A key-management scheme for distributed sensor networks
Proceedings of the 9th ACM conference on Computer and communications security
On Random Intersection Graphs: The Subgraph Problem
Combinatorics, Probability and Computing
Revisiting random key pre-distribution schemes for wireless sensor networks
Proceedings of the 2nd ACM workshop on Security of ad hoc and sensor networks
The vertex degree distribution of random intersection graphs
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
The Volume of the Giant Component of a Random Graph with Given Expected Degrees
SIAM Journal on Discrete Mathematics
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
Random intersection graphs with tunable degree distribution and clustering
Probability in the Engineering and Informational Sciences
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Given integers $n$, $m$ and a probability distribution $P_*$ on $[m]=\{1,\dots,m\}$, consider the random intersection graph on the vertex set $[n]$, where $i,j\in[n]$ are declared to be adjacent whenever $S(i)\cap S(j)\neq\emptyset$. Here $S(1),\dots,S(n)$ denote independent and identically distributed random subsets of $[m]$ with the distribution $\mathbf{P}(S(i)=A)={m\choose|A|}^{-1}P_*(|A|)$ for $A\subset[m]$. Assuming that $m$ is much larger than $n$, we show that the order of the largest connected component $N_1=n\rho+o_P(n)$ as $n,m\to\infty$. Here $\rho$ denotes the nonextinction probability of a related multitype Poisson branching process.