Motifs, Coherent Configurations and Second Order Network Generation

TL;DR

This paper reveals the algebraic structure behind the SONETS random graph model, connecting it to homogeneous coherent configurations, which simplifies computations and extends to other graph types.

Contribution

It identifies a deep algebraic connection between SONETS and homogeneous coherent configurations, leading to new insights and computational simplifications.

Findings

Identifies the link between SONETS and homogeneous coherent configurations.

Derives identities satisfied by covariance matrices in SONETS.

Extends the algebraic framework to generate subgraphs of various graph types.

Abstract

In this paper we illuminate some algebraic-combinatorial structure underlying the second order networks (SONETS) random graph model of Nykamp, Zhao and collaborators. In particular we show that this algorithm is deeply connected with a certain homogeneous coherent configuration, a non-commuting generalization of the classical Johnson scheme. This algebraic structure underlies certain surprising identities (that do not appear to have been previously observed) satisfied by the covariance matrices in the Nykamp-Zhao scheme. We show that an understanding of this algebraic structure leads to simplified numerical methods for carrying out the linear algebra required to implement the SONETS algorithm. We also show that this structure extends naturally to the problem of generating random subgraphs of graphs other than the complete directed graph.