Collusion Relations and their Applications to Balance Theory

2026-05-03

AI Summary

This study addresses polarization structures and group consensus in asymmetric social networks. By introducing the notion of “collusiveness” and analyzing quadruple relational patterns, the authors generalize classical balance theory to asymmetric signed networks and establish equivalent conditions between collusiveness and balance. The work proposes collusiveness as a novel criterion for identifying internally cohesive yet externally antagonistic groups, offering both a modal logical characterization and a labeled sequent calculus system. This framework not only effectively detects polarized structures in asymmetric networks but also unifies and extends existing modal logic formalizations of balance.

Abstract

We study quadrangular properties of binary relations on a set $X$~--i.e., properties defined on configurations of four elements--~within an agonistic interpretation, where $xRy$ is interpreted as $x$ ``attacks''~$y$. Such relations induce a suitable notion of ``protection,'' and we provide necessary and sufficient conditions for this notion to be consistent. We characterize the balance property in signed frames in terms of a specific quadrangular property, namely collusivity. In this way, we generalize a classical result in balance theory by offering an alternative method for determining whether a network is polarized. That is, one can identify well-formed groups of agents that agree with one another within the same group (a set of allies) while disagreeing with, or attacking, agents outside the group. Furthermore, we extend the balance theorem to non-symmetric relations, thereby relaxing a condition required in standard balance theory. We conclude by giving a modal characterization of collusive frames, together with corresponding rules in a labeled sequent calculus, and we show that previous modal characterizations of balance are derivable within this system.

Problem

• balance theory
• collusion
• quadrangular property
• signed frames
• non-symmetric relations

Innovation

• collusivity
• balance theory
• quadrangular properties
• modal characterization
• non-symmetric relations