k-Connectivity in Random Key Graphs with Unreliable Links

Random key graphs form a class of random intersection graphs and are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much interest recently, owing in part to their wide applicability in various domains including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of the edges are captured by independent Bernoulli random variables, rendering edges of the graph to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdos-Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model's parameters so that i) the minimum node degree in the graph is at least k, and ii) the graph is k-connected, both with high probability as the number of nodes becomes large. The results are given in the form of zeroone laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on the k-connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on the 1-connectivity of random key graphs with unreliable links.Comment: Published in IEEE Transactions on Information Theor

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

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

Tight Bounds for Connectivity of Random K-out Graphs

Random K-out graphs are used in several applications including modeling by sensor networks secured by the random pairwise key predistribution scheme, and payment channel networks. The random K-out graph with $n$ nodes is constructed as follows. Each node draws an edge towards $K$ distinct nodes selected uniformly at random. The orientation of the edges is then ignored, yielding an undirected graph. An interesting property of random K-out graphs is that they are connected almost surely in the limit of large $n$ for any $K \geq2$. This means that they attain the property of being connected very easily, i.e., with far fewer edges ($O(n)$) as compared to classical random graph models including Erd\H{o}s-R\'enyi graphs ($O(n \log n)$). This work aims to reveal to what extent the asymptotic behavior of random K-out graphs being connected easily extends to cases where the number $n$ of nodes is small. We establish upper and lower bounds on the probability of connectivity when $n$ is finite. Our lower bounds improve significantly upon the existing results, and indicate that random K-out graphs can attain a given probability of connectivity at much smaller network sizes than previously known. We also show that the established upper and lower bounds match order-wise; i.e., further improvement on the order of $n$ in the lower bound is not possible. In particular, we prove that the probability of connectivity is $1-\Theta({1}/{n^{K^2-1}})$ for all $K \geq 2$. Through numerical simulations, we show that our bounds closely mirror the empirically observed probability of connectivity

Parallel Graph Connectivity in Log Diameter Rounds

We study graph connectivity problem in MPC model. On an undirected graph with $n$ nodes and $m$ edges, $O(\log n)$ round connectivity algorithms have been known for over 35 years. However, no algorithms with better complexity bounds were known. In this work, we give fully scalable, faster algorithms for the connectivity problem, by parameterizing the time complexity as a function of the diameter of the graph. Our main result is a $O(\log D \log\log_{m/n} n)$ time connectivity algorithm for diameter-$D$ graphs, using $\Theta(m)$ total memory. If our algorithm can use more memory, it can terminate in fewer rounds, and there is no lower bound on the memory per processor. We extend our results to related graph problems such as spanning forest, finding a DFS sequence, exact/approximate minimum spanning forest, and bottleneck spanning forest. We also show that achieving similar bounds for reachability in directed graphs would imply faster boolean matrix multiplication algorithms. We introduce several new algorithmic ideas. We describe a general technique called double exponential speed problem size reduction which roughly means that if we can use total memory $N$ to reduce a problem from size $n$ to $n/k$, for $k=(N/n)^{\Theta(1)}$ in one phase, then we can solve the problem in $O(\log\log_{N/n} n)$ phases. In order to achieve this fast reduction for graph connectivity, we use a multistep algorithm. One key step is a carefully constructed truncated broadcasting scheme where each node broadcasts neighbor sets to its neighbors in a way that limits the size of the resulting neighbor sets. Another key step is random leader contraction, where we choose a smaller set of leaders than many previous works do

Random graphs with few disjoint cycles

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G-B has no cycles. We show that, amongst all such graphs on vertex set {1,..,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class; concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is still a blocker for all but at most k vertices v in B