# Two-Agent Games on Graphs

## TL;DR

This paper investigates the evolution of agent points on graphs through two-agent competitions, analyzing dynamics on general graphs and specifically on ring and torus structures, with connections to statistical physics models.

## Contribution

It introduces a model of competitive dynamics on graphs and explores its properties, especially on regular lattice structures like rings and tori.

## Key findings

- The model relates to vertex models and Potts models at zero temperature.
- Dynamics exhibit specific behaviors on ring and torus graphs.
- Theoretical analysis of evolution patterns on different graph types.

## Abstract

We study the dynamics of evolution of points of agents placed in the vertices of a graph within the rules of two-agent units of competition where an edge is randomly chosen and the agent with higher points gets a new point with a probability $p$. The model is closely connected to generalized vertex models and anti-ferromagnetic Potts models at zero temperature. After studying the most general properties for generic graphs we confine the study to discrete d-dimensional tori. We mainly focus on the ring and torus graphs.