random simplicial complexes - university of oxford...random simplicial complexes cat-school 2015...
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Random Simplicial Complexes
CAT-School 2015
Oxford
9/9/2015
Omer BobrowskiDuke University
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Part IIRandom Geometric Complexes
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Contents
Random Geometric Graphs
Random Geometric Complexes
Definitions
Probabilistic Ingredients
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Contents
Random Geometric Graphs
Random Geometric Complexes
Definitions
Probabilistic Ingredients
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Contents
Random Geometric Complexes
Definitions
Probabilistic Ingredients
Random Geometric Graphs
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
Random Graphs and Neworks
Random Geometric Graphs (RGG)
- undirected graph
• Vertices: P – either the binomial (Xn ) or the Poisson (Pn ) process
• Edges: for every x,y P , x~y iff d(x,y)r
Triangle condition:
Drawback – more difficult to analyze
Common application:
Wireless networks – transceivers with transmission range r are deployed in a region
r
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
Connectivity
From here on:
• - the uniform distribution ( )
•
• = volume of a d-dimensional unit ball
Connectivity threshold:
Other distributions – compact support same results (up to constants)
Compare to Erdős–Rényi :
Theorem [Penrose, 97]
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
Cycles in Random Geometric Graphs
From here on:
Acyclicity threshold:
In Erdős–Rényi:
Partial explanation:
The triangle condition “easier” to form tringles in G(n,r) than in G(n,p)
Theorem [?]
very different
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
Induced Subgraphs
Induced subgraphs of G(n,r):
All the graphs of the form G(S,r) where S Xn
G = a graph on k vertices
NG = the number induced subgraphs of G(n,r) that are isomorphic to G
Q: What is E{NG}?
a
b
c
d e
f
a
b
c
d a
b
c
d
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
The Expected Number of Subgraphs
Recall:
Example – triangles (k=3):
Theorem
acyclicity
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
Giant Component
Recall: L = largest connected component
Similar to Erdős–Rényi:
Continuum percolation
Theorem [Penrose & Pisztora, 96; Penrose 03]
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Contents
Random Geometric Graphs
Random Geometric Complexes
Probabilistic Ingredients
Definitions
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DEFINITIONS Omer Bobrowski
The Čech Complex
Take a set of vertices P in a metric space (0-simplexes)
Draw balls with radius r/2
Intersection of 2 balls an edge (1-simplex)
Intersection of 3 balls a triangle (2-simplex)
Intersection of k balls a (k-1)-simplex
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DEFINITIONS Omer Bobrowski
The Nerve Lemma
Lemma [Borsuk, 48]
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DEFINITIONS Omer Bobrowski
The Vietoris - Rips Complex
Take a set of vertices P in a metric space (0-simplexes)
Draw balls with radius r/2
Intersection of 2 balls an edge (1-simplex)
All pairwise intersections of 3 balls a triangle (2-simplex)
All pairwise intersections of k balls a (k-1)-simplex
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DEFINITIONS Omer Bobrowski
Some Useful Facts
Even if it is possible for to have simplexes
in any dimension
Still: for (Nerve Lemma)
Rips “approximates” Čech (de-Silva & Ghrist, 07):
The 1-skeleton of both is the same geometric graph
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Contents
Random Geometric Graphs
Definitions
Probabilistic Ingredients
Random Geometric Complexes
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Overview
Goal:
Study the limiting behavior of and as ,
Three main regimes:
Subcritical Critical Supercritical
~ vertex degree
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Connected Components (H0)
Random Geometric Graphs Theory (Penrose, Bollobás and others)
Subcritical ( ):
Critical ( ):
Supercritical ( ):
Connectivity threshold - :
(dust)
(continuum percolation)
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
The Subcritical Regime ( 0 )
Complex is very sparse
Homology is dominated by minimal & isolated cycles
Čech:
Rips:
k-cycle empty (k+1)-simplex k+2 vertices
k-cycle empty cross-polytope 2k+2 vertices
k=1
3 vertices
k=2
4 vertices
k=1
4 vertices
k=2
6 vertices
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Betti Numbers Approximation
Consider Čech the complex
Define:
Claim:
We can show:
(sparse regime)
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Similarly to subgraph counting:
Where:
Theorem [Kahle, 2011]
The Limiting Mean
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Limit Theorems
The Betti numbers:
- not independent
Poisson Approximation
The Central Limit Theorem (CLT)
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Limiting Distribution
Theorem [Kahle & Meckes, 2013]
Proofs use Stein’s method (limits of sums of “weakly dependent” variables)
Čech:
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
The Subcritical Regime ( 0 )
Exact limit values are known, as well as limit distributions
For example:
• Čech:
• Rips:
Phase transition - appearance:
Behavior is independent of f (constants might be different)
Čech: Rips:
Kahle, 2011Kahle & Meckes, 2013
B & Mukherjee, 2015
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
The Critical Regime ( l )
Cycles are neither minimal nor isolated
Inequality:
No longer true that:
Scale is known:
Law of large number & central limit theorems are proved
Limiting constants are unknown
Euler characteristic (later)
Behavior is independent of f and supp(f)
Kahle, 2011B. & Adler, 2014
Yogeshwaran, Subag & Adler, 2014B. & Mukherjee, 2015
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
The Subcritical Regime ( )
Highly connected, almost everything is covered
decays from (critical) to (coverage)
Recall: supp(f)=d-dimensional torus
Phase transition – “vanishing”:
Threshold: (connectivity threshold = logn)
NSW, 2008Kahle, 2011
B. & Mukherjee, 2015B. & Weinberger, 2015
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
The Expected Betti Numbers
Using Morse Theory, we can show in addition that:
Theorem [B. & Weinberger, 15]
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
sub-critical critical super-critical
Summary
(connectivity) (coverage)
B & Kahle –Topology of Random Geometric Complexes: a survey
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RANDOM GEOMETRIC GRAPHS Omer Bobrowski
Topological Inference
Objective: Study the homology of an unknown space from a set of samples
Example:
Problem: How should we choose r ?
Larger sample Smaller radius
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
More General Manifolds
- closed manifold, with
- a probability density function, with
- iid random samples, generated from f
Similar to the torus , but here – showing coverage is not enough
Morse Theory helps (later)
Theorem [B & Mukherjee]
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RANDOM GEOMETRIC COMPLEXES Omer Bobrowski
Clique complexes - if :
Geometric Complexes - if:
Geometric vs. Clique Complexes
Different degrees coexist
Different degrees are almost separated
Degree:
(past connectivity)
0(way before connectivity)
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Next:Extensions and Applications