# Critical branching processes in digital memcomputing machines

## TL;DR

This paper demonstrates that digital memcomputing machines naturally self-organize into a critical state characterized by avalanche distributions following a Borel distribution, which enhances their efficiency in solving complex combinatorial problems.

## Contribution

It introduces a mean-field theoretical prediction of avalanche size distribution in memcomputing machines and confirms it through numerical experiments on random 3-SAT instances.

## Key findings

- Avalanche sizes follow a power-law distribution with an exponent around 1.5.
- Memcomputing machines self-tune to a critical state during problem solving.
- Numerical results align with the theoretical critical branching process model.

## Abstract

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent $\tau= 3/2$. We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent $\tau = 1.51 \pm 0.02$, in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution.