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 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.
Highlights
In the last fifteen years, the exponential development of network theory has raised the practical problem of realizing and sampling large graphs with given degree sequences
The problem of determining if there exists a graph with given degree sequence and satisfying other specified conditions will be called in this paper the realization problem
The existence problem for the Joint Degree Matrix (JDM) model is not hard: already Patrinos and Hakimi [17] presented in 1976 an Erdos-Gallai-type theorem for joint degree matrices, essentially characterizing precisely those matrices which are the joint degree matrix for some graph, though using different terminology
Summary
In the last fifteen years, the exponential development of network theory has raised the practical problem of realizing and sampling large graphs with given degree sequences. The existence problem for the JDM model is not hard: already Patrinos and Hakimi [17] presented in 1976 an Erdos-Gallai-type theorem for joint degree matrices, essentially characterizing precisely those matrices which are the joint degree matrix for some graph, though using different terminology. Another proof for this result was given in [1], see [2]. The JDM model suggests a more general restricted degree sequence problem: the partition adjacency matrix model (or PAM for short). The answers in both cases are almost affirmative: the space is connected if we use swaps as well as an additional operation called double swaps
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