Graph realizations constrained by skeleton graphs

TL;DR

This paper introduces a new skeleton graph model for constrained graph realizations, generalizing existing models like JDM and PAM, and focuses on cases with few classes or specific topologies.

Contribution

The paper proposes the skeleton graph model as a more convenient alternative to PAM for certain network constraints, especially with few classes or specific topologies.

Findings

Model effectively describes networks with two classes or one-cycle skeletons.

Simplifies analysis of constrained graph realizations.

Provides a framework for handling topological restrictions.

Abstract

In 2008 Amanatidis, Green and Mihail introduced the Joint Degree Matrix (JDM) model to capture the fundamental difference in assortativity of networks in nature studied by the physical and life sciences and social networks studied in the social sciences. In 2014 Czabarka proposed a direct generalization of the JDM model, the Partition Adjacency Matrix (PAM) model. In the PAM model the vertices have specified degrees, and the vertex set itself is partitioned into classes. For each pair of vertex classes the number of edges between the classes in a graph realization is prescribed. In this paper we apply the new {\em skeleton graph} model to describe the same information as the PAM model. Our model is more convenient for handling problems with low number of partition classes or with special topological restrictions among the classes. We investigate two particular cases in detail: (i) when there are only two vertex classes and (ii) when the skeleton graph contains at most one cycle.