Random intersection graphs have received much attention for nearly two decades, and currently have a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. In this paper, we investigate the strengths of connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for k-connectivity and k-robustness, as well as the asymptotically exact probability of k-connectivity, for any positive integer k. The k-connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures. On the other hand, k-robustness measures the effectiveness of local diffusion strategies (that do not use global graph topology information) in spreading information over the graph in the presence of misbehaving nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, our results on asymptotically exact probabilities of k-connectivity and asymptotic zero-one laws for k-robustness are also novel in the literature.
For general random intersection graph @math , Godehardt and Jaworski @cite_7 investigate its degree distribution and Bloznelis @cite_14 explore its component evolution, but provides neither a zero--one law nor the exact probability of its @math -connectivity property reported in our work.
@cite_14 @cite_7
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We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we apply the coupling method used for establishing threshold functions for homogeneous random intersection graphs introduced by Karo 'nski, Scheinerman, and Singer--Cohen. In the case of inhomogeneous random intersection graphs the method has to be considerably modified and extended. By means of the altered method we are able to establish threshold functions for a general random intersection graph for such properties as @math -connectivity, matching containment or hamiltonicity. Moreover using the new approach we manage to sharpen the best known results concerning homogeneous random intersection graph. Graph concepts generally are usefulfor defining and detecting clusters. We consider basic properties of random intersection graphs generated by a random bipartite graph BG n, m on n+m vertices. In particular, we focus on the distr ibution of the number of isolated vertices, and on the distribution of the vertex degrees. These results are applied to study the asymptotic properties of such random intersection graphs for the special case that the distribution P (m) is degenerated. The application of this model to find clusters and to test their randomness especially for non-metric data is discussed.