text: Random @math -intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random @math -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 @math items (the graph is called a random intersection graph when @math ). A special case of particular interest is a uniform random @math -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 @math -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 the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random @math -intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random @math -intersection graphs). In this paper, for binomial uniform random @math -intersection graphs, we present results related to @math -connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties: (i) @math -vertex-connectivity, (ii) @math -edge-connectivity and (iii) the property of minimum vertex degree being at least @math .
  @cite_35: 
  @cite_18: 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.
  @cite_22: 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.
  @cite_8: 
  @cite_29: 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.
  @cite_39: We study isolated vertices and connectivity in the random intersection graph 𝐺(𝑛,𝑚,𝑝). A Poisson convergence for the number of isolated vertices is determined at the threshold for absence of isolated vertices, which is equivalent to the threshold for connectivity. When 𝑚=⌊𝑛𝛼⌋ and 𝛼>6, we give the asymptotic probability of connectivity at the threshold for connectivity. Analogous results are well known in Erdős-Renyi random graphs.
  @cite_5: 
  @cite_16: 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.
  @cite_10: We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we apply the coupling method used for establishing threshold functions for homogeneous random intersection graphs introduced by Karo 'nski, Scheinerman, and Singer--Cohen. In the case of inhomogeneous random intersection graphs the method has to be considerably modified and extended. By means of the altered method we are able to establish threshold functions for a general random intersection graph for such properties as @math -connectivity, matching containment or hamiltonicity. Moreover using the new approach we manage to sharpen the best known results concerning homogeneous random intersection graph.
summary: & 3 * -1.5pt @math -1.5pt & 2 * & 2 * -1.5pt exact probabilities -1.5pt & -1.5pt 5-5 & & & & 3-5 & & & -1.5pt exact probabilities & 2-5 & 4 * -1.5pt @math -1.5pt & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_22 4-5 & & & -1.5pt zero--one laws & -1.5pt @cite_35 @cite_5 3-5 & & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_18 4-5 & & & -1.5pt zero--one laws & -1.5pt @cite_16 @cite_29 8 * & 4 * -1.5pt @math -1.5pt & 2 * & -1.5pt exact probabilities -1.5pt & 4 * -1.5pt -1.5pt 4-4 & & & -1.5pt zero--one laws & 3-4 & & 2 * & -1.5pt exact probabilities -1.5pt & 4-4 & & & -1.5pt zero--one laws & 2-5 & 4 * -1.5pt @math -1.5pt & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_22 4-5 & & & -1.5pt zero--one laws -1.5pt & -1.5pt @cite_35 3-5 & & 2 * & -1.5pt exact probabilities -1.5pt & -1.5pt @cite_10 4-5 & & & -1.5pt zero--one laws & -1.5pt @cite_8 @cite_39 2.5