Geometry of a large random intersection graph inside the critical window

Título: Geometry of a large random intersection graph inside the critical window

Resumen: Random intersection graph is a simple random graph model that incorporates community structures. To build such a graph, imagine there are n individuals and m potential communities. Each individual joins a community independently with probability p. The graph G(n, m, p) has n nodes, corresponding to the n individuals. Each pair of these individuals share an edge between them if they belong to a common community. The critical threshold for the emergence of a giant component turns out to be at p^2 ~ 1/nm. I’ll discuss some results that can help us to understand what a large G(n, m, p) looks like at the critical threshold.