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).

翻译：图稀疏化是计算机科学与应用数学领域的一个重要研究方向。一般而言，图稀疏化旨在减少网络中的边数，同时保留图的特定性质（如割集和子图计数）。已知计算图的最稀疏割集是NP难的，现有的稀疏化算法能够在准二次运行时间 \(O(n^{2 + \epsilon})\) 内生成线性规模的稀疏化图。因此，获取稀疏化图可能是一项计算密集型任务，其复杂度随所需稀疏度水平而变化。本研究将稀疏化概念拓展至反应扩散复杂系统领域，旨在解决在保留底层流动动力学特性的前提下减少网络中边数的挑战。为解决该问题，我们采用一种松弛方法，仅考虑部分轨迹。我们将网络稀疏化问题映射为约化阶模型（ROM）空间上的数据同化问题，并施加约束以保留拉普拉斯矩阵在扰动下的特征模态。拉普拉斯矩阵（\(L = D - A\)）是度对角矩阵（\(D\)）与图邻接矩阵（\(A\)）的差值。我们提出了拉普拉斯矩阵在扰动下特征值与特征向量的近似方法以实现计算可行性，并将基于这些近似构造的自定义函数作为约束条件集成到数据同化框架中。我们进一步展示了该框架在神经常微分方程（neural ODEs）参数集稀疏化中的推广能力。