Uniform random intersection graphs have received much interest and been used in diverse applications. A uniform random intersection graph with @math nodes is constructed as follows: each node selects a set of @math different items uniformly at random from the same pool of @math distinct items, and two nodes establish an undirected edge in between if and only if they share at least one item. For such graph denoted by @math , we present the following results in this paper. First, we provide an exact analysis on the probabilities of @math having a perfect matching and having a Hamilton cycle respectively, under @math (all asymptotic notation are understood with @math ). The analysis reveals that just like ( @math -)connectivity shown in prior work, for both properties of perfect matching containment and Hamilton cycle containment, @math also exhibits phase transitions: for each property above, as @math increases, the limit of the probability that @math has the property increases from @math to @math . Second, we compute the phase transition widths of @math for @math -connectivity (KC), perfect matching containment (PMC), and Hamilton cycle containment (HCC), respectively. For a graph property @math and a positive constant @math , with the phase transition width @math defined as the difference between the minimal @math ensuring @math having property @math with probability at least @math or @math , we show for any positive constants @math and @math : (i) If @math and @math , then @math is either @math or @math for each @math sufficiently large. (ii) If @math , then @math . (iii) If @math , then @math . (iv) If @math , @math and @math are both @math .
A graph model related to uniform random intersection graphs is the so-called : each item in a pool is assigned to each node independently with the same probability, and two nodes establish an undirected edge upon sharing at least one item. Note that here the number of items on each node follows a binomial distribution. This graph model has also been studied in the literature as noted below. For connectivity, Rybarczyk presents a zero--one law @cite_17 and later obtains the stronger result of the asymptotically exact probability @cite_2 . For @math -connectivity, she establishes zero--one laws @cite_17 @cite_2 , and we compute the asymptotically exact probability @cite_11 . For perfect matching containment, Rybarczyk provides a zero--one law @cite_17 and derives the asymptotically exact probability @cite_2 . For Hamilton cycle containment, Efthymioua and Spirakis @cite_13 , and Rybarczyk @cite_17 @cite_2 show zero--one laws.
@cite_13 @cite_11 @cite_2 @cite_17
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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. 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. Random key graphs form a class of random intersection graphs that 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 attention 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 graph links is captured by independent Bernoulli random variables, rendering them to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdős–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: 1) the minimum node degree in the graph is at least @math and 2) the graph is @math -connected, both with high probability as the number of nodes becomes large. The results are given in the form of zero-one laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on @math -connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on one-connectivity of random key graphs with unreliable links. Let W1,…,Wn be independent random subsets of [m]= 1,…,m . Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m,d) on the vertex set W1,…Wn , defined by the adjacency relation: Wi∼Wj whenever |Wi∩Wj|≧s. For even n we show that as n,m→∞ the edge density threshold for the property that Gs(n,m,d) contains a perfect matching is asymptotically the same as that for Gs(n,m,d) being connected.