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تسجيل مستخدم جديدComment on `Phase transition in a network model of social balance with Glauber dynamics
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In a recent work [R. Shojaei et al, Physical Review E 100, 022303 (2019)] the Authors calculate numerically the critical temperature $T_c$ of the balanced-imbalanced phase transition in a fully connected graph. According to their findings, $T_c$ decreases with the number of nodes $N$. Here we calculate the same critical temperature using the heat-bath algorithm. We show that $T_c$ increases with $N$ as $N^{gamma}$, with $gamma$ close to 0.5 or 1.0. This value depends on the initial fraction of positive bonds.
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In recent numerical and analytical studies, Rabbani {it et al.} [Phys. Rev. E {bf 99}, 062302 (2019)] observed the first-order phase transition in social triads dynamics on complete graph with $N=50$ nodes. With Metropolis algorithm they found critic
al temperature on such graph equal to 26.2. In this comment we extend their main observation in more compact and natural manner. In contrast to the commented paper we estimate critical temperature $T^c$ for complete graph not only with $N=50$ nodes but for any size of the system. We have derived formula for critical temperature $T^c=(N-2)/a^c$, where $N$ is the number of graph nodes and $a^capprox 1.71649$ comes from combination of heat-bath and mean-field approximation. Our computer simulation based on heat-bath algorithm confirm our analytical results and recover critical temperature $T^c$ obtained earlier also for $N=50$ and for systems with other sizes. Additionally, we have identified---not observed in commented paper---phase of the system, where the mean value of links is zero but the system energy is minimal since the network contains only balanced triangles with all positive links or with two negative links. Such a phase corresponds to dividing the set of agents into two coexisting hostile groups and it exists only in low temperatures.
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