The shortest distance in random multi-type intersection graphs

TL;DR

This paper analyzes the typical distances in multitype random intersection graphs, revealing they follow a mixture of Gumbel distributions with a defect accounting for disconnected vertices.

Contribution

It introduces a new approximation method using branching processes to describe the distribution of distances in multitype intersection graphs.

Findings

Distances follow a mixture of Gumbel distributions

Disconnection probability is captured by a missing mass in the distribution

Provides a detailed probabilistic description of graph connectivity

Abstract

Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multitype random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph.