The q-composite key predistribution scheme [1] is used prevalently for secure communications in large-scale wireless sensor networks (WSNs). Prior work [2]-[4] explores topological properties of WSNs employing the q-composite scheme for q = 1 with unreliable communication links modeled as independent on off channels. In this paper, we investigate topological properties related to the node degree in WSNs operating under the q-composite scheme and the on off channel model. Our results apply to general q and are stronger than those reported for the node degree in prior work even for the case of q being 1. Specifically, we show that the number of nodes with certain degree asymptotically converges in distribution to a Poisson random variable, present the asymptotic probability distribution for the minimum degree of the network, and establish the asymptotically exact probability for the property that the minimum degree is at least an arbitrary value. Numerical experiments confirm the validity of our analytical findings.
 We show the asymptotic degree distribution of the typical vertex of a sparse inhomogeneous random intersection graph. 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 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. 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. This paper concerns constructing independent sets in a random intersection graph. We concentrate on two cases of the model: a binomial and a uniform random intersection graph. For both models we analyse two greedy algorithms and prove that they find asymptotically almost optimal independent sets. We provide detailed analysis of the presented algorithms and give tight bounds on the independence number for the studied models. Moreover we determine the range of parameters for which greedy algorithms give better results for a random intersection graph than this is in the case of an Erd?s-Renyi random graph G ( n , p ? ) . We investigate the existence and efficient algorithmic construction of close to optimal independent sets in random models of intersection graphs. In particular, (a) we propose a new model for random intersection graphs (G"n","m","p"->) which includes the model of [M. Karonski, E.R. Scheinerman, K.B. Singer-Cohen, On random intersection graphs: The subgraph problem, Combinatorics, Probability and Computing journal 8 (1999), 131-159] (the ''uniform'' random intersection graph models) as an important special case. We also define an interesting variation of the model of random intersection graphs, similar in spirit to random regular graphs. (b) For this model we derive exact formulae for the mean and variance of the number of independent sets of size k (for any k) in the graph. (c) We then propose and analyse three algorithms for the efficient construction of large independent sets in this model. The first two are variations of the greedy technique while the third is a totally new algorithm. Our algorithms are analysed for the special case of uniform random intersection graphs. Our analyses show that these algorithms succeed in finding close to optimal independent sets for an interesting range of graph parameters. 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.  A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this article a model is developed in which each vertex is given a random weight and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree, and—in the power-law case—tail exponent. In this paper we consider the degree of a typical vertex in two models of random intersection graphs introduced in [E. Godehardt, J. Jaworski, Two models of random intersection graphs for classification, in: M. Schwaiger, O. Opitz (Eds.), Exploratory Data Analysis in Empirical Research, Proceedings of the 25th Annual Conference of the Gesellschaft fur Klassifikation e.V., University of Munich, March 14-16, 2001, Springer, Berlin, Heidelberg, New York, 2002, pp. 67-81], the active and passive models. The active models are those for which vertices are assigned a random subset of a list of objects and two vertices are made adjacent when their subsets intersect. We prove sufficient conditions for vertex degree to be asymptotically Poisson as well as closely related necessary conditions. We also consider the passive model of intersection graphs, in which objects are vertices and two objects are made adjacent if there is at least one vertex in the corresponding active model ''containing'' both objects. We prove a necessary condition for vertex degree to be asymptotically Poisson for passive intersection graphs. 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. In a random key graph (RKG) of n nodes each node is randomly assigned a key ring of Kn cryptographic keys from a pool of Pn keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume that the n nodes are distributed uniformly in [0, l]2. In addition to the common key requirement, we require two nodes to also be within rn of each other to be able to have a direct edge. Thus we have a random graph in which the RKG is superposed on the familiar random geometric graph (RGG). For such a random graph, we obtain tight bounds on the relation between Kn, Pn and rn for the graph to be asymptotically almost surely connected. The notion of the random key graph, which originally appeared in models of secure communication in wireless sensor networks, has been used in other applications, some of which are unrelated to cryptographic-key predistribution or sensor networks. In this presentation, I will outline some of these applications, which exploit the connectivity property of random key graphs and its similarity with that of random graphs. I’d like to start with the zero-one law for randomgraph connectivity, then explain how (i.e., for what graph parameters) this law appears in random key graphs. Then, I will present three brief encounters with random-key-graph properties in new settings and perhaps speculate on other types of useful properties they might have.