Dynamic social balance and convergent appraisals via homophily and influence mechanisms

Abstract

Social balance theory describes allowable and forbidden configurations of the topologies of signed directed social appraisal networks. In this paper, we propose two discrete-time dynamical systems that explain how an appraisal network converges to social balance from an initially unbalanced configuration. These two models are based on two different socio-psychological mechanisms respectively: the homophily mechanism and the influence mechanism. Our main theoretical contribution is a comprehensive analysis for both models in three steps. First, we establish the well-posedness and bounded evolution of the interpersonal appraisals. Second, we fully characterize the set of equilibrium points; for both models, each equilibrium network is composed of an arbitrary number of complete subgraphs satisfying structural balance. Third, we establish the equivalence among three distinct properties: non-vanishing appraisals, convergence to all-to-all appraisal networks, and finitetime achievement of social balance. In addition to theoretical analysis, Monte Carlo validations illustrate how the non-vanishing appraisal condition holds for generic initial conditions in both models. Moreover, a numerical comparison between the two models indicates that the homophily-based model might be a more universal explanation for the emergence of social balance. Finally, adopting the homophily-based model, we present numerical results on the mediation and globalization of local conflicts, the competition for allies, and the asymptotic formation of a single versus two factions.

Structural Balance via Gradient Flows over Signed Graphs

“…The following facts are known about the Ku lakowski et al influencebased and the Traag et al homophily-based models: the set of well-behaved initial conditions that lead the social network towards social balance for the first model is a subset of the set of normal matrices, while the second model can work under generic initial conditions. Similar results are obtained by [23] for two discrete-time models based on influence and homophily respectively: influence-based processes do not perform well under generic initial conditions (in contrast to the homophily-based processes). Finally, only the models proposed in [23] and a variation of the model by Ku lakowski et al proposed in the early work [16], have a bounded evolution of appraisals, whereas the others have finite escape time.…”

“…Similar results are obtained by [23] for two discrete-time models based on influence and homophily respectively: influence-based processes do not perform well under generic initial conditions (in contrast to the homophily-based processes). Finally, only the models proposed in [23] and a variation of the model by Ku lakowski et al proposed in the early work [16], have a bounded evolution of appraisals, whereas the others have finite escape time.…”

“…, which entails (23). In the case of α = 0, one has p i = 2 √ β i , q i = √ β i for any i, and (29) implies that i q 2 i n i = 1.…”

The effect of social balance on social fragmentation

“…Heider's balance was later extended to a global balance, where a society is strongly (weakly) balanced 2 if it can be partitioned into two (multiple) mutually antagonistic groups [8][9][10]. The evolution of social networks towards a balanced state has been shown in [11][12][13][14][15][16][17][18][19][20][21][22]. For an extensive review on the use of social balance theory in sociology, social psychology and anthropology, see [23], for a list of its empirical evidence, e.g.…”

Minimum Memory-Based Sign Adjustment in Signed Social Networks

“…Many different assumptions on stress reduction in signed networks under local adjustment have been designed to check whether the network will reach a global balance [ 5 , 6 , 9 , 13 , 16 , 17 , 18 , 19 , 20 , 21 ]. Rules which mix imbalance stress reduction with homophily and other bilateral pressures have also been proposed, such as those found in [ 8 , 15 , 22 , 23 ].…”