Algebraic traits of structurally balanced nodes and structurally unbalanced nodes via a geometric-based method

Abstract

A node is a structurally balanced (respectively, unbalanced) node if and only if the absolute value of its corresponding entry in the right eigenvector associated with the zero eigenvalue of the Laplacian matrix is equal to (respectively, is less than) one. The corresponding entry in the right eigenvector is called the algebraic trait for the structurally balanced property of a node. This paper aims to explore the algebraic trait for the structurally balanced property of a node under a signed digraph with a spanning tree by using a geometric-based method. Such a geometric point of view reveals the relationships of the links in a network for structurally balanced and structurally unbalanced nodes. The structurally balanced property of a node is determined by its parent nodes, and the connections between itself and its parent nodes. First of all, a hierarchical structure of its parent nodes is proposed, based on which the Laplacian matrix of the underlying topology can be written as a lower triangular matrix. Then, according to the lower triangular Laplacian matrix, the mathematical expression of the right eigenvector associated with the zero eigenvalue is derived. Based on this, as well as the definition of the structurally balanced node, the algebraic trait for the structurally balanced property of a node is obtained. Finally, a numerical example is provided to verify theoretical results.