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Surrogate models for diffusion on graphs via sparse polynomials

Giuseppe Alessio D'Inverno, Kylian Ajavon, Simone Brugiapaglia

TL;DR

This work introduces sparse polynomial surrogate models for parametric diffusion on graphs with community structure and proves holomorphic regularity of the parametric diffusion map, enabling recovery guarantees for least squares and compressed sensing. By establishing that the parameter-to-solution map is holomorphic, the authors derive quasi-optimal best $s$-term convergence rates and provide probabilistic guarantees for both LS and CS recoveries, with explicit bounds that account for dimensionality and polylog factors. The methodology is validated on synthetic stochastic block model graphs and real networks (Twitter and Facebook), demonstrating robustness across dimension, basis size, and graph scale. The results offer a scalable framework for uncertainty quantification and fast surrogate modeling of diffusion processes on graphs, with potential extensions to time-dependent diffusion and connections to graph neural network architectures.

Abstract

Diffusion kernels over graphs have been widely utilized as effective tools in various applications due to their ability to accurately model the flow of information through nodes and edges. However, there is a notable gap in the literature regarding the development of surrogate models for diffusion processes on graphs. In this work, we fill this gap by proposing sparse polynomial-based surrogate models for parametric diffusion equations on graphs with community structure. In tandem, we provide convergence guarantees for both least squares and compressed sensing-based approximations by showing the holomorphic regularity of parametric solutions to these diffusion equations. Our theoretical findings are accompanied by a series of numerical experiments conducted on both synthetic and real-world graphs that demonstrate the applicability of our methodology.

Surrogate models for diffusion on graphs via sparse polynomials

TL;DR

This work introduces sparse polynomial surrogate models for parametric diffusion on graphs with community structure and proves holomorphic regularity of the parametric diffusion map, enabling recovery guarantees for least squares and compressed sensing. By establishing that the parameter-to-solution map is holomorphic, the authors derive quasi-optimal best -term convergence rates and provide probabilistic guarantees for both LS and CS recoveries, with explicit bounds that account for dimensionality and polylog factors. The methodology is validated on synthetic stochastic block model graphs and real networks (Twitter and Facebook), demonstrating robustness across dimension, basis size, and graph scale. The results offer a scalable framework for uncertainty quantification and fast surrogate modeling of diffusion processes on graphs, with potential extensions to time-dependent diffusion and connections to graph neural network architectures.

Abstract

Diffusion kernels over graphs have been widely utilized as effective tools in various applications due to their ability to accurately model the flow of information through nodes and edges. However, there is a notable gap in the literature regarding the development of surrogate models for diffusion processes on graphs. In this work, we fill this gap by proposing sparse polynomial-based surrogate models for parametric diffusion equations on graphs with community structure. In tandem, we provide convergence guarantees for both least squares and compressed sensing-based approximations by showing the holomorphic regularity of parametric solutions to these diffusion equations. Our theoretical findings are accompanied by a series of numerical experiments conducted on both synthetic and real-world graphs that demonstrate the applicability of our methodology.

Paper Structure

This paper contains 26 sections, 6 theorems, 39 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Outline
  4. Surrogate diffusion on graphs
  5. Graph theory: basic concepts and notation
  6. Diffusion on graphs
  7. Sparse polynomial approximation: basic concepts and methods
  8. Holomorphic regularity
  9. Community structure assumption for diffusion on graphs
  10. Theoretical guarantees
  11. Step 1: existence of a holomorphic extension
  12. Step 2: validity of the uniform bound \ref{['eq:unif_upper_bound_ext']}
  13. Best $s$-term approximation rates
  14. Convergence guarantees for least squares and compressed sensing
  15. Numerical experiments
  16. ...and 11 more sections

Key Result

Lemma 1

Let $T>0$, $n,d \in \mathbb{N}$, $\mathcal{K}\subset\mathbb{C}^d$ be a compact set, and $\bm{g} :[0,T]\times\mathcal{K}\to\mathbb{C}^n$ and $\bm{h} :[0,T]^2\times\mathbb{C}^n\times\mathcal{K}\to\mathbb{C}^n$ be continuous functions. Moreover, suppose that there exists a constant $L>0$ such that $\bm for some norm $\lVert\cdot\rVert$ over $\mathbb{C}^n$. Then the Volterra integral equation admits

Figures (10)

  • Figure 1: Multi-index sets of order $n=4$ in $\mathbb{N}_0^3$ (left: total degree; right: hyperbolic cross).
  • Figure 2: Convergence plot of average RMSE vs. number of sample points for $d=3$.
  • Figure 3: Convergence plot of average RMSE vs. number of sample points for $d=6$.
  • Figure 4: Convergence plot of average RMSE vs. cardinality of multi-index set for a fixed number of sample points $m=350$.
  • Figure 5: Convergence plot of average RMSE vs. cardinality of multi-index set for a fixed number of sample points $m=1000$.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Definition 1: Best $s$-term approximation error
  • Definition 2: Holomorphic function
  • Definition 3: Filled-in Berstein polyellipse
  • Example 1
  • Lemma 1: Holomorphic dependence; Volterra integral equations
  • Theorem 1: Holomorphy of parametric graph diffusion
  • proof
  • Corollary 1: Best $s$-term approximation rates for parametric graph diffusion
  • Corollary 2: Convergence of least squares-based surrogates
  • Corollary 3: Convergence of compressed sensing-based surrogates
  • ...and 8 more