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Accuracy criterion for mean field approximations of Markov processes on hypergraphs

Illes Horvath, Daniel Keliger

TL;DR

This work provides rigorous error bounds for the N-intertwined mean field approximation (NIMFA) of local density-dependent Markov population processes on hypergraphs, showing that NIMFA is accurate when typical vertices have many neighbors. By constructing an auxiliary process with independent neighborhoods, the authors derive uniform, local error bounds that scale with the network’s weight distribution characteristics ($w_{\max}^{*}$ and the normalized Frobenius norm $\frac{1}{N}\sum_{i,j} w_{ij}^2$) and hold across various network models. The paper further demonstrates how NIMFA can be reduced to simpler widely-used approximations, including homogeneous mean-field, metapopulation models (graphs and hypergraphs), annealed networks, activation-driven networks, and Szemerédi-regularity-based coarse-graining for dense graphs, under suitable regularity assumptions. These results clarify when and how higher-order interactions and network structure influence mean-field approximations, and they provide practical pathways to scalable analyses of large complex networks. Overall, the work offers a unified, theory-backed framework for applying NIMFA across diverse domains such as epidemiology, statistical physics, and opinion dynamics, with explicit error control and scalable reductions.

Abstract

We provide error bounds for the N-intertwined mean-field approximation (NIMFA) for local density-dependent Markov population processes with a well-distributed underlying network structure showing NIMFA being accurate when a typical vertex has many neighbors. The result justifies some of the most common approximations used in epidemiology, statistical physics and opinion dynamics literature under certain conditions. We allow interactions between more than 2 individuals, and an underlying hypergraph structure accordingly.

Accuracy criterion for mean field approximations of Markov processes on hypergraphs

TL;DR

This work provides rigorous error bounds for the N-intertwined mean field approximation (NIMFA) of local density-dependent Markov population processes on hypergraphs, showing that NIMFA is accurate when typical vertices have many neighbors. By constructing an auxiliary process with independent neighborhoods, the authors derive uniform, local error bounds that scale with the network’s weight distribution characteristics ( and the normalized Frobenius norm ) and hold across various network models. The paper further demonstrates how NIMFA can be reduced to simpler widely-used approximations, including homogeneous mean-field, metapopulation models (graphs and hypergraphs), annealed networks, activation-driven networks, and Szemerédi-regularity-based coarse-graining for dense graphs, under suitable regularity assumptions. These results clarify when and how higher-order interactions and network structure influence mean-field approximations, and they provide practical pathways to scalable analyses of large complex networks. Overall, the work offers a unified, theory-backed framework for applying NIMFA across diverse domains such as epidemiology, statistical physics, and opinion dynamics, with explicit error control and scalable reductions.

Abstract

We provide error bounds for the N-intertwined mean-field approximation (NIMFA) for local density-dependent Markov population processes with a well-distributed underlying network structure showing NIMFA being accurate when a typical vertex has many neighbors. The result justifies some of the most common approximations used in epidemiology, statistical physics and opinion dynamics literature under certain conditions. We allow interactions between more than 2 individuals, and an underlying hypergraph structure accordingly.
Paper Structure (22 sections, 14 theorems, 202 equations, 3 figures)

This paper contains 22 sections, 14 theorems, 202 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup
  3. The underlying hypergraph
  4. Local density dependent Markov population process
  5. N-intertwined mean field approximation
  6. Examples
  7. Error bounds for NIMFA
  8. Further reductions to NIMFA
  9. Homogeneous mean field approximation
  10. Metapopulation models
  11. Metapopulation models on graphs
  12. Metapopulation models on hypergraphs
  13. Annealed networks
  14. Activity-driven networks
  15. Dense graphs and Szemerédi's regularity lemma
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\Delta^{\mathcal{S}}$ denote the set of probability vectors from $\mathbb{R}^{\mathcal{S}}.$ For any initial condition $z_i(0) \in \Delta^{\mathcal{S}}$ for all $i$ the ODE system eq:NIMFA has a unique global solution such that $z_{i}(t) \in \Delta^S$ for all $i$ and $t>0$ as well.

Figures (3)

  • Figure 1: Edge (hyperedge) with weight $w^{(m)}_{i,j_1,\dots,j_{m}}$.
  • Figure 2: The ratio of infected based on the average of $1000$ simulations (triangles) compared to the estimate of NIMFA (solid line) on an $N=1000$ vertex modified cycle graphs with the closest $10$ (left) and $100$ (right) neighbors being connected. ($\beta=2, \gamma=1$) As we increase the degrees NIMFA performs better.
  • Figure 3: The ratio of infected based on the average of $10$ simulations (triangles) compared to the estimate of NIMFA (solid line) on an $N=5000$ vertex modified cycle graphs with the closest $10$ (left) and $100$ (right) neighbors being connected. ($\beta=2, \gamma=1$) As we increase the degrees NIMFA performs better.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 16 more