Continuous-time model of structural balance

Abstract

It is not uncommon for certain social networks to divide into two opposing camps in response to stress. This happens, for example, in networks of political parties during winner-takes-all elections, in networks of companies competing to establish technical standards, and in networks of nations faced with mounting threats of war. A simple model for these two-sided separations is the dynamical system dX/dt = X(2), where X is a matrix of the friendliness or unfriendliness between pairs of nodes in the network. Previous simulations suggested that only two types of behavior were possible for this system: Either all relationships become friendly or two hostile factions emerge. Here we prove that for generic initial conditions, these are indeed the only possible outcomes. Our analysis yields a closed-form expression for faction membership as a function of the initial conditions and implies that the initial amount of friendliness in large social networks (started from random initial conditions) determines whether they will end up in intractable conflict or global harmony.

Fig. 1.

Representative large-n plots of the model for (A) μ > 0 (μ = 3/10 in the plot shown), (B) μ = 0, and (C) μ < 0 (μ = -3 in the plot shown). For all three plots, σ = 1 and n = 90. To reduce image complexity, only one randomly sampled fifth of the trajectories is included. In the second plot, t^∗ denotes the time at which the system diverges, and ε denotes a sufficiently small displacement. The white curves superimposed on the three plots are the large-n trajectories x_ij(t) = x_ij(0) - μ + μ/(1 - μnct) for x_ij(0) = μ,μ ± 3σ/2, where c represents a rescaling of time. Because we want to fix the blow-up time t^∗ near 1 and because ct^∗ = 1/λ₁ as found in the text, we choose c = 1/(μn + ν - μ + σ²/μ) for A and for B and C using estimates of λ₁ taken from ref. . The black dotted lines mark the blow-up times t^∗ = 1/(cλ₁).

Fig. 2.

Tests of the model of Kułakowski et al. (Eq. 1) against two existing datasets. (A) The evolution of the model starting from Zachary’s capacity matrix with the capacity of each relationship reduced by 0.58. This is the minimal downward displacement necessary (to two significant figures) for the resulting separation to be correct for all but 1 of the 34 club members. For reasons described by Zachary (12), this is basically the best separation we can expect. (B) The evolution of the model from Zachary’s capacity matrix with the capacity of zero between the two club leaders replaced by -11; the resulting factions are identical to those in A. Substituted values less than -11 yield the same two factions, whereas greater values produce less accurate divisions. (C) The evolution of the model starting from Axelrod and Bennett’s 1939 propensity(i,j)·size(i)·size(j) matrix for the 17 countries involved in World War II (by Axelrod and Bennett’s definition). The model finds the correct split into Allied and Axis powers with the exceptions of Denmark and Portugal. Axelrod and Bennett’s own landscape theory of aggregation does slightly better—its only misclassification is Portugal.