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

TL;DR

This paper develops a theoretical framework for hyperbolic conservation laws on networks, establishing an analog of Kirchhoff's first law that links global and edgewise perspectives without explicit boundary conditions.

Contribution

It introduces a calculus for networks as metric spaces and proves an equivalence of Kirchhoff's law in hyperbolic conservation laws, simplifying their formulation.

Findings

Established an analog of Kirchhoff's first law for hyperbolic conservation laws on networks.

Proved the equivalence between global and edgewise formulations without explicit boundary conditions.

Provided a theoretical foundation for analyzing dynamical systems on networks.

Abstract

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.