We consider a model of power distribution in a social system where a set of agents plays 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 one-dimensional and
two-dimensional 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 (nonprogressive) 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 curveWe acknowledge financial support from the Spanish Government through
Grants No. FIS2015-69167-C2-1-P, No. FIS2015-66020-C2-
1-P, and No. PGC2018-094763-B-I0