Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SGRDN-Data learned sparsification of graph reaction-diffusion networks

Abhishek Ajayakumar, Soumyendu Raha

TL;DR

SGRDN introduces a spectral-preserving sparsification framework for reaction-diffusion dynamics on undirected graphs by framing edge-weight reduction as a data-assimilation problem in a reduced-order model. It combines adjoint sensitivity for gradient-based optimization, POD-based model reduction, and perturbation-aware eigenpair approximations to enforce preservation of Laplacian eigenmodes and isodynamic behavior. The method is validated on real graphs and extended to sparsify Neural ODEs, achieving substantial edge reductions while maintaining high dynamical fidelity (e.g., $R>0.85$) and enabling significant computational speedups through ROM evaluation and parallel constraint processing. This approach broadens sparse graph techniques to dynamical systems, offering practical benefits for efficient simulation and learning in complex networked processes.

Abstract

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 exist for generating linear-sized sparsifiers in almost quadratic running time $O(n^{2 + ε})$. Consequently, obtaining a sparsifier can be a computationally demanding task, and the complexity varies based on the level of sparsity required. We propose SGRDN to extend sparsification to complex reaction-diffusion systems. This approach seeks to sparsify the graph such that the inherent reaction-diffusion dynamics are strictly preserved on the resulting structure. By selectively considering a subset of trajectories, we frame the network sparsification issue as a data assimilation problem within a Reduced Order Model (ROM) space, imposing constraints to conserve the eigenmodes of the Laplacian matrix ($L = D - A$), the difference between the degree matrix ($D$) and the adjacency matrix ($A$) despite perturbations. We derive computationally efficient eigenvalue and eigenvector approximations for perturbed Laplacian matrices and integrate these as spectral preservation constraints in the optimization problem. To further validate the method's broad applicability, we conducted an additional experiment on Neural Ordinary Differential Equations (neural ODEs), where SGRDN successfully achieved parameter sparsity.

SGRDN-Data learned sparsification of graph reaction-diffusion networks

TL;DR

SGRDN introduces a spectral-preserving sparsification framework for reaction-diffusion dynamics on undirected graphs by framing edge-weight reduction as a data-assimilation problem in a reduced-order model. It combines adjoint sensitivity for gradient-based optimization, POD-based model reduction, and perturbation-aware eigenpair approximations to enforce preservation of Laplacian eigenmodes and isodynamic behavior. The method is validated on real graphs and extended to sparsify Neural ODEs, achieving substantial edge reductions while maintaining high dynamical fidelity (e.g., ) and enabling significant computational speedups through ROM evaluation and parallel constraint processing. This approach broadens sparse graph techniques to dynamical systems, offering practical benefits for efficient simulation and learning in complex networked processes.

Abstract

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 exist for generating linear-sized sparsifiers in almost quadratic running time . Consequently, obtaining a sparsifier can be a computationally demanding task, and the complexity varies based on the level of sparsity required. We propose SGRDN to extend sparsification to complex reaction-diffusion systems. This approach seeks to sparsify the graph such that the inherent reaction-diffusion dynamics are strictly preserved on the resulting structure. By selectively considering a subset of trajectories, we frame the network sparsification issue as a data assimilation problem within a Reduced Order Model (ROM) space, imposing constraints to conserve the eigenmodes of the Laplacian matrix (), the difference between the degree matrix () and the adjacency matrix () despite perturbations. We derive computationally efficient eigenvalue and eigenvector approximations for perturbed Laplacian matrices and integrate these as spectral preservation constraints in the optimization problem. To further validate the method's broad applicability, we conducted an additional experiment on Neural Ordinary Differential Equations (neural ODEs), where SGRDN successfully achieved parameter sparsity.
Paper Structure (20 sections, 5 theorems, 80 equations, 14 figures, 2 tables)

This paper contains 20 sections, 5 theorems, 80 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Laplacian matrix:($L$)
  4. Incidence matrix:($B$)
  5. Dynamical systems involving graphs
  6. Adjoint method for data assimilation
  7. Adjoint sensitivity with respect to parameter $\alpha$
  8. Adjoint method Algorithm
  9. Computing the Jacobians $D_{k} \left(M\right)$ and $D_{\alpha} \left(M\right)$
  10. Using reduced order model for dynamical systems
  11. POD method performance
  12. Methodology Outline for Sparse Graph Construction with Trajectory-Based Pattern Preservation and Eigenmode Approximations
  13. Ensuring isodynamic behaviour
  14. Eigenvalue and eigenvector computation for the Laplacian matrix under perturbations
  15. Approximations to eigenpair
  16. ...and 5 more sections

Key Result

Lemma 5.1

(Sherman-Morrison formula). If A is a nonsingular $n \times n$ matrix and x is a vector, then

Figures (14)

  • Figure 1: Computational performance analysis of the chemical Brusselator model (Equation \ref{['Rd dynamics']}) represented on real-world graphs with and without POD.
  • Figure 2: Computational performance analysis of the chemical Brusselator model (Equation \ref{['Rd dynamics']}) represented on random graphs with and without POD.
  • Figure 3: Plot showing the stable and unstable eigenmodes of reaction-diffusion system as described by Equation (\ref{['Rd dynamics']}) on an Erdős-R$\acute{e}$nyi random graph.
  • Figure 4: Example of imposing the connectivity constraint using the unsigned incidence matrix for an unweighted 4-node graph. Edge multipliers are shown in red.
  • Figure 5: Complete graph $\mathbb{K}_3$
  • ...and 9 more figures

Theorems & Definitions (7)

  • Lemma 5.1
  • Lemma 5.2
  • Theorem 5.3
  • Theorem 5.4
  • proof
  • Theorem 5.5
  • proof