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  1. Discrete & Computational Geometry
  2. Discrete & Computational Geometry : Volume 45
  3. Discrete & Computational Geometry : Volume 45, Issue 3, April 2011
  4. Random Geometric Complexes
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Discrete & Computational Geometry : Volume 57
Discrete & Computational Geometry : Volume 56
Discrete & Computational Geometry : Volume 55
Discrete & Computational Geometry : Volume 54
Discrete & Computational Geometry : Volume 53
Discrete & Computational Geometry : Volume 52
Discrete & Computational Geometry : Volume 51
Discrete & Computational Geometry : Volume 50
Discrete & Computational Geometry : Volume 49
Discrete & Computational Geometry : Volume 48
Discrete & Computational Geometry : Volume 47
Discrete & Computational Geometry : Volume 46
Discrete & Computational Geometry : Volume 45
Discrete & Computational Geometry : Volume 45, Issue 4, June 2011
Discrete & Computational Geometry : Volume 45, Issue 3, April 2011
The Overlay of Minimization Diagrams in a Randomized Incremental Construction
Simple Realizability of Complete Abstract Topological Graphs in P
The Ratio of the Perimeter and the Area of Unions of Copies of a Fixed Set
The f-Vector of the Descent Polytope
Hardness Results for Homology Localization
Smooth Fano Polytopes Arising from Finite Partially Ordered Sets
Relative (p,ε)-Approximations in Geometry
The List-Chromatic Number of Infinite Graphs Defined on Euclidean Spaces
On γ-Vectors Satisfying the Kruskal–Katona Inequalities
Dilation of Newton Polytope and p-adic Estimate
The Forest Hiding Problem
Random Geometric Complexes
New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture
Discrete & Computational Geometry : Volume 45, Issue 2, March 2011
Discrete & Computational Geometry : Volume 45, Issue 1, January 2011
Discrete & Computational Geometry : Volume 44
Discrete & Computational Geometry : Volume 43
Discrete & Computational Geometry : Volume 42
Discrete & Computational Geometry : Volume 41
Discrete & Computational Geometry : Volume 40
Discrete & Computational Geometry : Volume 39
Discrete & Computational Geometry : Volume 38
Discrete & Computational Geometry : Volume 37
Discrete & Computational Geometry : Volume 36
Discrete & Computational Geometry : Volume 35
Discrete & Computational Geometry : Volume 34
Discrete & Computational Geometry : Volume 33
Discrete & Computational Geometry : Volume 32
Discrete & Computational Geometry : Volume 31
Discrete & Computational Geometry : Volume 30
Discrete & Computational Geometry : Volume 29
Discrete & Computational Geometry : Volume 28
Discrete & Computational Geometry : Volume 27
Discrete & Computational Geometry : Volume 26
Discrete & Computational Geometry : Volume 25
Discrete & Computational Geometry : Volume 24
Discrete & Computational Geometry : Volume 23
Discrete & Computational Geometry : Volume 22
Discrete & Computational Geometry : Volume 21
Discrete & Computational Geometry : Volume 20
Discrete & Computational Geometry : Volume 19
Discrete & Computational Geometry : Volume 18
Discrete & Computational Geometry : Volume 17

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Random Geometric Complexes

Content Provider Springer Nature Link
Author Kahle, Matthew
Copyright Year 2011
Abstract We study the expected topological properties of Čech and Vietoris–Rips complexes built on random points in ℝd. We find higher-dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H k is not monotone when k>0.In particular, for every k>0, we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes.
Starting Page 553
Ending Page 573
Page Count 21
File Format PDF
ISSN 01795376
Journal Discrete & Computational Geometry
Volume Number 45
Issue Number 3
e-ISSN 14320444
Language English
Publisher Springer-Verlag
Publisher Date 2011-01-19
Publisher Place New York
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Random geometric graphs Probabilistic topology Topological data analysis Computational Mathematics and Numerical Analysis Combinatorics
Content Type Text
Resource Type Article
Subject Discrete Mathematics and Combinatorics Theoretical Computer Science Computational Theory and Mathematics Geometry and Topology
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