The S-metric, the Beichl-Cloteaux approximation, and preferential attachment

Abstract: The S-metric has grown popular in network studies, as a measure of ``scale-freeness'' restricted to the collection G(D) of connected graphs with a common degree sequence D=(d_1,\ldots,d_n). The calculation of S depends on the maximum possible degree assortativity r among graphs in G(D). The original method involves a heuristic construction of a maximally assortative graph g*. The approximation by Beichl and Cloteaux involves constructing a possibly disconnected graph g' with r(g') >= r(g*) and requires O(n^2) tests for the graphicality of a degree sequence. The present paper uses the Tripathi-Vijay test to streamline this approximation, and thereby to investigate two collections of graphs: Barabasi-Albert trees and coauthorship graphs of mathematical sciences researchers. Long-term trends in the coauthorship graphs are discussed, and contextualized by insights derived from the BA trees. It is known that greater degree-based preferential attachment produces greater variance in degree sequences, and these trees exhibited assortativities restricted to a narrow band. In contrast, variance in degree rose over time in the coauthorship graphs in spite of weakening degree-based preferential attachment. These observations and their implications are discussed and avenues of future work are suggested.