Assessing frustration in real-world signed networks: towards a statistical theory of balance

Presentation Slides (PDF)

Abstract

The abundance of data about social and economic relationships has opened an era in which social theories can be tested against empirical evidence, allowing human behaviour to be analysed like other natural phenomena. Here we focus on balance theory [1], which postulates coherence between positive and negative interactions in real, social networks. To make balance theory testable, one needs 1) a proper representation of social networks, 2) a definition of “frustration” (or imbalance) and 3) a set of null models to quantify the statistical significance of the latter. The first two ingredients have been already explored comprehensively: social interactions can be represented via signed graphs and “frustrated” configurations are defined as those having cycles with an odd number of negative links. The third ingredient, however, is way less developed, since the existing null models typically do not account for the different, intrinsic tendencies of individual actors to establish positive and negative interactions. To reduce this gap, here we extend the ERG framework to binary, undirected, signed networks with both global and local constraints. Moreover, we define two variants for each benchmark: one where the topology is kept fixed and one where it is left to vary along with the edge signs.

Our analysis shows that homogeneous null models with global constraints tend to favour the weak version of balance theory, according to which only the triangle with one, negative link should be under-represented in real networks; on the other hand, heterogeneous null models with local constraints tend to favour the strong version, according to which the triangle with all negative links should be under-represented as well. As a comparison, biological networks display almost inverted patterns, confirming that structural balance inherently distinguishes social networks from other types of signed networks.

The generalisation of the concept of balance at the mesoscopic scale leads to the so-called k-balance theory, which interprets a generic graph as structurally balanced if its vertex set can be partitioned into k subsets (or modules) with positive, intra-modular links and negative, inter-modular links [1]. If adopted as a strict “yes/no” criterion, however, k-balance theory would too quickly interpret the vast majority of real-world networks as “frustrated”; therefore, we again make use of our null models to assess the significance of the empirical deviation from ideal k-balanced configurations and propose a statistically grounded test of hypothesis. Along the way, we show that the signed analogue of the resolution limit for community structure can be naturally re-interpreted as a threshold-based criterion for discerning whether a given, signed configuration is balanced. Finally, we advance a proposal for the statistical characterisation of the so-called relaxed balance theory [2] and compare the relaxed and traditional variants in terms of their performance [3].

Relevant publications

[1] Davis. Human Relations 20(2), 181-187 (1967).

[2] Doreian, Mrvar. Social Networks 31, 1-11 (2009).

[3] Gallo, Garlaschelli, Lambiotte, Saracco, Squartini. arXiv:2303.07023 (2023).

[4] Gallo, Garlaschelli, Squartini. arXiv:2404.15914 (2024).