  Networks are essential models in many applications such as information
technology, chemistry, power systems, transportation, neuroscience, and social
sciences. In light of such broad applicability, a general theory of dynamical
systems on networks may capture shared concepts, and provide a setting for
deriving abstract properties. To this end, we develop a calculus for networks
modeled as abstract metric spaces and derive an analog of Kirchhoff's first law
for hyperbolic conservation laws. In dynamical systems on networks, Kirchhoff's
first law connects the study of abstract global objects, and that of a
computationally-beneficial edgewise-Euclidean perspective by stating its
equivalence. In particular, our results show that hyperbolic conservation laws
on networks can be stated without explicit Kirchhoff-type boundary conditions.
