# Báo cáo toán học: "Degree distributions in general random intersection graphs"

## Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Degree distributions in general random intersection graphs. | Degree distributions in general random intersection graphs Yilun Shang Department of Mathematics Shanghai Jiao Tong University 200240 Shanghai China shyl@sj Submitted Jun 22 2009 Accepted Jan 26 2010 Published Jan 31 2010 Mathematics Subject Classification 05C80 Abstract We study G n m F H a variant of the standard random intersection graph model in which random weights are assigned to both vertex types in the bipartite structure. Under certain assumptions on the distributions of these weights the degree of a vertex is shown to depend on the weight of that particular vertex and on the distribution of the weights of the other vertex type. 1 Introduction Random intersection graphs denoted by G n m p are introduced in 9 14 as opposed to classical Erdos-Renyi random graphs. Let us consider a set V with n vertices and another universal set W with m elements. Define a bipartite graph B n m p with independent vertex sets V and W. Edges between v G V and w G W exist independently with probability p. The random intersection graph G n m p derived from B n m p is defined on the vertex set V with vertices v1 v2 G V adjacent if and only if there exists some w G W such that both v1 and v2 are adjacent to w in B n m p . To get an interesting graph structure and bounded average degree the work 15 sets m na and p cn- 1 a 2 for some a c 0 and determines the distribution of the degree of a typical vertex. Some related properties for this model are recently investigated for example independent sets 11 and component evolution 1 10 . A generalized random intersection graph is introduced in 5 by allowing a more general connection probability in the underlying bipartite graph. The corresponding vertex degrees are also studied by some authors see . 2 7 8 and shown to be asymptotically Poisson distributed. In this paper we consider a variant model of random intersection graphs where each vertex and element are associated with a random weight in order to obtain a larger class of

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