The Micro-Macro Link for the Theory of Structural Balance

Abstract

I consider the principle of structural balance that is commonly characterized with the aphorisms: “The friend of a friend is a friend, the friend of an enemy is an enemy, the enemy of a friend is an enemy, and the enemy of an enemy is a friend.” I study what patterns of friendship and hostility emerge at the macro-level when actors at the micro-level make friends and enemies in accordance with this principle. Recent studies have drawn attention to configurations that are imbalanced with many triadic relations violating structural balance yet jammed because no change in sentiment in any one relation can accomplish a net reduction in the number of violations of structural balance. The existence of such jammed states suggests that individual behavior consistent with structural balance need not aggregate to system-wide satisfaction of the principle. To investigate this I employ a best-response model of sentiment change on a fixed social network. I show that under a broad set of model conditions only configurations in which all triadic relations satisfy the structural balance principle can emerge. In a close-knit community in which all actors maintain relations with all other actors such a balanced configuration must consist of either one friendship clique or multiple antagonistic groups.

Keywords:

• balance theory
• dynamic models
• social networks

Acknowledgments

I thank the editors Vincent Buskens, Werner Raub and Marcel van Assen for extraordinarily helpful comments on earlier drafts and for their dedication to this special issue. I also wish to extend my gratitude to Ko Kuwabara, Patricio Gallardo, Tadeusz Sozański and two anonymous reviewers for very useful suggestions.

Notes

¹Heider (1946) defined balance in a more general scenario in which triads can contain both individuals as well as objects and relationships are directed. A separate line of scholarship employs this original notion of balance (e.g., Hummon and Doreian, 2003; Montgomery, 2009).

²Cartwright and Harary (1956) also prove that a signed graph, which may be incomplete, is clusterable into one or two groups if and only if every cycle in the graph (not just every triad) contains an even number of negative edges.

³Cartwright and Harary (1956) consider a graph balanced if all cycles are positive, not just all triads. For complete graphs balance of all triads implies balance of all cycles but for incomplete graphs it does not.

⁴Although the theorem considers m = n − 1, many of the configurations that Antal et al. (2005) and Marvel et al. (2009) identify that are jammed for m = 1 already unjam for intermediate values of m.

⁵This form of consent where permission for a sentiment change is required by all new friends is analogous to the form of consent employed in the unilateral stability concept proposed by Buskens and van de Rijt (2008) where permission for a network change is required by all new contacts.