无序三元相互作用下的Heider平衡
Heider Balance under Disordered Triadic Interactions
摘要：
海德平衡解决了三体相互作用的假设，即三元组在网络的动力学中同样重要。在许多网络中，关系的强度不同，因此三元组的权重不同。现在，问题是社交网络如何在加权时减少不平衡三角形的数量？根据我们已经从未加权余额中了解到的情况，结果是否可以预见？为了找到解决方案，我们考虑一个完全连接的网络，其中三元组被分配不同的随机权重。权重来自具有均值$μ$和方差$σ$的高斯概率分布。我们在两个区域中研究了这个系统：（\RN{1}）$\fracμσ \ge 1 $的比率对应于弱无序（小方差），三元组的权重近似相同，（\RN{2}）$\fracμσ < 1 $计数对应于强无序（大方差），权重显著不同。研究这样一个网络的结构演化是我们的意图。我们看到无序在决定系统的临界温度方面起着关键作用。利用平均场方法给出了系统的解析解，表明系统经历了一级相变。对于弱无序，我们的模拟结果显示，系统达到全球最低的温度降低，而对于高度无序，我们设法看到一个三重相互作用的Eudo-近藤效应。近藤效应解释了低温下磁性杂质导致金属电阻率异常增加的现象。在我们的工作中，在第二个制度的能量与温度的关系中也看到了类似的行为，并且系统没有达到全局最小值。
英文摘要：
The Heider balance addresses three-body interactions with the assumption that triads are equally important in the dynamics of the network. In many networks, the relations do not have the same strength so, triads are differently weighted. Now, the question is how social networks evolve to reduce the number of unbalanced triangles when they are weighted? Are the results foreseeable based on what we have already learned from the unweighted balance? To find the solution, we consider a fully connected network in which triads are assigned with different random weights. Weights are coming from Gaussian probability distribution with mean $μ$ and variance $σ$. We study this system in two regimes : (\RN{1}) the ratio of $\fracμσ \ge 1 $ corresponds to weak disorder (small variance) that triads' weight are approximately the same, (\RN{2}) $\fracμσ < 1 $ counts for strong disorder (big variance) and weights are remarkably diverse. Investigating the structural evolution of such a network is our intention. We see disorder plays a key role in determining the critical temperature of the system. Using the mean-field method to present an analytic solution for the system represents that the system undergoes a first-order phase transition. For weak disorder, our simulation results display the system reaches the global minimum as temperature decreases whereas for high disorder we manage to see a seudo-Kondo effect in triadic interactions. The Kondo effect explains the anomalous increase in electrical resistivity of metals due to magnetic impurities at low temperatures. Similar behavior is seen in the energy versus temperature in our work for the second regime and the system does not reach a global minimum.
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