A General Formalism for Inhomogeneous Random Graphs

TL;DR

This paper introduces a comprehensive framework for inhomogeneous random graphs where vertex types influence edge formation, enabling analysis of diverse models and their phase transitions.

Contribution

It extends classical random graph theory to include vertex types and type-dependent edge probabilities, broadening analytical capabilities.

Findings

Derived the phase structure using generating functions

Established relations to other graph models

Provided a unified framework for inhomogeneous graphs

Abstract

We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its terminal vertices. This approach provides a general framework for the analysis of a large class of models. The generic phase structure is derived using generating function techniques, and relations to other classes of models are pointed out.