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Inference for decorated graphs and application to multiplex networks

Charles Dufour, Sofia C. Olhede

TL;DR

The paper extends graphon theory to decorated graphs, introducing decorated graphons to capture edge attributes and multiplex structures. It proposes the first inference method for finitely decorated graphons by transforming decorations into binary indicators and solving a least-squares problem within a stochastic shape model, yielding convergence rates that match classical nonparametric theory when the decoration set is finite. Theoretical results show predictable estimation error bounds and MISE rates, with Hölder-continuous graphons achieving near-optimal rates; simulations confirm the rates, and applications to multiplex datasets (human diseases and school contacts) demonstrate improved structural recovery and richer interpretations than standard graphon approaches. This work broadens graphon applicability to complex, edge-attributed networks, enabling more accurate modeling of real-world multiplex systems.

Abstract

A graphon is a limiting object used to describe the behaviour of large networks through a function that captures the probability of edge formation between nodes. Although the merits of graphons to describe large and unlabelled networks are clear, they traditionally are used for describing only binary edge information, which limits their utility for more complex relational data. Decorated graphons were introduced to extend the graphon framework by incorporating richer relationships, such as edge weights and types. This specificity in modelling connections provides more granular insight into network dynamics. Yet, there are no existing inference techniques for decorated graphons. We develop such an estimation method, extending existing techniques from traditional graphon estimation to accommodate these richer interactions. We derive the rate of convergence for our method and show that it is consistent with traditional non-parametric theory when the decoration space is finite. Simulations confirm that these theoretical rates are achieved in practice. Our method, tested on synthetic and empirical data, effectively captures additional edge information, resulting in improved network models. This advancement extends the scope of graphon estimation to encompass more complex networks, such as multiplex networks and attributed graphs, thereby increasing our understanding of their underlying structures.

Inference for decorated graphs and application to multiplex networks

TL;DR

The paper extends graphon theory to decorated graphs, introducing decorated graphons to capture edge attributes and multiplex structures. It proposes the first inference method for finitely decorated graphons by transforming decorations into binary indicators and solving a least-squares problem within a stochastic shape model, yielding convergence rates that match classical nonparametric theory when the decoration set is finite. Theoretical results show predictable estimation error bounds and MISE rates, with Hölder-continuous graphons achieving near-optimal rates; simulations confirm the rates, and applications to multiplex datasets (human diseases and school contacts) demonstrate improved structural recovery and richer interpretations than standard graphon approaches. This work broadens graphon applicability to complex, edge-attributed networks, enabling more accurate modeling of real-world multiplex systems.

Abstract

A graphon is a limiting object used to describe the behaviour of large networks through a function that captures the probability of edge formation between nodes. Although the merits of graphons to describe large and unlabelled networks are clear, they traditionally are used for describing only binary edge information, which limits their utility for more complex relational data. Decorated graphons were introduced to extend the graphon framework by incorporating richer relationships, such as edge weights and types. This specificity in modelling connections provides more granular insight into network dynamics. Yet, there are no existing inference techniques for decorated graphons. We develop such an estimation method, extending existing techniques from traditional graphon estimation to accommodate these richer interactions. We derive the rate of convergence for our method and show that it is consistent with traditional non-parametric theory when the decoration space is finite. Simulations confirm that these theoretical rates are achieved in practice. Our method, tested on synthetic and empirical data, effectively captures additional edge information, resulting in improved network models. This advancement extends the scope of graphon estimation to encompass more complex networks, such as multiplex networks and attributed graphs, thereby increasing our understanding of their underlying structures.
Paper Structure (22 sections, 9 theorems, 56 equations, 10 figures, 2 tables)

This paper contains 22 sections, 9 theorems, 56 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Decorated graphs and graphons
  3. Finitely decorated graphs
  4. Inference of finitely decorated graphons
  5. Properties of estimator
  6. Numerical experiments
  7. Data Analysis
  8. Multiplex of human diseases
  9. High School contact network
  10. Transformation
  11. Technical details of the rate of convergence
  12. Optimisation: practical consideration
  13. Supplemental Material
  14. Proofs of the main theorems
  15. Notation
  16. ...and 7 more sections

Key Result

Theorem 4.1

Let $W$ be a $\mathcal{K}$-decorated $(s,k)$-stochastic shape model, then for any $C'>0$ there exists $C>0$ such that with probability at least $1-\exp\left(-C'n\log s\right)$ uniformly over $\bm{\theta} \in \Theta_{s,k}$. Furthermore, we have with some universal constant $C_1>0$ and $n>\max \left(0, k^2-s\right)$.

Figures (10)

  • Figure 1: Stochastic Block Model approximation (SBM) of $W_3$ (see \ref{['tab:sim']}) in the first row and estimated decorated graphon using a decorated $(s=27,k=14)$-Stochastic Shape Model (SSM) in the second. The smoothing effect of using shapes instead of blocks is particularly visible in $w^{(4)}$ on the right-hand side. The estimator was computed based on an observation with $300$ nodes.
  • Figure 2: The MSE error of our block-estimator with for the independent $W_1$ and dependent layer $W_3$ case (see \ref{['tab:sim']} for more details). Each point represents the average of MSE over $10$ independent repetitions, and the standard errors were of the order of $10^{-5}$. The constants $C_1$, $C_{0.5}$ were picked for visibility.
  • Figure 3: Ground truth $W_3$ (see \ref{['tab:sim']}) in the first column. The two columns on the right show the presented estimator with increasing nodes observed.
  • Figure 4: Visual Representation of the Multiplex Network of Human Diseases: This figure displays an ordered matrix of the network, organised by the refined categorisations derived from our decorated graphon estimation method, illustrating clearer patterns of disease interrelations. Fitted correlation matrix between the two layers on the right.
  • Figure 5: Marginal probability of having a certain connection between two students based on our fitted model. The data was sorted first by the class of the students and then by the grouping from the estimator. Notice how we retrieve the similarity between the different classes (indicated by the ticks) and specialisations (dashed lines).
  • ...and 5 more figures

Theorems & Definitions (23)

  • Definition 2.1: $\mathcal{K}$-graphon lovasz_limits_2010
  • Definition 3.1: $\mathcal{K}$-decorated $(s,k)$-Stochastic Shape Model (SSM)
  • Theorem 4.1
  • Theorem 4.2
  • Theorem B.1
  • Proof B.1: of \ref{['theorem:holder-rate']}
  • Proposition B.1
  • Proof B.2: of \ref{['proposition:function_estimation']}
  • Remark C.1
  • Remark D.1
  • ...and 13 more