 13-14, 2010. Proceedings PDF
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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
Thus, two nodes are adjacent if and only if they p1 have at least s attributes in common. p2 vn One limitation of our analysis is that pm for simplicity, we fix s = 1. wm Our model generalizes the uniform model G(n, m, p), studied in [4,6], where all pw take on the same value p. Different generalizations and special Fig. 1. Random intersection graph. V is set of nodes and W is set of attributes. A particular atcases have been studied in [13,15,17,8]. tribute wi is associated with every node indepenTo complete the picture of previous dently at random with probability p .
U ↔ v0 |Ht ] = P[W (u) ∩ W (vt ) = ∅, W (u) ∩ Wt−1 = ∅|Ht ] = P[W (u) ∩ W (vt ) = ∅, W (u) ∩ Wt−1 = ∅|W (vt ), Wt−1 ] = P[W (u) ∩ W (vt ) = ∅|W (vt )] P[W (u) ∩ Wt−1 = ∅|Wt−1 ] = 1− (1 − pα ) α∈W (vt ) (1 − pβ ). β∈Wt−1 Component Evolution in General Random Intersection Graphs 41 The last expression can be rewritten as rt = qβ − β∈Wt−1 qα α∈Wt = φt−1 − φt , (12) where we set qw = 1 − pw for all w ∈ W and φt = α∈Wt qα , and use the convention W−1 = ∅ and φ−1 = 1. Observe that the probability (12) does not depend on u.
2 Supercritical Regime We now turn to the study of the supercritical regime in which limn→∞ n c > 1. w∈W p2w = Component Evolution in General Random Intersection Graphs 45 Theorem 2. Suppose that pw = o log n n for all w p3w = o and w∈W log n . n2 For any constant c > 1, if w∈W p2w = c/n, then whp there exists a unique largest component in G(n, m, p), of order Θ(n). Moreover, the size of the giant component is given by nζc (1 + o(1)), where ζc is the solution in (0, 1) of the equation 1 − e−cζ = ζ, while all other components are of size O(log n).