Graph-Theoretic Diffusion Modelling of Soil Organic Carbon Propagation in Paddy-Field Networks

Abstract

Estimating field-scale indigenous nitrogen supply from soil organic carbon (SOC) requires propagating sparse point observations across a spatially connected landscape. This paper formalises that propagation as a diffusion process on a graph. We represent a paddy-field monitoring network as a weighted graph G = (V, E), define the combinatorial graph Laplacian L = D − A, and show that the heat equation on G has closed-form solution x(t) = exp(−tL)x0, converging to the graph mean at a rate governed by the algebraic connectivity

(the Fiedler value). We then pose field-scale SOC/indigenous- nitrogen-supply reconstruction from partial observations as a

Tikhonov-regularised graph-diffusion problem, derive its closed- form solution and its relationship to Gaussian-field/ harmonic- function semi-supervised learning. A worked numerical example

on a twelve-node illustrative paddy-catchment network seeded within the SOC range documented for Sri Lankan paddy top

soils demonstrates the full pipeline: Laplacian spectral decom- position, heat-kernel diffusion, and Tikhonov reconstruction at

four withheld nodes, achieving a mean absolute error of 0.189 percentage points of SOC. We compare this to empirical Bayesian Kriging cross-validation error reported for a real national soil database and discuss the conditions under which graph diffusion is complementary to, rather than a replacement for, geostatistical interpolation.