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

A Proof of Kirchhoff's First Law for Hyperbolic Conservation Laws on Networks