论文标题
完整网络中同步的Heider动力学中的完美循环
Perfect cycles in the synchronous Heider dynamics in complete network
论文作者
论文摘要
我们讨论一个蜂窝自动机，以模拟在完全连接的网络中达到Heider平衡的过程。自动机的动力学由确定性，同步和全局更新规则定义。动力学具有非常丰富的吸引子，包括固定点和极限周期，其长度和数量随系统的大小而变化。在本文中，我们集中于保留连续状态能量光谱的一类极限循环。我们称这样的极限周期完美。完美循环中的连续状态彼此之间相同的锤子距离分开。同样，在完美周期中，任何两个州分开的任意两个州之间的锤距距离对于所有这对各对国家来说都是相同的。完美循环的状态在配置空间中形成了非常对称的轨迹。我们认为，轨迹的对称性植根于网络顶点的置换对称性以及一定能量函数的局部对称性，以测量三合会的平衡/挫折水平。
We discuss a cellular automaton simulating the process of reaching Heider balance in a fully connected network. The dynamics of the automaton is defined by a deterministic, synchronous and global update rule. The dynamics has a very rich spectrum of attractors including fixed points and limit cycles, the length and number of which change with the size of the system. In this paper we concentrate on a class of limit cycles that preserve energy spectrum of the consecutive states. We call such limit cycles perfect. Consecutive states in a perfect cycle are separated from each other by the same Hamming distance. Also the Hamming distance between any two states separated by $k$ steps in a perfect cycle is the same for all such pairs of states. The states of a perfect cycle form a very symmetric trajectory in the configuration space. We argue that the symmetry of the trajectories is rooted in the permutation symmetry of vertices of the network and a local symmetry of a certain energy function measuring the level of balance/frustration of triads.
