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 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. We consider the random graph induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. We show the existence of a zero-one law for the absence of isolated nodes, and complement it by a Poisson convergence for the number of isolated nodes. Leveraging earlier results and analogies with Erdos-Renyi graphs, we explore similar results for the property of graph connectivity. In this paper, we consider a wireless network with unreliable links and investigate its minimum node degree and k- connectivity. In such network, n nodes are uniformly distributed in a region, and edges are established for any two nodes within a certain distance and with a probabilistically active link in between. On a torus topology, we present a zero-one law for the property of minimum degree being at least k, leading to a zero-law for k-connectivity and an intermediate result towards a one-law. On a square with boundary effect, we establish a one- law for minimum degree. Our results are derived from rigorous analysis and also confirmed by the simulation, and they provide guidelines for the design of wireless networks. Index Terms—Connectivity, Erdos-Renyi graph, geometric graph, graph intersection, minimum node degree, random graph, unreliable link, wireless network, zero-one law. 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.