Heider balance under disordered triadic interactions

M. Bagherikalhor, A. Kargaran, A. H. Shirazi, and G. R. Jafari

Phys. Rev. E 103, 032305 – Published 9 March 2021

Abstract

The Heider balance addresses three-body interactions with the assumption that triads are equally important in the dynamics of the network. In many networks, the relations do not have the same strength, so triads are differently weighted. Now, the question is how social networks evolve to reduce the number of unbalanced triangles when they are weighted? Are the results foreseeable based on what we have already learned from the unweighted balance? To find the solution, we consider a fully connected network in which triads are assigned with different random weights. Weights are coming from Gaussian probability distribution with mean $μ$ and variance $σ$ . We study this system in two regimes: (I) the ratio of $\frac{μ}{σ} \geq 1$ corresponds to weak disorder (small variance) that triads' weight are approximately the same; (II) $\frac{μ}{σ} < 1$ counts for strong disorder (big variance) and weights are remarkably diverse. Investigating the structural evolution of such a network is our intention. We see disorder plays a key role in determining the critical temperature of the system. Using the mean-field method to present an analytic solution for the system represents that the system undergoes a first-order phase transition. For weak disorder, our simulation results display the system reaches the global minimum as temperature decreases, whereas for the second regime, due to the diversity of weights, the system does not manage to reach the global minimum.

Received 4 November 2020
Revised 4 February 2021
Accepted 11 February 2021

DOI:https://doi.org/10.1103/PhysRevE.103.032305

Physics Subject Headings (PhySH)

Evolving networks Network phase transitions

NetworksStatistical Physics & Thermodynamics

Authors & Affiliations

M. Bagherikalhor ^1,*, A. Kargaran ¹, A. H. Shirazi¹, and G. R. Jafari ^1,2,†

¹Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
²Institute for Cognitive and Brain Sciences, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran

^*mahsa.bagherikalhor@gmail.com
^†g_jafari@sbu.ac.ir

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Issue

Vol. 103, Iss. 3 — March 2021

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Images

Figure 1
Solid line (dashed line) represents positive (negative) relationship. $J > 0$ : ${+, +, +}$ , ${+, -, -}$ are balanced, and ${-, -, -}$ , ${+, +, -}$ are unbalanced states. $J < 0$ : ${-, -, -}$ , ${+, +, -}$ are balance, and ${+, +, +}$ , ${+, -, -}$ are unbalanced states. A triangle is called balanced if the product of its weight and links signs, $J_{i j k} S_{i j} S_{j k} S_{k i}$ , is positive and unbalanced if the product is negative.
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Figure 2
Graphical analysis of Eq. (15) for $N = 50$ , $T_{c} ≃ 28$ . For $T_{1} < T_{c}$ the equation has three fixed points which two of them are stable and one is unstable. For $T_{2} > T_{c}$ the equation has only one stable fixed point.
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Figure 3
(a) Bifurcation diagram shows whether the normalized solutions of Eq. (15), $Q^{★}$ , are stable or not. The diagram displays $T^{★ ★}$ , the temperature that the zero changes from a stable to an unstable fixed point, and $T_{c}$ , the critical temperature that indicates the phase transition of the system. The simulation result is also shown to confirm the accuracy of our analytical solution. (b) Bifurcation diagram of Eq. (18) with normalized values and its corresponding simulation result.
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Figure 4
(a) The effective field (the mean value of weighted two-stars), $Q$ , vs. temperature. In high temperatures, each link feels a zero field because we have equal number of positive and negative two-stars. The final state of positive initial condition in low temperature is heaven so, $Q = 1$ . For other initial conditions, the final state is bipolar and an equal ratio of positive and negative effective field results in $Q = 0$ . (b) The energy of network vs. temperature for different initial states. In high temperature energy is zero since the system is in the random state and by decreasing temperature system is led to the balanced state.
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Figure 5
(a) The effective field, Q, vs. temperature. In high temperatures, each link feels a zero field due to the random distribution of two-stars. Because of the diversity of weights in big variance, the behavior of this quantity deviates from the weak disorder case in low temperatures. (b) The energy of the network vs. temperature. The interaction between triads with various weights causes zero energy in high temperatures. In low temperatures, the existence of unbalanced triads causes energy increases so, the system does not reach a balanced state.
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Figure 6
A sample unbalanced triad, $E_{i j k} = 39$ , is depicted. Each link shares in several triads (here, we have shown only three of them). The total energy of triangles on each link is shown, e.g., $E_{i j}$ is the sum of the energy of triangles on the link $i, j$ . The negative value of energy on each link shows that the majority of triads were balanced.
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