Wandering in the state space

TL;DR

This paper explores the topology of state spaces in two systems, revealing structural insights through network analysis, with potential applications in understanding complex system dynamics.

Contribution

It introduces a general method for analyzing state space topology using network representations, applicable to diverse systems like spin models and traffic scenarios.

Findings

Identified classes and subclasses of states based on network connectivity.

Characterized clusters by anisotropy in spin systems and road occupancy in traffic.

Provided a framework for applying random walk theory to complex state spaces.

Abstract

We analyse the topology of the state space of two systems: i) N Ising spins +/-1 with the antiferromagnetic interactions on a triangular lattice, with the condition of minimum of energy, ii) a roundabout of three access roads and three exit roads, with up to 2 cars on each road. The state space is represented by a network, and states - as nodes; two nodes are linked if an elementary process (spin flip or car shift) transforms the respective states one into another. Information is collected on the number of neighbours of states, what allows to distinguish classes and subclasses of states, and on the cluster structure of the state space. In the Ising systems, the clusters are characterized by anisotropy of the spin-spin correlation functions. In the case of a roundabout, the clusters differ by the number of empty or full roads. The method is general and it provides a basis for applications of the random walk theory.