Algebraic topological characterizations of structural balance in signed graphs

Abstract

Algebraic topological characterizations of the structural balance in signed graphs are considered in this work. Leveraging tools from the algebraic topology, simplicial complexes are used instead of conventional graphical approaches to model signed graphs and capture their global topological properties. Topological invariants, such as homology and cohomology, are then explored to extract topological characterizations of the structural balance from the simplicial-complex-based models. The developed topological characterizations reveal that the structural balance is closely related to the first homology and cohomology of the modeled simplicial complex. Examples are provided to demonstrate the developed topological insights of the structural balance and how it can be leveraged to construct networks with desired topologies.