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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 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 $a^capprox 1.71649$ 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.
With the availability of cell phones, internet, social media etc. the interconnectedness of people within most societies has increased drastically over the past three decades. Across the same timespan, we are observing the phenomenon of increasing levels of fragmentation in society into relatively small and isolated groups that have been termed filter bubbles, or echo chambers. These pose a number of threats to open societies, in particular, a radicalisation in political, social or cultural issues, and a limited access to facts. In this paper we show that these two phenomena might be tightly related. We study a simple stochastic co-evolutionary model of a society of interacting people. People are not only able to update their opinions within their social context, but can also update their social links from collaborative to hostile, and vice versa. The latter is implemented such that social balance is realised. We find that there exists a critical level of interconnectedness, above which society fragments into small sub-communities that are positively linked within and hostile towards other groups. We argue that the existence of a critical communication density is a universal phenomenon in all societies that exhibit social balance. The necessity arises from the underlying mathematical structure of a phase transition phenomenon that is known from the theory of a kind of disordered magnets called spin glasses. We discuss the consequences of this phase transition for social fragmentation in society.
Heterogeneous adoption thresholds exist widely in social contagions, but were always neglected in previous studies. We first propose a non-Markovian spreading threshold model with general adoption threshold distribution. In order to understand the effects of heterogeneous adoption thresholds quantitatively, an edge-based compartmental theory is developed for the proposed model. We use a binary spreading threshold model as a specific example, in which some individuals have a low adoption threshold (i.e., activists) while the remaining ones hold a relatively high adoption threshold (i.e., bigots), to demonstrate that heterogeneous adoption thresholds markedly affect the final adoption size and phase transition. Interestingly, the first-order, second-order and hybrid phase transitions can be found in the system. More importantly, there are two different kinds of crossover phenomena in phase transition for distinct values of bigots adoption threshold: a change from first-order or hybrid phase transition to the second-order phase transition. The theoretical predictions based on the suggested theory agree very well with the results of numerical simulations.
In this paper, we propose a Boltzmann-type kinetic description of opinion formation on social networks, which takes into account a general connectivity distribution of the individuals. We consider opinion exchange processes inspired by the Sznajd model and related simplifications but we do not assume that individuals interact on a regular lattice. Instead, we describe the structure of the social network statistically, assuming that the number of contacts of a given individual determines the probability that their opinion reaches and influences the opinion of another individual. From the kinetic description of the system, we study the evolution of the mean opinion, whence we find precise analytical conditions under which phase transitions, i.e. changes of sign between the initial and the asymptotic mean opinions, occur. Furthermore, we show that a non-zero correlation between the initial opinions and the connectivity of the individuals is necessary to observe phase transitions. Finally, we validate our analytical results through Monte Carlo simulations of the stochastic opinion exchange processes on the social network.
Social network based information campaigns can be used for promoting beneficial health behaviours and mitigating polarisation (e.g. regarding climate change or vaccines). Network-based intervention strategies typically rely on full knowledge of network structure. It is largely not possible or desirable to obtain population-level social network data due to availability and privacy issues. It is easier to obtain information about individuals attributes (e.g. age, income), which are jointly informative of an individuals opinions and their social network position. We investigate strategies for influencing the system state in a statistical mechanics based model of opinion formation. Using synthetic and data based examples we illustrate the advantages of implementing coarse-grained influence strategies on Ising models with modular structure in the presence of external fields. Our work provides a scalable methodology for influencing Ising systems on large graphs and the first exploration of the Ising influence problem in the presence of ambient (social) fields. By exploiting the observation that strong ambient fields can simplify control of networked dynamics, our findings open the possibility of efficiently computing and implementing public information campaigns using insights from social network theory without costly or invasive levels of data collection.
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