Euclidean embeddings of acyclic hypergraph metrics with improved distortion bounds
Euclidean Embeddings of Acyclic Hypergraph Line Metrics
Discrete Mathematics
Summary
This work looks at a special kind of network called a hypergraph, where connections can link more than two points. The authors study how to represent the distances in these hypergraphs as points in normal flat space with as little distortion as possible. They focus on a particular class of hypergraphs called α-acyclic, showing these can be embedded with much less distortion compared to general cases. Their findings give a precise measure of how closely these complex relationships can be represented in everyday geometric space.
What this means in practice
- •For network designers: Use better geometric representations of complex network metrics for more efficient routing and visualization in acyclic hypergraph-based models.
- •For data structure engineers: Improve algorithms that rely on low-distortion embeddings of hypergraph metrics by focusing on α-acyclic hypergraphs for more accurate geometric approximations.
A theory result. No direct application yet.
Authors
Nicholas Popescu
Abstract
We study Euclidean embeddings of higher-order line metrics arising from hypergraphs. For a fixed threshold \(s\geq 1\), the \(s\)-line graph \(L_s(H)\) has one vertex for each hyperedge of \(H\), with two hyperedges adjacent when they intersect in at least \(s\) vertices; its shortest-path metric is the native \(s\)-line metric. In the unrestricted setting, every finite graph occurs as the \(s\)-line graph of some hypergraph, so the worst-case Euclidean distortion is \(Θ(\log n)\). We show that \(α\)-acyclicity yields a sharp improvement to \(Θ(\sqrt{\log\log n})\). Although the canonical join tree of an \(α\)-acyclic hypergraph may have arbitrarily large stretch relative to the native metric, we prove that for every \(s\), the \(s\)-line graphs of \(α\)-acyclic hypergraphs are exactly the dually chordal graphs. Using their constant-factor spanning-tree approximations together with classical tree-embedding bounds, we obtain an \(O(\sqrt{\log\log n})\) Euclidean embedding bound. We then show this is asymptotically tight by realizing every finite tree as the \(s\)-line graph of an \(α\)-acyclic hypergraph. Thus, the worst-case distortion is \(Θ(\sqrt{\log\log n})\), including for hypergraphs of rank at most \(3s+1\).