Large Cliques in a Power-law Random Graph
نویسندگان
چکیده
We study the size of the largest clique ω(G(n, α)) in a random graphG(n, α) on n vertices which has power-law degree distribution with exponent α. We show that for ‘flat’ degree sequences with α > 2 whp the largest clique in G(n, α) is of a constant size, while for the heavy tail distribution, when 0 < α < 2, ω(G(n, α)) grows as a power of n. Moreover, we show that a natural simple algorithm whp finds in G(n, α) a large clique of size (1 + o(1))ω(G(n, α)) in polynomial time.
منابع مشابه
Parameterized clique on inhomogeneous random graphs
Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. In typical applications like social networks or biological networks, however, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We pr...
متن کاملParameterized Clique on Scale-Free Networks
Finding cliques in graphs is a classical problem which is in general NP-hard and parameterized intractable. However, in typical applications like social networks or protein-protein interaction networks, the considered graphs are scale-free, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more e...
متن کاملLarge Cliques in Sparse Random Intersection Graphs
Given positive integers n and m, and a probability measure P on {0, 1, . . . ,m}, the random intersection graph G(n,m,P ) on vertex set V = {1, 2, . . . , n} and with attribute set W = {w1, w2, . . . , wm} is defined as follows. Let S1, S2, . . . , Sn be independent random subsets of W such that for any v ∈ V and any S ⊆ W we have P(Sv = S) = P (|S|)/ ( m |S| ) . The edge set of G(n,m,P ) consi...
متن کاملEmbedding large graphs into a random graph
In this paper we consider the problem of embedding bounded degree graphs which are almost spanning in a random graph. In particular, let ∆ ≥ 5 and let H be a graph on (1−o(1))n vertices and with maximum degree ∆. We show that a random graph Gn,p with high probability contains a copy of H, provided that p (n−1 log n). Our assumption on p is optimal (including the powers in the log term) with res...
متن کاملEigenvalues of Random Power Law Graphs
Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power l...
متن کامل