On the Number of Non-zero Elements of Joint Degree Vectors

TL;DR

This paper establishes bounds on the number of non-zero elements in joint degree vectors of graphs, impacting the understanding of parameter estimability in exponential random graph models.

Contribution

It provides the first bounds on the maximum non-zero entries in joint degree vectors, linking graph structure to statistical model complexity.

Findings

Derived bounds for non-zero elements in joint degree vectors

Connected bounds to parameter estimation in exponential random graph models

Enhanced understanding of graph degree distribution structures

Abstract

Joint degree vectors give the number of edges between vertices of degree $i$ and degree $j$ for $1\le i\le j\le n-1$ in an $n$-vertex graph. We find lower and upper bounds for the maximum number of nonzero elements in a joint degree vector as a function of $n$. This provides an upper bound on the number of estimable parameters in the exponential random graph model with bidegree-distribution as its sufficient statistics.