Low-Rank Graphon Estimation: Theory and Applications to Graphon Games

Abstract

This paper tackles the challenge of estimating a low-rank graphon from sampled network data, employing a singular value thresholding (SVT) estimator to create a piecewise-constant graphon based on the network's adjacency matrix. Under certain assumptions about the graphon's structural properties, we establish bounds on the operator norm distance between the true graphon and its estimator, as well as on the rank of the estimated graphon. In the second part of the paper, we apply our estimator to graphon games. We derive bounds on the suboptimality of interventions in the social welfare problem in graphon games when the intervention is based on the estimated graphon. These bounds are expressed in terms of the operator norm of the difference between the true and estimated graphons. We also emphasize the computational benefits of using the low-rank estimated graphon to solve these problems.

Figures (1)

• Figure 1: Upper Left: difference between target functions for optimal interventions of a network on $n$ vertices sampled from $(1, 1/2)$-Hölder graphon $W_1(x, y) = \sqrt{|x - y|}$ and interventions based on hard-thresholding estimator. Lower Left: Rank of the hard-thresholding estimator for a network on $n$ vertices sampled from $W_1(x, y)$. Midlle: Difference between target functions for optimal interventions of a network on $n$-vertices sampled from SBM with $4$ comminties and interventions computed using a) graphon b) hard-thersholding estimator. Right: The difference between target functions for optimal interventions of a network $\bm{A}$ of size 10000 sampled from SBM with $4$ communities and interventions based on a) graphon b) hard-thersholding estimator comupted for other network $\bm{A}'$ of size $n$.

Theorems & Definitions (38)

• Lemma 1
• Theorem 1
• Remark 1: Computational cost
• Remark 2
• Theorem 2
• Remark 3
• Remark 4
• Corollary 1
• Definition 1
• Theorem 3