完全网络中同步Heider动力学的完美圈
Perfect cycles in the synchronous Heider dynamics in complete network
摘要：
本文讨论了一种元胞自动机来模拟全连通网络达到Heider平衡的过程。自动机的动态性由确定性的、同步的和全局的更新规则定义。动力学有非常丰富的吸引子谱，包括不动点和极限环，其长度和数量随系统的大小而变化。本文主要研究一类保持连续态能谱的极限环。我们称这样的极限环为完美的。在一个完美的循环中，连续的状态彼此之间被相同的汉明距离分开。而且，在一个完美循环中，由$k$步隔开的任何两个状态之间的汉明距离对于所有这样的状态对都是相同的。完美循环的状态在位形空间中形成非常对称的轨迹。我们认为，对称性的轨迹是植根于网络的顶点的置换对称性和一个局部对称性的一定的能量函数测量的水平的平衡/挫折的三合会。
英文摘要：
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.
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