# Expansive Automata Networks

## TL;DR

This paper introduces the concept of expansivity in automata networks, characterizes the interaction graphs that support it, and explores its prevalence in linear and non-linear cases, linking it to coding theory.

## Contribution

It provides a characterization of graphs allowing expansivity, analyzes its genericity in linear networks, and investigates conditions for non-linear networks with fixed alphabet sizes.

## Key findings

- Expansivity is characterized for certain interaction graphs.
- It is generic among linear automata networks with large alphabets.
- Non-linear solutions exist for fixed alphabet sizes, enabling strong observability.

## Abstract

An Automata Network is a map ${f:Q^n\rightarrow Q^n}$ where $Q$ is a finite alphabet. It can be viewed as a network of $n$ entities, each holding a state from $Q$, and evolving according to a deterministic synchronous update rule in such a way that each entity only depends on its neighbors in the network's graph, called interaction graph. A major trend in automata network theory is to understand how the interaction graph affects dynamical properties of $f$. In this work we introduce the following property called expansivity: the observation of the sequence of states at any given node is sufficient to determine the initial configuration of the whole network. Our main result is a characterization of interaction graphs that allow expansivity. Moreover, we show that this property is generic among linear automata networks over such graphs with large enough alphabet. We show however that the situation is more complex when the alphabet is fixed independently of the size of the interaction graph: no alphabet is sufficient to obtain expansivity on all admissible graphs, and only non-linear solutions exist in some cases. Finally, among other results, we consider a stronger version of expansivity where we ask to determine the initial configuration from any large enough observation of the system. We show that it can be achieved for any number of nodes and naturally gives rise to maximum distance separable codes.