Rethinking Structural Balance in Signed Social Networks

Estrada, E.
Discrete Applied Mathematics, , 268, 2019, 70-90.

Signed social networks account for the situation in which friendship, trust, alliance – represented by positive edges – coexist with relations of dislike, distrust,conflict — represented by negative edges. According to Heider’s hypothesis of balance certain combinations of positive and negative edges in a network confer stability-balance-to the networks, while others – imbalanced networks – are unstable and tend towards balance. Here, we review the concept of structural balance starting from “first principles” based on the original formulation of Heider. Based on Harary’s “tendency towards completeness” hypothesis we formulate that a network is balanced if it can give rise to a balanced complete signed graph. We then propose an algorithm for edge completion of networks and prove several results concerning the balance of cycles, and graphs. We then consider balance as a dynamic process in which the entities of the system try to make agreements by pairs to reach consensus or “agreed upon dissensus”. At this point we arrive at a few general conditions that any index quantifying the degree of balance – or degree of imbalance – has to have. We then analyze a degree of balance index proposed by Estrada and Benzi (2014) and show that it fulfills all these requirements. In contrast, we show how some other approaches to quantify the degree of balance are incomplete and we provide examples of the difficulties found with their use. Using all these developed conceptions about the level of balance in signed social networks we proceed to the study of the international relation among countries in the world for the period 1938–2008. We conclude that the system of international relations is highly imbalanced for the whole period with no trend to increasing balance with the passage of time. Finally, we consider the use of degree of balance indices relative to null models and clarify that they do not account for the degree of balance of a signed network but for the “effort” needed to create such degree of balance among a large set of randomly shuffled networks.

Signed social networks account for the situation in which friendship, trust, alliance – represented by positive edges – coexist with relations of dislike, distrust,conflict — represented by negative edges. According to Heider’s hypothesis of balance certain combinations of positive and negative edges in a network confer stability-balance-to the networks, while others – imbalanced networks – are unstable and tend towards balance. Here, we review the concept of structural balance starting from “first principles” based on the original formulation of Heider. Based on Harary’s “tendency towards completeness” hypothesis we formulate that a network is balanced if it can give rise to a balanced complete signed graph. We then propose an algorithm for edge completion of networks and prove several results concerning the balance of cycles, and graphs. We then consider balance as a dynamic process in which the entities of the system try to make agreements by pairs to reach consensus or “agreed upon dissensus”. At this point we arrive at a few general conditions that any index quantifying the degree of balance – or degree of imbalance – has to have. We then analyze a degree of balance index proposed by Estrada and Benzi (2014) and show that it fulfills all these requirements. In contrast, we show how some other approaches to quantify the degree of balance are incomplete and we provide examples of the difficulties found with their use. Using all these developed conceptions about the level of balance in signed social networks we proceed to the study of the international relation among countries in the world for the period 1938–2008. We conclude that the system of international relations is highly imbalanced for the whole period with no trend to increasing balance with the passage of time. Finally, we consider the use of degree of balance indices relative to null models and clarify that they do not account for the degree of balance of a signed network but for the “effort” needed to create such degree of balance among a large set of randomly shuffled networks.