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 together, we examine the thermal behavior of the structural balance model defined on Erdős-Rényi random networks. We provide a Mean-Field solution for the model. We observe a first-order phase transition with temperature, for both the sparse and densely connected networks. We detect two transition temperatures, $T_{cold}$ and $T_{hot}$, characterizing a hysteresis loop. We find that with increasing the network sparsity, both $T_{cold}$ and $T_{hot}$ decrease. But the slope of decreasing $T_{hot}$ with sparsity is larger than the slope of decreasing $T_{cold}$. Hence, the hysteresis region gets narrower, until, in a certain sparsity, it disappears. We provide a phase diagram in the temperature-tie density plane to observe the meta-stable/coexistence region behavior more accurately. Then we justify our Mean-Field results with a series of Monte-Carlo simulations.
Heider balance of a chain of actors as dependent on the interaction range and a thermal noise
在周期边界条件和值域$r$的邻域内，以$r$为参数，研究了一系列作用体中的Heider平衡.应用了两种模型动力学：确定性元胞自动机（Malarz等人，Physica D 411（2020）132556）和热浴算法，其中不平衡-平衡三元组的密度起能量的作用。其结果是在稳态和闪烁状态下的能量谱以及由热噪声驱动的平衡-不平衡网络转换。临界点$T_c$随着范围$r$而增加，并且它不依赖于系统大小。
On the dynamics of Social Balance on general networks (with an application to XOR-SAT)
我们研究离散动力系统的非确定性和概率版本（由于T。Antal，P. L. Krapivsky和S. Redner）的社会平衡理论的启发。我们研究了几类图的收敛时间的动态。我们的贡献包括：1.我们指出了三重动力学和广义的零化走超图之间的连接。特别是，这种连接使我们能够完全表征图中的递归状态，其中每条边最多属于两个三角形。2.我们还解决了超图不包含由一个或两个顶点组成的边的情况。3.我们证明了在所谓的“三元圈”图上，收敛时间是线性的。4.我们得到了2-正则三元单形G的收敛时间的三次上界。这个界可以进一步改进到一个依赖于G的Cheeger常数的量。特别是这提供了一些严格的对应以前的实验观察。我们还指出了一个应用程序的随机游走算法的分析在某些情况下的3-异或-SAT问题。
