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

Abstract

Joint degree vectors give the number of edges between vertices of degree i and degree j for 1 ≤ i ≤ j ≤ 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.

Assortativity and bidegree distributions on Bernoulli random graph superpositions

“…Assortativity and bidegree distributions have earlier been analyzed in the context of random intersection graph models [8,10], inhomogeneous Bernoulli graphs and their extensions [13,35,42], preferential attachment models [32,44], and configuration models in [44][45][46]. Extremal properties of bidegree correlations in general graphs have been reported in [17,44].…”

Assortativity and bidegree distributions on Bernoulli random graph superpositions

“…Assortativity and bidegree distributions have earlier been analyzed in the context of random intersection graph models [8,9], inhomogeneous Bernoulli graphs and their extensions [12,31,37], preferential attachment models [28,39], and configuration models in [39,40,41]. Extremal properties of bidegree correlations in general graphs have been reported in [16,39].…”

An algebraic Monte-Carlo algorithm for the Partition Adjacency Matrix realization problem

“…Note that the JDM also specifies the degree sequence itself, uniquely [5]. The JDM received considerable attention in the literature [1,2,5,9,7,21,23,25,27] and it is well understood [1,2,5,21,27]. Reference [3] presents an exact algorithm for constructing simple graphs with a prescribed JDM.…”