Algebraic criteria for structure identification and behaviour analysis of signed networks

This paper deals with the problem on how to algebraically identify structure properties and characterise dynamic behaviours of signed networks with cooperative and antagonistic interactions. A signed digraph is employed to represent the two competitive classes of interactions, and an algebraic criterion based on the determinant of its Laplacian matrix is exploited to examine whether it is r-structurally balanced or unbalanced. Given the r-structural balance condition, an indicator vector is constructed to calculate the steady states of signed networks. This vector particularly helps to distinguish rooted nodes and non-rooted nodes from all nodes of signed digraphs, determine the connectivity of signed digraphs and develop algebraic criteria for weight balance and sign-average consensus of signed networks. Further, an algorithm via algebraic manipulations is proposed to identify the structural balance and unbalance of signed digraphs, which contributes to achieving behaviour analysis of signed networks. Simulations are included to illustrate validity of the proposed algebraic criteria.