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 random intersection graphs in this paper, two nodes have an edge in between if their object sets share at least one object. A natural variant is to define graphs with edges only between nodes which have at least @math objects in common (instead of just @math ) for some positive integer @math . Zhao @cite_5 @cite_25 @cite_11 consider @math -connectivity in graphs under this definition. In addition, ( @math )-connectivity of other random graphs have also been investigated in the literature @cite_4 @cite_34 .
@cite_4 @cite_5 @cite_34 @cite_25 @cite_11
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Random s-intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random s-intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least s items (the graph is called a random intersection graph when s = 1). A special case of particular interest is a uniform random s-intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random s-intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and modeling recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random s-intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random s-intersection graphs). In this paper, for binomial uniform random s-intersection graphs, we present results related to k-connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero--one laws for the following three properties: (i) k-vertex-connectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k. 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. Random Intersection Graphs, G"n","m","p, is a class of random graphs introduced in Karonski (1999) [7] where each of the n vertices chooses independently a random subset of a universal set of m elements. Each element of the universal sets is chosen independently by some vertex with probability p. Two vertices are joined by an edge iff their chosen element sets intersect. Given n, m so that [email protected]?n^@[email protected]?, for any real @a different than one, we establish here, for the first time, a sharp threshold for the graph property ''Contains a Hamilton cycle''. Our proof involves new, nontrivial, coupling techniques that allow us to circumvent the edge dependencies in the random intersection graph model. 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.