Private Graphon Estimation for Sparse Graphs

TL;DR

This paper introduces algorithms for private estimation of sparse graph models, ensuring privacy while accurately approximating the underlying graphon as the network size grows, with proven consistency and explicit error bounds.

Contribution

It presents the first node-differentially-private algorithms for nonparametric graphon estimation in sparse graphs with theoretical guarantees.

Findings

Algorithms achieve consistency in estimating the graphon as network size increases.

Explicit error bounds are provided, matching or improving nonprivate results.

Methods work under conditions of bounded graphon and growing average degree.

Abstract

We design algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members. Given a sparse input graph $G$, our algorithms output a node-differentially-private nonparametric block model approximation. By node-differentially-private, we mean that our output hides the insertion or removal of a vertex and all its adjacent edges. If $G$ is an instance of the network obtained from a generative nonparametric model defined in terms of a graphon $W$, our model guarantees consistency, in the sense that as the number of vertices tends to infinity, the output of our algorithm converges to $W$ in an appropriate version of the $L_2$ norm. In particular, this means we can estimate the sizes of all multi-way cuts in $G$. Our results hold as long as $W$ is bounded, the average degree of $G$ grows at least like the log of the number of vertices, and the number of blocks goes to infinity at an appropriate rate. We give explicit error bounds in terms of the parameters of the model; in several settings, our bounds improve on or match known nonprivate results.