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 connectivity in uniform random intersection graph @math , Rybarczyk @cite_12 derives the exact probability and a zero--one law, while Blackburn and Gerke @cite_10 , Ya g an and Makowski @cite_20 , and Zhao @cite_3 @cite_0 also obtain zero--one laws. Rybarczyk @cite_24 implicitly shows a zero--one law for @math -connectivity in @math . Our Theorem also gives a zero--one law. In addition, it gives the exact probability to provide an accurate understanding of @math -connectivity.
@cite_3 @cite_0 @cite_24 @cite_10 @cite_12 @cite_20
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Random key predistribution scheme of Eschenauer and Gligor (EG) is a typical solution for ensuring secure communications in a wireless sensor network (WSN). Connectivity of the WSNs under this scheme has received much interest over the last decade, and most of the existing work is based on the assumption of unconstrained sensor-to-sensor communications. In this paper, we study the k-connectivity of WSNs under the EG scheme with physical link constraints; k-connectivity is defined as the property that the network remains connected despite the failure of any (k - 1) sensors. We use a simple communication model, where unreliable wireless links are modeled as independent on off channels, and derive zero-one laws for the properties that i) the WSN is k-connected, and ii) each sensor is connected to at least k other sensors. These zero-one laws improve the previous results by Rybarczyk on the k-connectivity under a fully connected communication model. Moreover, under the on off channel model, we provide a stronger form of the zero-one law for the 1-connectivity as compared to that given by Yagan. We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for @math –connectivity, perfect matching containment and Hamilton cycle containment. A uniform random intersection graphG(n,m,k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when modelling the network graph of the well-known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph G(n,m,k) when n-> in many situations. For example, when k is a function of n such that k>=2 and [email protected]?n^@[email protected]? for some fixed positive real number @a then G(n,m,k) is almost surely connected when lim infk^2n mlogn>1, and G(n,m,k) is almost surely disconnected when lim supk^2n mlogn<1. We study properties of the uniform random intersection graph model G(n,m,d). We find asymptotic estimates on the diameter of the largest connected component of the graph near the phase transition and connectivity thresholds. Moreover we manage to prove an asymptotically tight bound for the connectivity and phase transition thresholds for all possible ranges of d, which has not been obtained before. The main motivation of our research is the usage of the random intersection graph model in the studies of wireless sensor networks. The random key graph is a random graph naturally associated with the random key predistribution scheme introduced by Eschenauer and Gligor in the context of wireless sensor networks (WSNs). For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to WSNs.