  Graph-based approximation methods are of growing interest in many areas,
including transportation, biological and chemical networks, financial models,
image processing, network flows, and more. In these applications, often a basis
for the approximation space is not available analytically and must be computed.
We propose perturbations of Lagrange bases on graphs, where the Lagrange
functions come from a class of functions analogous to classical splines. The
basis functions we consider have local support, with each basis function
obtained by solving a small energy minimization problem related to a
differential operator on the graph. We present $\ell_\infty$ error estimates
between the local basis and the corresponding interpolatory Lagrange basis
functions in cases where the underlying graph satisfies a mild assumption on
the connections of vertices where the function is not known, and the
theoretical bounds are examined further in numerical experiments. Included in
our analysis is a mixed-norm inequality for positive definite matrices that is
tighter than the general estimate $\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$.
