Motifs, coherent configurations and second order network generation

In this paper we illuminate some algebraic-combinatorial structure underlying the second order networks (SONETS) random graph model of Zhao, Beverlin, Netoff and Nykamp and collaborators (Fuller, 2016; Zhao, 2012; Zhao et al. 2011). 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, previously unobserved, identities satisfied by the covariance matrices in the SONETS model. 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 the SONETS method can be substantially generalized to allow different types of vertices and/or edges, and that these generalizations enjoy similar algebraic structure. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This paper reveals the algebraic-combinatorial structure underlying the second order networks (SONETS) random graph model proposed by Zhao et al., and demonstrates its connection with a non-commuting generalization of the classical Johnson scheme. The study shows that the algebraic structure leads to simplified numerical methods for implementing the SONETS algorithm, and that the method can be generalized to different types of vertices and edges while maintaining similar algebraic structure.