NDLI: Metric Embedding, Hyperbolic Space, and Social Networks

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A Framework for Computing the Greedy Spanner

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On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Progressive Geometric Algorithms

On the Impossibility of Dimension Reduction for Doubling Subsets of ℓp

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The Connect-The-Dots Family of Puzzles: The Video

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Trajectory Grouping Structure: the Video

Visualizing hyperbolic Voronoi diagrams

Animation of an Algorithm for Drawing Graphs in 3D

Visualization of Floater and Gotsman's Morphing Algorithm

Geometric kth Shortest Paths: the Applet

Polytope Offsets and Straight Skeletons in 3D

Optimal Geometric Flows via Dual Programs

Sub-linear Time Hybrid Approximations for Least Trimmed Squares Estimator and Related Problems

Quasi-Polynomial Time Approximation Scheme for Sparse Subsets of Polygons

A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs

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Improved bounds for the conflict-free chromatic art gallery problem

Linear transformation distance for bichromatic matchings

Delaunay triangulations with disconnected realization spaces

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Higher Dimensional Cheeger Inequalities

Incidences between points and lines in R4: Extended Abstract

On triple intersections of three families of unit circles

Finding the Largest Disk Containing a Query Point in Logarithmic Time with Linear Storage

Smallest enclosing ball for probabilistic data

Peeling Potatoes Near-Optimally in Near-Linear Time

On the Complexity of Randomly Weighted Voronoi Diagrams

A Census of Plane Graphs with Polyline Edges

Polynomials vanishing on grids: The Elekes-Rónyai problem revisited

Using Convex Relaxations for Efficiently and Privately Releasing Marginals

Near-Linear Algorithms for Geometric Hitting Sets and Set Covers

Better embeddings for planar Earth-Mover Distance over sparse sets

Higher-Quality Tetrahedral Mesh Generation for Domains with Small Angles by Constrained Delaunay Refinement

On the reconstruction of convex sets from random normal measurements

Intersection of paraboloids and application to Minkowski-type problems

Efficient Random-Walk Methods for Approximating Polytope Volume

The Persistent Homology of Distance Functions under Random Projection

Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces

Computing Topological Persistence for Simplicial Maps

Induced Matchings of Barcodes and the Algebraic Stability of Persistence

Towards Expressive 3D Modeling: new challenges for geometric computing

Computing the Fréchet distance with shortcuts is NP-hard

The Discrete Fréchet Distance with Shortcuts via Approximate Distance Counting and Selection

Light spanners for Snowflake Metrics

Continuously Flattening Polyhedra Using Straight Skeletons

Two-Point L1 Shortest Path Queries in the Plane

Better ϵ-Dependencies for Offline Approximate Nearest Neighbor Search, Euclidean Minimum Spanning Trees, and ϵ-Kernels

Free Edge Lengths in Plane Graphs

Weight Balancing on Boundaries and Skeletons

Separation by Convex Pseudo-Circles

Euclidean Steiner Shallow-Light Trees

Measuring Distance between Reeb Graphs

Stochastic Convergence of Persistence Landscapes and Silhouettes

The Morse Theory of Čech and Delaunay Filtrations

Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph

Metric Embedding, Hyperbolic Space, and Social Networks

Sampling with Removal in LP-type Problems

Liftings and stresses for planar periodic frameworks

The Opaque Square

The JS-graphs of Join and Split Trees

Distinct distances on algebraic curves in the plane

Lower bounds on geometric Ramsey functions

Curves in Rd intersecting every hyperplane at most d + 1 times

Metric Embedding, Hyperbolic Space, and Social Networks

Content Provider	ACM Digital Library
Author	Verbeek, Kevin Suri, Subhash
Abstract	We consider the problem of embedding an undirected graph into hyperbolic space with minimum distortion. A fundamental problem in its own right, it has also drawn a great deal of interest from applied communities interested in empirical analysis of large-scale graphs. In this paper, we establish a connection between distortion and quasi-cyclicity of graphs, and use it to derive lower and upper bounds on metric distortion. Two particularly simple and natural graphs with large quasi-cyclicity are n-node cycles and n × n square lattices, and our lower bound shows that any hyperbolic-space embedding of these graphs incurs a multiplicative distortion of at least Ω(n/log n). This is in sharp contrast to Euclidean space, where both of these graphs can be embedded with only constant multiplicative distortion. We also establish a relation between quasi-cyclicity and Δ-hyperbolicity of a graph as a way to prove upper bounds on the distortion. Using this relation, we show that graphs with small quasi-cyclicity can be embedded into hyperbolic space with only constant additive distortion. Finally, we also present an efficient (linear-time) randomized algorithm for embedding a graph with small quasi-cyclicity into hyperbolic space, so that with high probability at least a (1 ™ &epsis;) fraction of the node-pairs has only constant additive distortion. Our results also give a plausible theoretical explanation for why social networks have been observed to embed well into hyperbolic space: they tend to have small quasi-cyclicity.
Starting Page	501
Ending Page	510
Page Count	10
File Format	PDF
ISBN	9781450325943
DOI	10.1145/2582112.2582139
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-06-08
Publisher Place	New York
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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