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.