  A `symbolic dynamical system' is a continuous transformation F:X-->X of a
closed perfect subset X of A^V, where A is a finite set and V is countable.
(Examples include subshifts, odometers, cellular automata, and automaton
networks.) The function F induces a directed graph structure on V, whose
geometry reveals information about the dynamical system (X,F). The `dimension'
dim(V) is an exponent describing the growth rate of balls in the digraph as a
function of their radius. We show: if X has positive entropy and dim(V)>1, and
the system (A^V,X,F) satisfies minimal symmetry and mixing conditions, then
(X,F) cannot be positively expansive; this generalizes a well-known result of
Shereshevsky about multidimensional cellular automata. We also construct a
counterexample to a version of this result without the symmetry condition.
Finally, we show that network dimension is invariant under topological
conjugacies which are Holder-continuous.
