Component Evolution in General Random Intersection Graphs

TL;DR

This paper studies how components form and grow in general random intersection graphs, providing conditions for the emergence of a giant component using advanced branching process techniques.

Contribution

It extends existing methods to analyze component evolution in RIGs with dependent, inhomogeneous structures, addressing the complex interplay of nodes and attributes.

Findings

Conditions for the existence of a giant component

Analysis of survival and extinction in branching processes

Generalization of Erdős-Rényi component analysis techniques

Abstract

Random intersection graphs (RIGs) are an important random structure with applications in social networks, epidemic networks, blog readership, and wireless sensor networks. RIGs can be interpreted as a model for large randomly formed non-metric data sets. We analyze the component evolution in general RIGs, and give conditions on existence and uniqueness of the giant component. Our techniques generalize existing methods for analysis of component evolution: we analyze survival and extinction properties of a dependent, inhomogeneous Galton-Watson branching process on general RIGs. Our analysis relies on bounding the branching processes and inherits the fundamental concepts of the study of component evolution in Erd\H{o}s-R\'enyi graphs. The major challenge comes from the underlying structure of RIGs, which involves its both the set of nodes and the set of attributes, as well as the set of different probabilities among the nodes and attributes.