  Graph sparsification is an area of interest in computer science and applied
mathematics. Sparsification of a graph, in general, aims to reduce the number
of edges in the network while preserving specific properties of the graph, like
cuts and subgraph counts. Computing the sparsest cuts of a graph is known to be
NP-hard, and sparsification routines exists for generating linear sized
sparsifiers in almost quadratic running time $O(n^{2 + \epsilon})$.
Consequently, obtaining a sparsifier can be a computationally demanding task
and the complexity varies based on the level of sparsity required. In this
study, we extend the concept of sparsification to the realm of
reaction-diffusion complex systems. We aim to address the challenge of reducing
the number of edges in the network while preserving the underlying flow
dynamics. To tackle this problem, we adopt a relaxed approach considering only
a subset of trajectories. We map the network sparsification problem to a data
assimilation problem on a Reduced Order Model (ROM) space with constraints
targeted at preserving the eigenmodes of the Laplacian matrix under
perturbations. The Laplacian matrix ($L = D - A$) is the difference between the
diagonal matrix of degrees ($D$) and the graph's adjacency matrix ($A$). We
propose approximations to the eigenvalues and eigenvectors of the Laplacian
matrix subject to perturbations for computational feasibility and include a
custom function based on these approximations as a constraint on the data
assimilation framework. We demonstrate the extension of our framework to
achieve sparsity in parameter sets for Neural Ordinary Differential Equations
(neural ODEs).
