Enumeration and structure of inhomogeneous graphs

TL;DR

This paper studies a weighted inhomogeneous graph model, analyzing their structure and enumeration, especially for sparse graphs, and applies findings to properly colored graphs and social network models.

Contribution

It introduces a generalized weighted inhomogeneous graph model, providing new enumeration proofs and structural insights, with applications to social networks.

Findings

Almost all sparse graphs contain no component with more than one cycle.

Provides a new proof of Wright's enumeration theorem for properly colored graphs.

Discusses applications to social network modeling.

Abstract

We analyze a general model of weighted graphs, introduced by de Panafieu and Ravelomanana (2014) and similar to the "inhomogeneous graph model" of S\"oderberg (2002). Each vertex receives a "type" among a set of $q$ possibilities as well as a "weight" corresponding to this type, and each edge is weighted according to the types of the vertices it links. The weight of the graph is then the product of the weights of its vertices and edges. We investigate the sum of the weights of such graphs and prove that when the number of edges is small, almost all of them contain no component with more than one cycle. Those results allow us to give a new proof in a more general setting of a theorem of Wright (1961) on the enumeration of properly colored graphs. We also discuss applications related to social networks.