  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.
