On the relationship between Heider links and Ising spins

Abstract

We show that the Heider model with an external field is equivalent, in the limit of structural balance, to the Ising model with nearest-neighbor interactions without an external field. More precisely, we claim that the signs of the Heider relations that maintain structural equilibrium in the system can be represented as nearest neighbor Ising spin products. We demonstrate this explicitly for a complete graph and provide a general argument for an arbitrary graph. A consequence of the equivalence is that the system of balanced Heider states undergoes a phase transition, inherited from the Ising model, at a critical value of the social field at which the fluctuations of edge magnetization are maximal.

Figures (4)

• Figure 1: Susceptibility (normalized variance) $\chi_m=\left(\langle M^2 \rangle - \langle M \rangle^2\right)/N$ for balanced states in the Heider model on the complete graph $K_N$. The solid lines were calculated analytically for $N=20,100,500$ (red, blue, yellow), thanks to the equivalence with the Ising model, from \ref{['eq:Zfs1']} for finite $N$. The green solid line was calculated from the saddle point equations (\ref{['eq:sp1']}, \ref{['eq:sp2']}) for $N=\infty$. The symbols, with the corresponding colors, were obtained using Monte Carlo simulations of the Heider model with a very large $\varepsilon'=1000$ to mimic $\varepsilon'=\infty$, where the system is in Heider balance, and the equivalence to the Ising model is exact
• Figure 2: Evolution of the normalized variance $\chi_m$ with $\varepsilon'$, for $N=20$. The curves are obtained by Monte Carlo simulations. For $N=20$, the critical value is $\varepsilon'_{cr} \approx 6$, see Fig. \ref{['fig:epscr']}. For $\varepsilon' > \varepsilon'_{cr}$, the curves quickly converge to a limit curve. For $\varepsilon' < \varepsilon'_{cr}$, the curves broaden when $\varepsilon'$ increases; they also broaden when $N$ increases although this is not shown in the figure
• Figure 3: Histogram of the probability density function $\rho(p)$ with the bin size $0.04$ for $N=20$, for three different values of $\varepsilon'=5.88$, $6.02$, and $6.18$. The mean values (which correspond to the positions of the histogram mass centers) are $\langle p \rangle = 0.3$, $0.5$, and $0.7$ respectively. The functions have a bimodal shape typical for the first order phase transition. Changing $\varepsilon'$ changes proportions of the two phases. The pseudo-critical value $\varepsilon'_{cr}$ is defined as a value of $\varepsilon'$ for which $\langle p \rangle=1/2$
• Figure 4: The pseudo-critical value $\varepsilon'_{cr}$ for different values of $N$, for $h'=0$. The phase transition is first order. At the phase transition two phases co-exist: one with $p<1/2$ and the other with $p>1/2$, see Fig. \ref{['fig:hist']}. The pseudo-critical values $\varepsilon'_{cr}$ shown as points for different $N$, are obtained from the condition that the mean (the center of mass of the histogram) is $\langle p \rangle =1/2$. The line $\varepsilon'_{cr}=(N-2)/3$, drawn as a solid line, passes through the points