Dynamical models explaining social balance and evolution of cooperation

Abstract

Social networks with positive and negative links often split into two antagonistic factions. Examples of such a split abound: revolutionaries versus an old regime, Republicans versus Democrats, Axis versus Allies during the second world war, or the Western versus the Eastern bloc during the Cold War. Although this structure, known as social balance, is well understood, it is not clear how such factions emerge. An earlier model could explain the formation of such factions if reputations were assumed to be symmetric. We show this is not the case for non-symmetric reputations, and propose an alternative model which (almost) always leads to social balance, thereby explaining the tendency of social networks to split into two factions. In addition, the alternative model may lead to cooperation when faced with defectors, contrary to the earlier model. The difference between the two models may be understood in terms of the underlying gossiping mechanism: whereas the earlier model assumed that an individual adjusts his opinion about somebody by gossiping about that person with everybody in the network, we assume instead that the individual gossips with that person about everybody. It turns out that the alternative model is able to lead to cooperative behaviour, unlike the previous model.

Figure 1. Social Balance.

The two upper triads are balanced, while the two lower triads are unbalanced. According to the structure theorem , a complete graph can be split into (at most) two opposing factions, if and only if all triads are balanced. This is represented by the coloured matrix on the right, where blue indicates positive entries, and red negative entries.

Figure 2. The two models compared.

The first row illustrates what happens generically for the model , while the second row displays the results for . Each row contains from left to right: (1) an illustration of the model; (2) the random initial state; (3) the dynamics of the model; and (4) the final state to which the dynamics converge. Blue indicates positive entries, and red negative entries. Although the first model converges to a rank one matrix, it is not socially balanced. The second model does converge generically to social balance. The small bumps in the dynamics for are due to complex eigenvalues that show circular behaviour (see Fig. S1).

Figure 3. Prisoner's Dilemma.

Both players have the option to either Cooperate or Defect. Whenever an agent cooperates, it costs him c while his partners receives a benefit b>c, leading to the indicated payoffs.

Figure 4. Evolution of Cooperation.

(A) The fixation probability (probability to be the sole surviving species) is higher for model than . This implies that the model is more viable against defectors, and has an evolutionary advantage compared to . (B) The point b* at which the model has an evolutionary advantage against defectors (i.e. the fixation probability ρ >1/2) depends on the number of agents n. The condition for the model to defeat defectors can be approximated by , with β ≈ 1.72.