# On absence of steady state in the Bouchaud-M\'ezard network model

## TL;DR

This paper investigates the Bouchaud-Mézard network model, revealing that finite networks do not reach a true steady state, but exhibit time-dependent distributions, contrasting with the infinite network case.

## Contribution

It demonstrates that finite Bouchaud-Mézard networks lack a true steady state, showing instead a time-dependent distribution, which challenges previous assumptions based on the infinite network limit.

## Key findings

- Finite networks have time-dependent lognormal means.
- Inverse gamma distribution is only quasi-stationary.
- Infinite networks reach a steady state with a time-independent distribution.

## Abstract

In the limit of infinite number of nodes (agents), the It\^o-reduced Bouchaud-M\'ezard network model of economic exchange has a time-independent mean and a steady-state inverse gamma distribution. We show that for a finite number of nodes the mean is actually distributed as a time-dependent lognormal and inverse gamma is quasi-stationary, with the time-dependent scale parameter.