We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the @math sensors in the network is classified into @math classes according to some probability distribution @math . Before deployment, a class- @math sensor is assigned @math cryptographic keys that are selected uniformly at random from a common pool of @math keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters @math and @math so that this graph: 1) has no isolated nodes and 2) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors @math growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by and for the same model, therein referred to as the general random intersection graph.
The random graph model @math considered here is also known as general random intersection graphs in the literature; e.g., see @cite_16 @cite_14 @cite_20 . To the best of our knowledge this model has been first considered by Godehardt and Jaworski @cite_20 and by @cite_3 . Results for both the existence of isolated nodes and graph connectivity have been established; see below for a comparison of these results with those established here. Later, @cite_14 analyzed the component evolution problem in the general random intersection graph and provided scaling conditions for the existence of a giant component . There, they also established that under certain conditions @math behaves very similarly with a standard Erd o s-R 'enyi graph @cite_9 . Taking advantage of this similarity, @cite_16 established various results for the @math -connectivity and @math -robustness of the general random intersection graph by means of a coupling argument.
@cite_14 @cite_9 @cite_3 @cite_16 @cite_20
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We study a connectivity property of a secure wireless network that uses random pre-distribution of keys. A network is composed of n sensors. Each sensor is assigned a collection of d different keys drawn uniformly at random from a given set of m keys. Two sensors are joined by a communication link if they share a common key. We show that for large n with high probability the connected component of size Ω(n) emerges in the network when the probability of a link exceeds the threshold 1-n. Similar component evolution is shown for networks where sensors communicate if they share at least s common keys. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009 We study properties of random intersection graphs generated by a random bipartite graph. We focus on the connectedness of these random intersection graphs and give threshold functions for this property and results for the size of the largest components in such graphs. The application of intersection graphs to find clusters and to test their randomness in sets of non-metric data is shortly discussed. 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. 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.