  We show that a graph class $\cal G$ has bounded expansion if and only if it
has bounded $r$-neighbourhood complexity, i.e. for any vertex set $X$ of any
subgraph $H$ of $G\in\cal G$, the number of subsets of $X$ which are exact
$r$-neighbourhoods of vertices of $H$ on $X$ is linear to the size of $X$. This
is established by bounding the $r$-neighbourhood complexity of a graph in terms
of both its $r$-centred colouring number and its weak $r$-colouring number,
which provide known characterisations to the property of bounded expansion.
