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On the accuracy of population level approximation of network processes

Noémi Nagy, Sándor Horváth, Balázs Maga, Péter L. Simon

TL;DR

The paper analyzes simple contagion dynamics on networks by contrasting the full NIMFA (individual-based) system with a one-dimensional population-level approximation on regular Turán graphs. It derives an explicit error term $H$ that captures inter-group heterogeneity and proves an exponential decay bound for $H$, enabling a perturbation-based bound on the relative error between the full and reduced models. A Bernoulli-type perturbation lemma yields a concrete, integrable upper bound on the discrepancy between the two models, which is shown to be sharp in numerical experiments for dense graphs, particularly SIS on Turán graphs. The work provides the first theoretical guarantee on the accuracy of a population-level approximation for network processes and points to extensions to broader graph classes and higher-dimensional coarse-grainings.

Abstract

The individual-based model of simple contagion processes is considered on regular graphs. This model explicitly incorporates the adjacency matrix of the network enabling us to study the effect of network structure on the dynamic of the propagation process. While the asymptotic behaviour of the model is well known, the transient behaviour has been less studied. Our goal in this paper is to give a theoretical estimate on the accuracy of the one-dimensional population-level approximation. This is carried out for arbitrary simple contagion processes and regular Turán graphs. Numerical evidence is shown that the theoretical estimate is rather sharp for dense graphs.

On the accuracy of population level approximation of network processes

TL;DR

The paper analyzes simple contagion dynamics on networks by contrasting the full NIMFA (individual-based) system with a one-dimensional population-level approximation on regular Turán graphs. It derives an explicit error term that captures inter-group heterogeneity and proves an exponential decay bound for , enabling a perturbation-based bound on the relative error between the full and reduced models. A Bernoulli-type perturbation lemma yields a concrete, integrable upper bound on the discrepancy between the two models, which is shown to be sharp in numerical experiments for dense graphs, particularly SIS on Turán graphs. The work provides the first theoretical guarantee on the accuracy of a population-level approximation for network processes and points to extensions to broader graph classes and higher-dimensional coarse-grainings.

Abstract

The individual-based model of simple contagion processes is considered on regular graphs. This model explicitly incorporates the adjacency matrix of the network enabling us to study the effect of network structure on the dynamic of the propagation process. While the asymptotic behaviour of the model is well known, the transient behaviour has been less studied. Our goal in this paper is to give a theoretical estimate on the accuracy of the one-dimensional population-level approximation. This is carried out for arbitrary simple contagion processes and regular Turán graphs. Numerical evidence is shown that the theoretical estimate is rather sharp for dense graphs.

Paper Structure

This paper contains 10 sections, 5 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General model for simple contagion on networks
  3. Approximation error for regular Turán graphs
  4. Derivation of the error term
  5. Estimation of the error term H
  6. Estimate for the difference of the individual and population-level models
  7. A perturbation result for differential equations of Bernoulli type
  8. Upper bound on $|A(t)-\overline{A}(t)|$
  9. Numerical investigation of the sharpness of the estimate in the case of epidemic spread
  10. Discussion

Key Result

Lemma 1

Let $K\geq 2$ be an integer and let $c_k\in\mathbb{R}$ be arbitrary numbers for $k=1,2, \ldots, K$. Then the following identity holds

Figures (2)

  • Figure 1: The sharpness of the estimate of Corollary \ref{['cor1']} in the case of SIS epidemic propagation. The relative difference $\frac{I(t)-\overline{I}(t)}{\overline{I}(t)}$ (green) on a Turán graph $T(100,5)$ starting with one infected node and the upper bound (red) given by Corollary \ref{['cor1']}. The function $I$ satisfies equation \ref{['NIMFA_SIS']} and $\overline{I}$ is the solution of the population-level model \ref{['MF_NIMFA']}. The epidemic parameters are $\gamma=1$, $\tau=\frac{2\gamma}{d}=\frac{2}{95}$ (chosen in such a way that half of the nodes in the graph become infected asymptotically.)
  • Figure 2: The sharpness of the estimate of Corollary \ref{['cor1']} in the case of SIS epidemic propagation. The relative difference $\frac{I(t)-\overline{I}(t)}{\overline{I}(t)}$ (green) on a complete bi-partite graph with $N=100$ nodes starting with one infected node and the upper bound (red) given by Corollary \ref{['cor1']}. The function $I$ satisfies equation \ref{['NIMFA_SIS']} and $\overline{I}$ is the solution of the population-level model \ref{['MF_NIMFA']}. The epidemic parameters are $\gamma=1$, $\tau=\frac{2\gamma}{d}=\frac{2}{50}$ (chosen in such a way that half of the nodes in the graph become infected asymptotically.)

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof : Lemma \ref{['lem2']}
  • Theorem 1
  • proof
  • Corollary 1