Modeling advection on distance-weighted directed networks

TL;DR

The paper addresses discrete transport (advection) on distance-weighted graphs by constructing a rigorous axiomatic framework that yields a unique operator $A_G$ on $\ell^\infty(V)$, governing the dynamics $\frac{d}{dt}f_t = -A_G f_t$ with $f_t=e^{-tA_G}f_0$. It introduces locality, mass transfer, and forward-advection axioms (including a splitting rule) and proves existence and uniqueness of the operator under these axioms, along with extensions to general graph classes. Through analytical and numerical examples on infinite and finite graphs, including a road-network application, it demonstrates how the operator captures directionality, mass conservation, and branch-splitting behavior that align with physical intuition of advection. The work provides a principled discrete analogue of continuous transport, with potential impact on traffic modeling and networked transport phenomena, by connecting edge lengths, node flows, and long-time mass distribution to a single uniquely defined operator. $A_G$ thus serves as a robust tool for simulating and analyzing transport on complex networks with guaranteed mass conservation and forward-directed dynamics.

Abstract

In this paper we propose a model for describing advection dynamics on distance-weighted directed graphs. To this end we establish a set of key properties, or axioms, that a discrete advection operator should satisfy, and prove that there exists an essentially unique operator satisfying all such properties. Both infinite and finite networks are considered, as well as possible variants and extensions. We illustrate the proposed model through examples, both analytical and numerical, and we describe an application to the simulation of a traffic network.

Figures (9)

• Figure 1: Examples of neighbourhoods: the elements that belong to the set written below the graph are depicted as white squares.
• Figure 2: An example of a finite graph $G\in\mathcal{G}$. The number on each edge specifies its length.
• Figure 3: Long-term comparison of the advection process on the graph in (\ref{['f:advection']}), according to operators $A^{(3)}$ and $A^{(4)}$. The latter, which satisfies Advection II, exhibits resonance behavior.
• Figure 4: Example of the advection process on a simple tree, with initial unit mass on $v$, according to the operator $A^{(4)}$.
• Figure 5: Depiction of the infinite grid and simulation on a $40\times 120$ grid of the mass distribution according to three different advection operators. The simulation was conducted on a sufficiently large truncated grid to minimize border effects.

Theorems & Definitions (47)

• Definition 1: Graph
• Definition 2: Oriented graph
• Definition 3: Walk, path and cycle
• Definition 4: Oriented tree
• Definition 5: Neighbourhood
• Definition 6
• Definition 7
• Definition 8
• Remark 1
• Proposition 1