The largest component in an inhomogeneous random intersection graph with clustering
The electronic journal of combinatorics, Tome 17 (2010)
Given integers $n$ and $m=\lfloor\beta n \rfloor$ and a probability measure $Q$ on $\{0, 1,\dots, m\}$, consider the random intersection graph on the vertex set $[n]=\{1,2,\dots, n\}$ where $i,j\in [n]$ are declared adjacent whenever $S(i)\cap S(j)\neq\emptyset$. Here $S(1),\dots, S(n)$ denote the iid random subsets of $[m]$ with the distribution $\bf{P}(S(i)=A)={{m}\choose{|A|}}^{-1}Q(|A|)$, $A\subset [m]$. For sparse random intersection graphs, we establish a first-order asymptotic as $n\to \infty$ for the order of the largest connected component $N_1=n(1-Q(0))\rho+o_P(n)$. Here $\rho$ is the average of nonextinction probabilities of a related multitype Poisson branching process.
@article{10_37236_382,
author = {Mindaugas Bloznelis},
title = {The largest component in an inhomogeneous random intersection graph with clustering},
journal = {The electronic journal of combinatorics},
year = {2010},
volume = {17},
doi = {10.37236/382},
zbl = {1193.05144},
url = {http://geodesic.mathdoc.fr/articles/10.37236/382/}
}
Mindaugas Bloznelis. The largest component in an inhomogeneous random intersection graph with clustering. The electronic journal of combinatorics, Tome 17 (2010). doi: 10.37236/382
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