# Power Accretion in Social Systems

## TL;DR

This paper models power dynamics in social networks, showing how inequality naturally arises and can be mitigated through taxation, with the system's geometry playing a crucial role in the distribution and transition to equality.

## Contribution

It introduces a simple game-based model of power evolution on networks and analyzes the effects of redistribution mechanisms, highlighting the role of system geometry in inequality.

## Key findings

- Inequality stabilizes at a stationary level with a clear class division.
- Taxation can induce a transition to a more equal society.
- Distribution roughness and entropy provide additional inequality measures.

## Abstract

We consider a model of power distribution in a social system where a set of agents play a simple game on a graph: the probability of winning each round is proportional to the agent's current power, and the winner gets more power as a result. We show that, when the agents are distributed on simple 1D and 2D networks, inequality grows naturally up to a certain stationary value characterized by a clear division between a higher and a lower class of agents. High class agents are separated by one or several lower class agents which serve as a geometrical barrier preventing further flow of power between them. Moreover, we consider the effect of redistributive mechanisms, such as proportional (non-progressive) taxation. Sufficient taxation will induce a sharp transition towards a more equal society, and we argue that the critical taxation level is uniquely determined by the system geometry. Interestingly, we find that the roughness and Shannon entropy of the power distributions are a very useful complement to the standard measures of inequality, such as the Gini index and the Lorenz curve.