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.
Random intersection graphs (RIGs) are an important random structure with algorithmic applications in social networks, epidemic networks, blog readership, and wireless sensor networks. RIGs can be interpreted as a model for large randomly formed non-metric data sets. We analyze the component evolution in general RIGs, giving conditions on the existence and uniqueness of the giant component. Our techniques generalize existing methods for analysis of component evolution: we analyze survival and extinction properties of a dependent, inhomogeneous Galton-Watson branching process on general RIGs. Our analysis relies on bounding the branching processes and inherits the fundamental concepts of the study of component evolution in Erdős-Renyi graphs. The major challenge comes from the underlying structure of RIGs, which involves both a set of nodes and a set of attributes, with different probabilities associated with each attribute.  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.  It is shown that almost all labelled r -regular graphs are r -connected, for any fixed r ≥3. This is an instance of a more general result which applies to labelled locally restricted graphs whose points have degrees lying between r and R , where r and R are fixed, 3≤ r ≤ R . The proof employs a known asymptotic formula for the numbers of these locally restricted graphs. It is also demonstrated that for fixed k >0 and r ≥3, almost all labelled r -regular graphs with girth at least [ k ( r −2)] are cyclically k -connected. This provides an asymptotic formula for the number of labelled cyclically k -connected r -regular graphs with p points, for fixed k and r . 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.