  We continue our study of the dependence of the density of states measure and
related spectral functions of Schr\"odinger operators on the potential. Whereas
our earlier work focused on random Schr\"odinger operators, we extend these
results to Schr\"odinger operators on infinite graphs with deterministic
potentials and ergodic potentials, and improve our results for random
potentials. In particular, we prove the Lipschitz continuity of the DOSm for
random Schr\"odinger operators on the lattice, recovering results of
\cite{kachkovskiy, shamis}. For our treatment of deterministic potentials, we
first study the density of states outer measure (DOSoM), defined for all
Schr\"odinger operators, and prove a deterministic result of the modulus of
continuity of the DOSoM with respect to the potential. We apply these results
to Schr\"odinger operators on the lattice $Z^d$ and the Bethe lattice. In the
former case, we prove the Lipschitz continuity of the DOSoM, and in the latter
case, we prove that the DOSoM is $\frac{1}{2}$-log-H\"older continuous. Our
technique combines the abstract Lipschitz property of one-parameter families of
self-adjoint operators with a new finite-range reduction that allows us to
study the dependency of the DOSoM and related functions on only finitely-many
variables and captures the geometry of the graph at infinity.
