Comment on "Mean-field solution of structural balance dynamics in nonzero temperature"

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 critical 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 $a^c\approx 1.71649$ 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.Comment: 7 pages, 6 figures, 1 tabl

Glassy States of Aging Social Networks

Individuals often develop reluctance to change their social relations, called “secondary homebody”, even though their interactions with their environment evolve with time. Some memory effect is loosely present deforcing changes. In other words, in the presence of memory, relations do not change easily. In order to investigate some history or memory effect on social networks, we introduce a temporal kernel function into the Heider conventional balance theory, allowing for the “quality” of past relations to contribute to the evolution of the system. This memory effect is shown to lead to the emergence of aged networks, thereby perfectly describing—and what is more, measuring—the aging process of links (“social relations”). It is shown that such a memory does not change the dynamical attractors of the system, but does prolong the time necessary to reach the “balanced states”. The general trend goes toward obtaining either global (“paradise” or “bipolar”) or local (“jammed”) balanced states, but is profoundly affected by aged relations. The resistance of elder links against changes decelerates the evolution of the system and traps it into so named glassy states. In contrast to balance configurations which live on stable states, such long-lived glassy states can survive in unstable states