Iterative Equitable Partition of Graph As a Model of Constant Structure Discrete Time Closed Semantic System

Abstract

Constant structure closed semantic systems are the systems each element of which receives its denition through the correspondent unchangeable set of other elements (neighbors) of the system. The denitions of the elements change iteratively and simultaneously based on the neighbor portraits from the previous iteration. In this paper, I study the behavior of such model systems, starting from the zero state, where all the system's elements are equal. The development of constant-structure discrete time closed semantic systems may be modelled as a discrete time coloring process on a connected graph. Basically, I consider the iterative redenition process on the vertices only, assuming that the edges are plain connectors, which do not have their own colors and do not participate in the denition of the incident vertices. However, the iterative coloring process for both vertices and edges may be converted to the vertices-only coloring case by the addition of virtual vertices corresponding to the edges assuming the colors for the vertices and for the edges are taken from the same palette and assigned in accordance with the same laws. I prove that the iterative coloring (redenition) process in the described model will quickly degenerate into a series of pairwise isomorphic states and discuss some directions of further research.