Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Low-Complexity and Consistent Graphon Estimation from Multiple Networks

Roland Boniface Sogan, Tabea Rebafka

Abstract

Recovering the random graph model from an observed collection of networks is known to present significant challenges in the setting, where the networks do not share a common node set and have different sizes. More specifically, the goal is the estimation of the graphon function that parametrizes the nonparametric exchangeable random graph model. Existing methods typically suffer from either limited accuracy or high computational complexity. We introduce a new histogram-based estimator with low algorithmic complexity that achieves high accuracy by jointly aligning the nodes of all graphs, in contrast to most conventional methods that order nodes graph by graph. Consistency results of the proposed graphon estimator are established. A numerical study shows that the proposed estimator outperforms existing methods in terms of accuracy, especially when the dataset comprises only small and variable-size networks. Moreover, the computing time of the new method is considerably shorter than that of other consistent methodologies. Additionally, when applied to a graph neural network classification task, the proposed estimator enables more effective data augmentation, yielding improved performance across diverse real-world datasets.

Low-Complexity and Consistent Graphon Estimation from Multiple Networks

Abstract

Recovering the random graph model from an observed collection of networks is known to present significant challenges in the setting, where the networks do not share a common node set and have different sizes. More specifically, the goal is the estimation of the graphon function that parametrizes the nonparametric exchangeable random graph model. Existing methods typically suffer from either limited accuracy or high computational complexity. We introduce a new histogram-based estimator with low algorithmic complexity that achieves high accuracy by jointly aligning the nodes of all graphs, in contrast to most conventional methods that order nodes graph by graph. Consistency results of the proposed graphon estimator are established. A numerical study shows that the proposed estimator outperforms existing methods in terms of accuracy, especially when the dataset comprises only small and variable-size networks. Moreover, the computing time of the new method is considerably shorter than that of other consistent methodologies. Additionally, when applied to a graph neural network classification task, the proposed estimator enables more effective data augmentation, yielding improved performance across diverse real-world datasets.
Paper Structure (30 sections, 7 theorems, 101 equations, 11 figures, 4 tables)

This paper contains 30 sections, 7 theorems, 101 equations, 11 figures, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. Main contributions
  3. Related Works
  4. BACKGROUND
  5. Graphon
  6. Graphon identifiability
  7. JOINT GRAPH SORTING ESTIMATOR
  8. Estimation
  9. Optional smoothing step.
  10. Selection of the number of blocks $k$.
  11. Complexity.
  12. Theoretical results
  13. NUMERICAL EXPERIMENTS
  14. Comparison of JGS with the state of the art
  15. Scaling behavior of the JGS
  16. ...and 15 more sections

Key Result

Proposition 3.1

Let the normalized degree function $g(u)=\int_0^1 W(u,v)\,dv$ be strictly increasing. Assume that there exists a constant $L_1>0$ such that Then, with probability at least $1 - \delta$, for all $m \in \llbracket M\rrbracket$ and for all $i \in \llbracket n^{(m)}\rrbracket$, the following holds where $\varepsilon_j^{(\ell)}= \sqrt{\frac{\log(2/\delta)}{2(\,n^{(\ell)}-1)}}$ and $\varepsilon^{(N)

Figures (11)

  • Figure 1: Illustration of the Joint Graph Sorting (JGS) estimator. (a) Networks are generated from a graphon. (b) Nodes are jointly sorted across networks using normalized degrees, yielding a merged adjacency matrix with missing entries (grey). Dark blue and white denote edges and non-edges in the first graph, respectively, while dark red and misty rose denote edges and non-edges in the second graph, respectively. (c) JGS graphon estimates with different resolutions $k$, where colors represent estimated edge probabilities.
  • Figure 2: Time as a function of the graph size $n$
  • Figure 3: Time depending on the number of graphs $M$
  • Figure 4: MISE as a function of the graph size $n$
  • Figure 5: MISE depending on the number of graphs $M$
  • ...and 6 more figures

Theorems & Definitions (13)

  • Proposition 3.1: Consistency of latent positions' estimators
  • Corollary 3.2: Uniform consistency of the latent positions' estimators
  • Theorem 3.3: Consistency of JGS graphon estimate
  • Definition A.1: Notations
  • Lemma A.2: Control of the rank discrepancy
  • proof
  • proof : Proof of Proposition \ref{['prop:latent_consistency']}
  • Lemma A.3: Discretization error
  • Lemma A.4: Oracle vs. block approximation
  • proof
  • ...and 3 more