Quartic balance theory: Global minimum with imbalanced triangles

Abstract

Balance theory proposed by Heider for the first time modeled triplet interaction in a signed network, stating that relationships between two people, friendship or enmity, is dependent on a third person. The Hamiltonian of this model has an implicit assumption that all triads are independent, meaning that state of each triad, being balanced or imbalanced, is ineffective to others. This independence forces the network to have completely balanced final states. However, there exists evidence indicating that real networks are partially balanced raising the question of what is the mechanism preventing the system to be perfectly balanced. Our suggestion is to consider a quartic interaction which dissolves the triad's independence. We use mean field method to study thermal behavior of such systems where the temperature is a parameter that allows the stochastic behavior of agents. We show that under a certain temperature, the symmetry between balanced and imbalanced triads will spontaneously break and we have a discrete phase transition. As consequence stability arises where either similar balanced or imbalanced triads dominate, hence the system obtains two new imbalanced stable states. In this model, the critical temperature depends on the second power of the number of nodes, which was a linear dependence in thermal balance theory. Our simulations are in good agreement with the results obtained by the mean field method.

Topological analysis of interaction patterns in cancer-specific gene regulatory network: persistent homology approach

“…Based on this observation, we asked how triads with different energies are connected to one another and how schematic diagrams of distribution of frustrated triangles in the normal and cancer network differ. To address this, the concept of exceeding the length of interaction from triplet interactions towards higher-order interactions, quartic interactions or Energy-Energy Correlation between triangles can be proposed, allowing one to study the very influence of units of four entities on the final degree of balance 61 . Considering a simple pairwise interaction term between triads with a common edge in previous reports, the model Hamiltonian to treat the states of balanced and imbalanced triads is defined where represents a triad shaped by i , j , k nodes.…”

“…In quartic balance theory now the number of squares, i.e., s ( G ), is an essential parameter for the specific graph configuration and according to structural balance, the corresponding energies can be compared. This formalism examines the probability distribution of the jammed states’ levels of energy, assuming that for the triads, the shift from balanced to imbalanced can be determined based on all triads that share a common link 61 . As discussed, constructing 3rd and 4th order interaction networks and examining their corresponding energy for normal and cancer networks provide us with practical insights and can be used to compare stability, energy, and the tendency toward changing their states in cancer and normal samples.…”

Topological Analysis of Interaction Patterns in Cancer-Specific Gene Regulatory Network: Persistent Homology Approach

“…Based on this observation, we asked how triads with different energies are connected to one another and how schematic diagrams of distribution of frustrated triangles in the normal and cancer network differ. To address this, the concept of exceeding the length of interaction from triplet interactions towards higher-order interactions, quartic interactions or Energy-Energy Correlation between triangles can be proposed, allowing one to study the very influence of units of four entities on the final degree of balance 62 . Considering a simple pairwise interaction term between triads with a common edge in previous reports, the model Hamiltonian to treat the states of balanced and imbalanced triads is defined…”

“…In quartic balance theory now the number of squares, i.e., s(G), is an essential parameter for the specific graph configuration and according to structural balance, the corresponding energies can be compared. This formalism examines the probability distribution of the jammed states' levels of energy, assuming that for the triads, the shift from balanced to imbalanced can be determined based on all triads that share a common link 62 . As discussed, constructing 3rd and 4th order interaction networks and examining their corresponding energy for normal and cancer networks provide us with practical insights and can be used to compare stability, energy, and the tendency toward changing their states in cancer and normal samples.…”

The structure balance of gene-gene networks beyond pairwise interactions

“…Consequently, there have been two main trends in the literature of SBT: 1) Studying the analytical aspects theoretically [19,[29][30][31][32][33][34][35], 2) Applying it to a wide variety of real-signed social, economic, ecologic, and political networks empirically to clarify their structures [36][37][38][39][40][41][42][43]. Amongst these applications, it should be mentioned that understanding the structure entirely, not partially, calls for considering not only short-range interactions but also longer-range cycles [44][45][46][47].…”