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_19: 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.
  @cite_45: 
  @cite_13: We consider sparse random intersection graphs with the property that the clustering coefficient does not vanish as the number of nodes tends to infinity. We find explicit asymptotic expressions for the correlation coefficient of degrees of adjacent nodes (called the assortativity coefficient), the expected number of common neighbours of adjacent nodes, and the expected degree of a neighbour of a node of a given degree k. These expressions are written in terms of the asymptotic degree distribution and, alternatively, in terms of the parameters defining the underlying random graph model.
summary: Bloznelis @cite_45 investigates the clustering coefficient of graph @math and show that the chance of two neighbors of a given vertex @math to be adjacent decays as @math , where @math is a positive constant and @math is the degree of vertex @math . Bloznelis @cite_13 study the correlation coefficient of degrees of adjacent vertices in @math . Bloznelis @cite_19 demonstrate that a connected component in @math with at at least a constant fraction of @math emerges as @math when the edge probability @math exceeds @math .