By Ageev A.A., Baburin A.E., Gimandi E.K.

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3 1 2 3 The density of a subgraph with k vertices is the fraction of the k(k − 1)/2 possible edges that exist in the subgraph. A more careful implementation, using a bucket-based priority queue, gives complexity O(n log n). , take a collection of m k-cliques and connect them via m additional edges. All nodes have a degree of k or k + 1 so the core is the whole graph. As m grows the density vanishes. Finding a Dense-Core in Jellyfish Graphs 31 Subramanian et al. [32] suggested a 5-tier hierarchical layering of the AS-graph.

Return C, H Our main result is the following: Theorem 2. Let G = (V, E) be a sparse graph that contains a (k, Ω(n1−β ), O(n1−β/2 ), c, )-Jellyfish subgraph. t. 5 For k = O(log n) and β ≤ 2/5, the time complexity of ˜ 1−β/2 )6 . Algorithm 2 is O(n The proof of Theorem 2 appears in the full version of the paper [20]. 5 Implementation To demonstrate the usefulness of our algorithms beyond their theoretical contribution, we conducted a performance evaluation of our algorithm in comparison with the GreedyMaxClique algorithm of Siganos et al.

Xt−1 .... 1 Introduction During the last decade a large body of research has centered on understanding and modeling the structure of large-scale networks like the Internet and the World Wide Web. Several recent books provide a general introduction to this topic [37] and [40]. , the proportion of vertices of degree at least k is proportional to k −α for some constant α). This has driven the investigation of random graph distributions which generate heavy-tailed degree distributions, including the ﬁxed degree sequence model, the copying model, and the preferential attachment model.