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Consider a network embedded in the 2D plane, where a particle diffuses alongthe edges of the network. It is clear that over short length scales a particlemoves along a single edge and thus undergoes one-dimensional diffusion.However, on larger length scales it is no longer immediately clear how thetransport will behave. One could intuit that as the network is embedded in twodimensions for "large enough" length scales the transport will also appeartwo-dimensional. Is this true for all networks? Can we quantify the lengthscales upon which this transition occurs? What is the transport behaviour onintermediate spatial scales? In this paper, we answer these question bypresenting a numerical linear algebra approach that provides the exact momentsof first passage times for a given network. Comparing these networkedfirst-passage times to first-passage times for planar diffusion reveals severalinteresting properties of networked transport. In particular we can directlyquantify the length scale upon which networked diffusion will appear planar ifit does at all. Finally, we introduce an adaptation of the method of maximumentropy to use the moments of first-passage times to construct an analyticalapproximation to the first-passage times entire probability distribution.

First passage times of transport on planar spatial networks and their connections to off-network planar diffusion