Modal logics and group polarization

This paper proposes different ways of modally defining properties related to the concept of balance in signed social networks where relations can be either positive or negative. The motivation is to be able to formally reason about the social phenomenon of group polarization based on balance theory. The starting point is a recently developed basic modal logic that axiomatizes the class of social networks that are balanced up to a certain degree. This property is not modally definable but can be captured using a deduction rule. In this work, we examine different possibilities for extending this basic language to define frame properties such as balance and related properties such as non-overlapping positive and negative relations and collective connectedness as axioms. Furthermore, we define the property of full balance rather than balanced-up-to-a-degree. We look into the complexity of the model checking problem and show a non-compactness result of the extended language. Along the way, we provide axioms for weak balance. We also look at a full hybrid extension and reason about network changes with dynamic modalities. Then, to explore the measures of how far a network is from polarization, we consider variations of measures in relation to balance.