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 collaborators (Fuller, 2016; 2012; Zhao et al. 2011). particular show that algorithm is deeply connected with a certain homogeneous coherent configuration, non-commuting generalization classical Johnson scheme. This algebraic underlies surprising, previously unobserved, identities satisfied by covariance matrices in SONETS model. We an understanding leads to simplified numerical methods for carrying out linear algebra required implement algorithm. also method can be substantially generalized allow different types vertices and/or edges, these generalizations enjoy similar structure.