Discrete Poincar\'e inequalities and universal approximators for random graphs

TL;DR

This paper proves that the optimal nonlinear Poincaré constant for maps between random graphs is independent of graph size, leading to universal approximators for random graphs with broad applications.

Contribution

It resolves Kleinberg's open problem by showing the dimension-free nature of the Poincaré constant for random graphs of any degree.

Findings

Optimal nonlinear Poincaré constant is dimension-free for random graphs.

Constructs universal approximators for random graphs.

Answers a longstanding open problem in graph embeddings.

Abstract

Nonlinear Poincar\'e inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon Kleinberg (2013), asks whether the optimal nonlinear Poincar\'e constant for maps between two independent $3$-regular random graphs is dimension-free, i.e., independent of vertex-set sizes. We give a complete and affirmative resolution to Kleinberg's problem, also allowing for arbitrary graph degrees. As a corollary, we obtain a stochastic construction of $O(1)\text{-universal}$ approximators for random graphs, answering a question of Mendel and Naor.