Power-law distribution functions derived from maximum entropy and a symmetry relationship

TL;DR

This paper demonstrates that power-law distributions can naturally emerge from a maximum entropy principle under a shared cost framework, providing a new interpretation of their origin in social systems.

Contribution

It introduces a variational principle based on shared costs that explains the emergence of power-law distributions in social communities.

Findings

Power-law distributions arise from entropy maximization with equal cost sharing.

Unequal sharing of costs leads to a smooth transition from exponential to power-law.

The framework offers a new interpretation of power-laws in social physics.

Abstract

Power-law distributions are common, particularly in social physics. Here, we explore whether power-laws might arise as a consequence of a general variational principle for stochastic processes. We describe communities of 'social particles', where the cost of adding a particle to the community is shared equally between the particle joining the cluster and the particles that are already members of the cluster. Power-law probability distributions of community sizes arise as a natural consequence of the maximization of entropy, subject to this 'equal cost sharing' rule. We also explore a generalization in which there is unequal sharing of the costs of joining a community. Distributions change smoothly from exponential to power-law as a function of a sharing-inequality quantity. This work gives an interpretation of power-law distributions in terms of shared costs.