On the Isolated Vertices and Connectivity in Random Intersection Graphs

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.

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