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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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ERDOS-RENYIRANDOMCLIQUECOMPLEXE
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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ERDOS-RENYIRANDOMCLIQUECOMPLEXE
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
If p n and p o n then
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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ERDOS-RENYIRANDOMCLIQUECOMPLEXE
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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RANDOMCECH& RIPSCOMPLEX
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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RANDOMCECH& RIPSCOMPLEX
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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RANDOMCECH& RIPSCOMPLEX
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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RANDOMCECH& RIPSCOMPLEX
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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RANDOMCECH& RIPSCOMPLEX
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
If p n and p o n then
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