Mean-field solution for critical behavior of signed networks in competitive balance theory

Response of the competitive balance model to the external field

The competitive balance model was proposed as an extension of the structural balance theory, aiming to account for heterogeneities observed in real-world networks. In this model, different paradigms lead to form different friendship and enmity. As an example, friendship or enmity between countries can have a political or religious basis. The suggested Hamiltonian is symmetrical between paradigms. Our analyses show that a balanced state can be achieved if just one paradigm prevails in the network and the paradigm shift is possible only by imposing an external field. In this paper, we investigate the influence of the external field on the evolution of the network. We drive the mean-field solutions of the model and verify the accuracy of our analytical solutions by performing Monte-Carlo simulations. We observe that the external field breaks the symmetry of the system. The response of the system to this external field, contingent upon temperature, can be either paramagnetic or ferromagnetic. We observed a hysteresis behavior in the ferromagnetic regime. Once communities are formed based on a certain paradigm, then they resist change. We found that to avoid wasting energy we need to know the level of stochastic behavior in the network. Analogous to magnetic systems, we observe that susceptibility adheres to Curie's law.

Financial crisis in the framework of non-zero temperature balance theory

In financial crises, assets see a deep loss of value, and the financial markets experience liquidity shortages. Although they are not uncommon, they may cause by multiple contributing factors which makes them hard to study. To discover features of the financial network, the pairwise interaction of stocks has been considered in many pieces of research, but the existence of the strong correlation between stocks and their collective behavior in crisis made us address higher-order interactions. Hence, in this study, we investigate financial networks by triplet interaction in the framework of balance theory. Due to detecting the contribution of higher-order interactions in understanding the complex behavior of stocks we take the advantage of the order parameter of the higher-order interactions. Looking at real data of the financial market obtained from S&P500 index(SPX) through the lens of balance theory for the quest of network structure in different periods (on and off-crisis) faces us with the existence of a structural difference of networks corresponding to the periods. Addressing two well-known crises the Great regression (2008) and the Covid-19 recession (2020), our results show an ordered structure forms in the on-crisis period in the financial network while stocks behave independently far from a crisis. The formation of the ordered structure of stocks in crisis makes the network more resilient to disorder (thermal fluctuations). The resistance of the ordered structure against applying the disorder measure the crisis strength and determine the temperature at which the network transits. There is a critical temperature, Tc, in the language of statistical mechanics and mean-field approach which above, the ordered structure destroys abruptly and a first-order phase transition occurs. The stronger the crisis, the higher the critical temperature.

Modified Heider balance on Erdös-Rényi networks

The lack of signed random networks in standard balance studies has prompted us to extend the Hamiltonian of the standard balance model. Random networks with tunable parameters are suitable for better understanding the behavior of standard balance as an underlying dynamics. Moreover, the standard balance model in its original form does not allow preserving tensed triads in the network. Therefore, the thermal behavior of the balance model has been investigated on a fully connected signed network recently. It has been shown that the model undergoes an abrupt phase transition with temperature. Considering these two issues, we examine the thermal behavior of the structural balance model defined on Erdös-Rényi random networks within the range of their connected regime. We provide a mean-field solution for the model. We observe a first-order phase transition with temperature for a wide range of connection probabilities. We detect two transition temperatures, T_{cold} and T_{hot}, characterizing a hysteresis loop. We find that with decreasing the connection probability, both T_{cold} and T_{hot} decrease. However, the slope of decreasing T_{hot} with decreasing connection probability is larger than the slope of decreasing T_{cold}. Hence, the hysteresis region gets narrower until it disappears in a certain connection probability. We provide a phase diagram in the temperature-tie density plane to accurately observe the metastable or coexistence region behavior. Then we justify our mean-field results with a series of Monte Carlo simulations.

Hybrid balance theory: Heider balance under higher-order interactions

Heider's balance theory in signed networks, which consists of friendship or enmity relationships, is a model that relates the type of relationship between two people to the third person. In this model, there is an assumption of the independence of triadic relations, which means that the balance or imbalance of one triangle does not affect another and the energy only depends on the number of each type of triangle. There is evidence that in real network data, in addition to third-order interactions (Heider balance), higher-order interactions also play a role. One step beyond the Heider balance, the effect of quartic balance has been studied by removing the assumption of triangular independence. The application of quartic balance results in the influence of the balanced or imbalanced state of neighboring triangles on each specific one. Here, a question arises as to how the Heider balance is affected by the existence of quartic balance (fourth order). To answer this question, we presented a model which has both third- and fourth-order interactions and we called it a hybrid balance theory. The phase diagram obtained from the mean-field approximation shows there is a threshold for higher-order interaction strength, below which a third-order interaction dominates and there are no imbalance triangles in the network, and above this threshold, squares effectively determine the balance state in which the imbalance triangles can survive. The solution of the mean-field indicates that we have a first-order phase transition in terms of the random behavior of agents (temperature) which is in accordance with the Monte Carlo simulation results.

The structure balance of gene-gene networks beyond pairwise interactions

Despite its high and direct impact on nearly all biological processes, the underlying structure of gene-gene interaction networks is investigated so far according to pair connections. To address this, we explore the gene interaction networks of the yeast Saccharomyces cerevisiae beyond pairwise interaction using the structural balance theory (SBT). Specifically, we ask whether essential and nonessential gene interaction networks are structurally balanced. We study triadic interactions in the weighted signed undirected gene networks and observe that balanced and unbalanced triads are over and underrepresented in both networks, thus beautifully in line with the strong notion of balance. Moreover, we note that the energy distribution of triads is significantly different in both essential and nonessential networks compared to the shuffled networks. Yet, this difference is greater in the essential network regarding the frequency as well as the energy of triads. Additionally, results demonstrate that triads in the essential gene network are more interconnected through sharing common links, while in the nonessential network they tend to be isolated. Last but not least, we investigate the contribution of all-length signed walks and its impact on the degree of balance. Our findings reveal that interestingly when considering longer cycles, not only, both essential and nonessential gene networks are more balanced compared to their corresponding shuffled networks, but also, the nonessential gene network is more balanced compared to the essential network.

Thermal properties of structurally balanced systems on diluted and densified triangulations

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) are discussed for various networks of interacting actors under the influence of the temperature modeling the social noise level. For that purpose, two main types of lattices are considered. The first is created by removing some links from a regular triangular lattice to produce a diluted triangular lattice, and the second by adding more links to create an enhanced triangular lattice. In both those cases, the full range of possible graph densities is discussed, limited by the extreme cases of networks which consist of a small number of separated triads and fully connected networks. It is shown that the existence of the balanced state is not possible if the average node degree is too close to the value characterizing the regular triangular lattice. Otherwise, both balanced (or partially balanced) and imbalanced states are possible, depending on the temperature. However, only for graphs which are dense enough a phase transition of the first kind is observed, while less enhanced networks (and all diluted) indicate a smooth crossover between the two states. The crossover temperatures are size independent only for the diluted triangular lattices and depend on the size of the system for the enhanced triangular lattices, as is the case also for the critical temperatures of the phase transition observed in denser enhanced lattices.