On the strengths of connectivity and robustness in general random intersection graphs

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.Comment: This paper about random graphs appears in IEEE Conference on Decision and Control (CDC) 2014, the premier conference in control theor

On Topological Properties of Wireless Sensor Networks under the q-Composite Key Predistribution Scheme with On/Off Channels

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.Comment: Best Student Paper Finalist in IEEE International Symposium on Information Theory (ISIT) 201

k-connectivity of Random Graphs and Random Geometric Graphs in Node Fault Model

k-connectivity of random graphs is a fundamental property indicating reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of sensor nodes with limited power resources are modeled by random graphs with unreliable nodes, which is known as the node fault model. In this paper, we investigate k-connectivity of random graphs in the node fault model by evaluating the network breakdown probability, i.e., the disconnectivity probability of random graphs after stochastic node removals. Using the notion of a strongly typical set, we obtain universal asymptotic upper and lower bounds of the network breakdown probability. The bounds are applicable both to random graphs and to random geometric graphs. We then consider three representative random graph ensembles: the Erdos-Renyi random graph as the simplest case, the random intersection graph for WSNs with random key predistribution schemes, and the random geometric graph as a model of WSNs generated by random sensor node deployment. The bounds unveil the existence of the phase transition of the network breakdown probability for those ensembles.Comment: 6 page