A symbolic dynamical system is a continuous transformation Φ : X ⟶ X of closed subset X ⊆ A V , where A is a finite set and V is countable (examples include subshifts, odometers, cellular automata, and automaton networks). The function Φ induces a directed graph (‘network’) structure on V , whose geometry reveals information about the dynamical system ( X , Φ ) . The dimension dim ( V ) is an exponent describing the growth rate of balls in this network as a function of their radius. We show that, if X has positive entropy and dim ( V ) > 1 , and the system ( A V , X , Φ ) satisfies minimal symmetry and mixing conditions, then ( X , Φ ) 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 Hölder-continuous.