Collusion Relations and their Applications to Balance Theory

TL;DR

This paper generalizes balance theory by characterizing collusivity in signed relations, providing conditions for network polarization, and extending the theory to non-symmetric relations with modal logic tools.

Contribution

It introduces a quadrangular property called collusivity, extends balance theory to non-symmetric relations, and offers a modal framework for analyzing collusive frames.

Findings

Characterizes balance in signed frames via collusivity.

Extends balance theorem to non-symmetric relations.

Provides a modal characterization with sequent calculus rules.

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.