A continuum limit for dense networks

Publication date: 17 January 2023

Abstract: Metric graphs -- with continuous real-valued edges -- have been studied extensively as domains for solutions of PDEs. We study the action of the Laplace operator on each edge as a metric graph fills in a

n -

dimensional Riemannian manifold, e.g., a very dense spider web in the planar disc or ball of wire within the

2 -

sphere. We derive an

n

-dimensional limiting differential operator whose low-order eigenmodes approximate the low-order eigenmodes of the metric graph Laplace operator. High-density metric graphs exhibit nontrivial, continuum-like behaviour at low frequencies and purely graph-like behaviour at high frequencies. In the continuum limit, we find the emergence of a new symmetric tensor field but with an unusual distance scaling compared to the traditional Riemannian metric. The limiting operator from a metric graph implies a possible new kind of continuous geometry.

Has companion code repository: https://github.com/sidneyholden1/laplace_operator_metric_graph

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