Download Algorithms and Models for the Web Graph: 10th International by Jeannette Janssen, Paweł Prałat, Rory Wilson (auth.), PDF

By Jeannette Janssen, Paweł Prałat, Rory Wilson (auth.), Anthony Bonato, Michael Mitzenmacher, Paweł Prałat (eds.)

This ebook constitutes the refereed lawsuits of the tenth overseas Workshop on Algorithms and types for the net Graph, WAW 2013, held in Cambridge, MA, united states, in December 2013. The 17 papers offered have been conscientiously reviewed and chosen for inclusion during this quantity. They deal with issues concerning graph-theoretic and algorithmic facets of comparable complicated networks, together with quotation networks, social networks, organic networks, molecular networks and different networks bobbing up from the Internet.

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Additional resources for Algorithms and Models for the Web Graph: 10th International Workshop, WAW 2013, Cambridge, MA, USA, December 14-15, 2013, Proceedings

Example text

3 3 1+φ This last step follows under the assumption that φ ≤ 1/2. We can do this without loss of generality since the guarantee on h(S) in the theorem is trivial for φ > 1/2. The above shows that the algorithm will not experience a failure due to the volume becoming too large, and we have seen that conditions (2) and (3) will be satisfied by the output. Finally, to show condition (1), we apply the Sharp Drop Lemma. We know that kf is the smallest index such that vol(Skf +1 ) ≥ (1+φ) vol(Skf ), and q(vkf +1 ) ≤ q(vkf ) − 2α/φ vol(Ski ).

5 Further Work In [8], the authors find small separators for the SPA model. They deduce that the spectral gap of the normalised Laplacian of the graph Gt converges to 1, yielding bad expansion properties for Gt . The separators they found are simply obtained by cutting the hypercube in half. We expect that the same strategy would yield similar results for our generalised model, with the slight difference that the separators will not be as small, depending on the tail of the profile function ϕ. Existence of a Giant Component.

3. vol(S ∩ C) ≥ (3/4) vol(S). Proof. Claim. There exist an index j such that q(vj ) ≥ 1 γ vol(Sj ) . 1 for every index j. Since v ∈ Cα , Proof. Suppose that q(vj ) < γ vol(S j) h(C), and h(C) ≤ α/64γ then we know that p(C) ≥ pr(α, χv )(C) − δ vol(C) ≥ 1 − since x ≥ vol(C). 2(h(C) + ) 1 1 1 1 − vol(C) ≥ 1 − − =1− α 16γx 16γ 16γ 8γ < 38 F. Chung and M. Kempton On the other hand, under our assumption, n p(C) ≤ p(V ) = n p(vi ) = i=1 q(vi ) dvi i=1 n < dvi γ vol(Sj ) i=1 2m ≤ 1 γ k=1 1 . k Putting these together, we have 1− 1 1 < 8γ γ 2m k=1 1 .

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