Graph Dynamical System - Formal Definition

Formal Definition

A graph dynamical system is constructed from the following components:

• A finite graph Y with vertex set v = {1,2, ..., n}. Depending on the context the graph can be directed or undirected.

• A state x_v for each vertex v of Y taken from a finite set K. The system state is the n-tuple x = (x₁, x₂, ..., x_n), and x is the tuple consisting of the states associated to the vertices in the 1-neighborhood of v in Y (in some fixed order).

• A vertex function f_v for each vertex v. The vertex function maps the state of vertex v at time t to the vertex state at time t + 1 based on the states associated to the 1-neighborhood of v in Y.

• An update scheme specifying the mechanism by which the mapping of individual vertex states is carried out so as to induce a discrete dynamical system with map F: Kn → Kn.

The phase space associated to a dynamical system with map F: Kn → Kn is the finite directed graph with vertex set Kn and directed edges (x, F(x)). The structure of the phase space is governed by the properties of the graph Y, the vertex functions (f_i)_i, and the update scheme. The research in this area seeks to infer phase space properties based on the structure of the system constituents. The analysis has a local-to-global character.