Perfect cycles in the synchronous Heider dynamics in complete network

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 and frustration of triads.

Figure 1

An example of a perfect limit cycle on a complete graph on $N=9$ nodes. The cycle consists of $c=12$ configurations. They are drawn in the order they appear in the cycle. Edges for $s_{ij}=1$ are in red and for $s_{ij}=-1$ in blue. Labels of vertices are not displayed. Vertices are numbered $i=1,...,9$ clockwise, starting from the vertex 1 on the top. The configuration in the left upper corner is equivalent to that given by the matrix Eq. (9).