ariel topology

8/12/2019 Ariel Topology

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BETTINUMBERSOFRANDOM

SIMPLICIALCOMPLEXESMATTHEW KAHLE & ELIZABETH MECKE

Presented by Ariel Szapiro


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INTRODUCTION : BETTINUMBERS

Similarity to bar codes method, Betti numbers can also tell

you a lot about the topology of an examined space or

object. Suppose we sample random points from a given

object. Its corresponding Betti numbers are a vector of

random variables k.

Understanding how k is distributed can shed a lot of light

about the original space or object. Shown here are some

interesting bounds and relation of k for three well

known random objects.


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ERDOS-RENYIRANDOMCLIQUECOMPLEXE

Erdos-Renyi random graph

Definition : The Erdos-Renyirandom graph G(n, p) is the

probability space of all graphs on vertex set [n] = {1, 2, . . . ,

n} with each edge included independently with probability p.

clique complex

The clique complexX(H)of a graphHis the simplicial complex

with vertex set V(H) and a face for each set of vertices

spanning a complete subgraph of H i.e . clique.

Erdos-Renyi random clique complex

is simplyX(G(n, p))


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EXAMPLE

Let say we are in an instance of Erdos-Renyi random

graph with n=5 andp=0.5

1

3

2

4

5

Simplexes complex with dimension: 0 are all the dots

1 are all the lines

2 are all the triangels

What are the Betti numbers ?


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RANDOMCECH& RIPSCOMPLEXRandom geometric graph

Definition: Letf : Rd

R be a probability density function, letx1, x2, . . ., xnbe a sequence of independent and identically

distributed d-dimensional random variables with common density

f, and letXn= {x1, x2, . . ., xn}.

The geometric random graph G(Xn; r) is the geometric graph with

verticesXn, and edges between every pair of vertices u, v withd(u, v) r.

The random Cech complex ; is a simplicial complex

with vertex set , and a face of ; if ,i

n

n n x i

C X r

X C X r B x r

The random Rips complex R ; is a simplicial complex with

vertex set ,and a face of R ; if , ,

for every pair , x

n

n n i j

i j

X r

X X r B x r B x r

x

The random Rips complex

The random Cech complex


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RANDOMCECH& RIPSCOMPLEXEXAMPLE

ANDDIFFERENCES

Let say we are in an instance of random geometricgraph with n=5 and r = 1

1

32

4

5

In Cech configuration the Simplexes are:In Rips configuration the Simplexes are:


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MAINRESULTS

Theorem on Expectation

1/ 1/ 1

0,1

k k

k k

k

If p n and p o n then

E

Var

N

Central limit theorem

1/ 1/ 1

1

2

1lim

1 !

k k

k

k

n k


E

kn p

1/ 2 11/

k

In particular it is shown that if or

for some constant 0, then a.a.s. 0.

kkp O n p n


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MAINRESULTS

1/

Lower bound

0.01kp n 1/2 1Upper bound

0.215k

p n 1/Lower bound

0.1kp n 1/2 1Upper bound

0.398k

p n 1/Lower bound

0.215k

p n 1/2 1Upper bound

0.517k

p n


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RANDOMCECH& RIPSCOMPLEX

MAINRESULTS

There are four main ranges i.e. regimes, with qualitativelydifferent behavior in each, for different values of r, the

ranges are :

SUBCRITICAL -

CRITICAL -

SUPERCRITICAL -

CONNECTED

1/d

r o n

1/dr n

1/ 1/d d

r n o r n

1/log / dr n n

Notesince the results for cech and rips complexes are very similarwe will ignore the former.


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MAINRESULTS- SUBCRITICAL

In the Subcriticalregime the simplicial complexes that isconstructed from the random geometric graph G(Xn; r)

intuitively, hasmany disconnected pieces.

In this regime the writes shows:Theorem on Expectation and Variance (for Rips

Complexes)

1/

2 1 2 12 2 2 2

For any 2, 1, 0, and

as where is a constant that depends only on and the

underlying density function .

d

k k

k kd k d k k k

k

d k r O n

E VarC Cn r n r

n C k

f


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MAINRESULTS- SUBCRITICAL

1/For 2, 1, 0, and this limit holds

0,1

as .

d

k k

k

d k r O n

E

Var

n

N

Central limit Theorem

A very interesting outcome from the previous Theorem is

that you can know a.a.s in this regime that:

1 1

If 1 then 02

1 1And if 1 then 0

2

k

k

k d

kd


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MAINRESULTS- CRITICAL

In the Criticalregime the expectation of all the Bettinumbers grow linearly, we will see that this is the maximal

rate of growth for every Betti number from r = 0 to infinty.

In this regime the writes shows:Theorem on Expectation (for Rips Complexes)

For any density on and 0 fixed,d kk E n


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MAINRESULTS- SUPERCRITICAL

In the Supercriticalregime the writes shows an upperbound on the expectation of Betti numbers. This illustrate

that it grows sub-linearly, thus the linear growth

of the Betti numbers in the critical regime is maximal

In this regime the writes shows:

Theorem on Expectation (for Rips Complexes)

1/

Let n points taken i.i.d. uniformly from a smoothly bounded convex

body C. Let r= , where as , and k 0 is fixed.

then

dn n

0

k c

kE O e n

for same c


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MAINRESULTS- CONNECTED

In the Connectedregime the graph becomes fullyconnected w.h.p for the uniform distribution on a convex

body

In this regime the writes shows:Theorem on connectivity

1For a smoothly bounded convex body in , endowed with

a uniform distribution, and fixed 0, if log then

the random Rips complex ( ; ) is a.a.s. k-connected.

d

d

n

C

k r n n

R X r


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METHODSOFWORK

The main techniques/mode of work to obtain the nicetheorems presented here are:

First move the problem topology into a combinatorial one

-this is done mainly with the help of Morse theory

Second use expectation and probably properties to obtainthe requested theorem

Lets take for Example the Theorem on Expectation for

Erdos-Renyi random clique complexes :

1/ 1/ 1

1

2

1lim

1 !

k k

k

kn

k


E

kn p


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METHODSOFWORKFIRSTSTAGE

The writers uses the following inequality (proven by AllenHatcher. In Algebraic topology) :

Wherefi donates the number of i-dimensional simplexes.

In the Erdos-Renyi case this is simply the number of (k +

1)-cliques in the original graph.

Thus we obtain:

1 1k k k k k f f f f

11 212

1 1 !

kkk

kn

n n pE f p

k k


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METHODSOFWORKSECONDSTAGE

Now we only need to finish the proof, we know by nowthat :

Thus we only need to squeeze the k-Betti number and

obtain the desire result.

1

1

21

12

1

1 ! 1

1

!

k

k

k

kn

k

kk p n

kk

kn

n pE f

E fk

oE f np

n pE f

k

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