On attractors in finite dynamic systems of complete graphs orientations

Finite dynamic systems of complete graphs orientations are considered. The states of such a system S = (Г_Кп, a), n > 1, are all possible orientations G of the complete graph K_n, and evolutionary function a transforms a given state G by reversing all arcs in G that enter into sinks, and there are no other differences between the given (G) and the next (a(G)) states. The following criterion for belonging states to attractors in S is given: a state G belongs to an attractor if and only if it hasn't a sink or its indegrees vector is a permutation of numbers 0,1,... ,n - 1. All attractors in S are the attractors of length 1, each of which consists of states without sinks, and the attractors of length n, each of which consists of states with indegrees vectors being permutations of numbers 0, 1, . . . , n - 1. Any such an attractor represents a circuit, for every state G in which if the indegrees vector of G is (d(v₁ ),d(v₂),..., d(v_n)), then the indegrees vector of a(G) is (d(v₁) + 1, d(v₂) + 1,..., d(v_n) +1), where the addition is calculated modulo n. Note that in system S, the number of attractors of length n is equal to (n - 1)! and the number of states belonging to them is equal to n!.

Keywords

Authors

References