四次平衡理论：不平衡三角形的全局极小值
Quartic Balance Theory: Global Minimum With Imbalanced Triangles
摘要：
海德提出的平衡理论第一次在一个有符号的网络中模拟了三重互动，指出两个人之间的关系，无论是友谊还是敌意，都取决于第三人。这个模型的哈密顿量有一个隐含的假设，即所有三元组都是独立的，这意味着每个三元组的状态，无论是平衡的还是不平衡的，对其他三元组都是无效的。这种独立性迫使网络具有完全平衡的最终状态。然而，有证据表明，真实的网络是部分平衡的，这就提出了一个问题，即是什么机制阻止了系统的完美平衡。我们的建议是考虑一个四次相互作用，它消除了三元组的独立性。我们使用平均场方法来研究这类系统的热行为，其中温度是一个参数，允许随机行为的代理。结果表明，在一定温度下，平衡和非平衡三元组之间的对称性将自发地破缺，出现离散的相变。当相似的平衡或不平衡三元组占主导地位时，系统就会出现稳定性，从而得到两个新的不平衡稳定状态。在该模型中，临界温度依赖于节点数的二次幂，这是一个线性依赖于热平衡理论。我们的模拟结果与平均场方法得到的结果吻合得很好。
英文摘要：
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.
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