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

TL;DR

Calculus is extended to networks, modeled as abstract metric spaces, and Kirchhoff's first law for hyperbolic conservation laws is derived, showing that hyperbola conservation laws on networks can be stated without explicit Kirch Hoffman-type boundary conditions.

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.

Figures (6)

• Figure 1: The networks that represent the drivable roads of Berkeley, CA, USA (left) and Bochum, Germany (right) projected onto $\mathbb{R}^2$. Created from OpenStreetMap data OpenStreetMap using the tool OSMnx boeing2017.
• Figure 2: An embedding of the Wheatstone network into $\mathbb{R}^2$ that locally preserves vertex distances.
• Figure 3: Examples motivating \ref{['def:regularnetwork']}: Even a locally finite, connected network may not be complete (left). Even if edge lengths have a lower bound, a network may not be locally compact (middle). A network which is not connected (right).
• Figure 4: The discrete measure representing New York City taxi drop-offs on January 15, 2015. Data taken from NYC Taxi and Limousine Commission. Inspired by and created with kepler.gl.
• Figure 5: Differentiability across vertices with more than two edges: In vertices, derivatives along isometric paths must be zero. Assuming the derivatives of a function along the yellow paths have the same non-zero sign, it can not be differentiated along the red path.

Theorems & Definitions (35)

• Definition 2.1: Positively weighted graph
• Lemma 2.1
• proof
• Definition 2.2: Network
• Remark 1
• Example 2.1: Wheatstone network
• Definition 2.3: Regularity
• Theorem 2.1: Regularity
• Remark 2
• proof