Quasi-Isometric Graph Simplifications

TL;DR

This paper introduces the concept of quasi-isometries into graph simplification, exploring their properties and constructions, and demonstrating their ability to preserve key graph features like centers and medians.

Contribution

It presents a novel application of quasi-isometries to graph simplification, including new constructions and theoretical guarantees.

Findings

Quasi-isometric graph simplifications can preserve key graph features.

Two constructions of quasi-isometric simplifications are proposed.

The grouping vertices method preserves centers and medians of trees.

Abstract

Quasi-isometries are mappings on graphs, with distance-distortions parameterized by a multiplicative factor and an additive constant. The distance-distortions of quasi-isometries are in a general form that captures a wide range of distance-approximating graph simplifications. This paper introduces quasi-isometries into the field of graph simplifications, which is becoming increasingly important as large-scale graphs gain more and more prevalence. We discuss some general goals of graph simplification under the framework of quasi-isometries, and investigate several constructions of quasi-isometric graph simplifications, namely one based on maximal independent sets and one based on grouping vertices. For the latter construction, we prove that it preserves the centers and medians of trees.