Critical scaling limits of the random intersection graph

Abstract: We analyse the scaling limit of the sizes of the largest components of the Random Intersection Graph $G(n,m,p)$ close to the critical point $p=\frac{1}{\sqrt{nm}}$, when the numbers $n$ of individuals and $m$ of communities have different orders of magnitude. We find out that if $m \gg n$, then the scaling limit is identical to the one of the \ER Random Graph (ERRG), while if $n \gg m$ the critical exponent is similar to that of Inhomogeneous Random Graphs with heavy-tailed degree distributions, yet the rescaled component sizes have the same limit in distribution as in the ERRG. This suggests the existence of a wide universality class of inhomogeneous random graph models such that in the critical window the largest components have sizes of order $n^{\rho}$ for some $\rho \in (1/2,2/3]$, which depends on some parameter of the graph.