The largest component of the random intersection graph in the subcritical regime

Abstract

We study the size of the largest component of the randomintersection graph $G(n,m,p)$ with $m=[n^r]$ in the subcritical regime, i.e., the case where the expected vertex degree is smaller than one. If $r>1$, then the largest component and the largest tree component have the same type of weak law of large numbers, and both of them are of order $\Theta(\log~n)$; if $r=1$, then the largest component is not a tree, but the largest component and the largest tree component are also of order $\Theta(\log~n)$; if $0<r<1$, then the largest tree component is of order $o(\log~n)$while the largest component is of order $\Theta(np\log~n)$.