Graph eigenvalues give new bounds on complex network partitions
A polylogarithmic higher-order Cheeger inequality
Data Structures and Algorithms
Summary
This paper explores how to break a network into multiple groups with small connections between them. The authors show a new mathematical guarantee that links certain eigenvalues of the network’s structure to how well it can be split into many parts. Their method improves earlier estimates, especially when dividing the graph into many sections. The result involves creating functions with special properties related to the graph’s structure to prove these partition quality bounds. This offers a stronger tool for understanding and working with complex networks.
What this means in practice
- •For network engineers: Improve algorithms that identify clusters in large networks by using tighter theoretical bounds on partition quality derived from spectral graph properties.
- •For data scientists: Develop better tools for community detection in graph-structured data with new guarantees linking spectral features to group separations.
A theory result. No direct application yet.
Authors
Yunpeng Li
Abstract
Let $λ_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph, and let $ρ_G(k)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove \[ ρ_G(k)\le C[1+\log(k+1)]^5\sqrt{λ_k(G)} \] for an absolute constant $C$. The construction gives exactly $k$ sets and a bound in terms of $λ_k(G)$, with all boundaries and volumes measured in the original graph. More strongly, it yields $k$ nonnegative functions with pairwise disjoint supports and Rayleigh quotients $O([1+\log(k+1)]^{10}λ_k(G))$. The proof uses independent local cutoffs whose lost covariance is controlled by conditioning on the loss along a principal direction in each cell. A regularized spectral embedding bounds cutoff energy on the entire original low eigenspace, while an adaptive construction reduces the remaining coefficient dimension by at least half at each stage. A dimension argument then converts almost rank-one local covariances into exactly $k$ scalar witnesses.