Betti numbers of random simplicial complexes

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BETTI NUMBERS OF RANDOM
SIMPLICIAL COMPLEXES
MATTHEW KAHLE & ELIZABETH MECKE
Presented by Ariel Szapiro
INTRODUCTION : BETTI NUMBERS
Informally, the kth Betti number refers to the number of
unconnected k-dimensional surfaces. The first few Betti
numbers have the following intuitive definitions:
 β0 is the number of connected components
 β1 is the number of two-dimensional holes or “handles”
 β2 is the number of three-dimensional holes or “voids”
 etc …
INTRODUCTION : BETTI NUMBERS
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 βk.
Understanding how βk is distributed can shed a lot of light
about the original space or object. Shown here are some
interesting bounds and relation of βk for three well
known random objects.
ERDOS-R’ENYI RANDOM CLIQUE COMPLEXE
Erdos-R’enyi random graph
Definition : The Erdos-R’enyi random 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 complex X(H) of a graph H is 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-R’enyi random clique complex
is simply X(G(n, p))
ERDOS-R’ENYI RANDOM CLIQUE COMPLEXE
EXAMPLE

Let say we are in an instance of Erdos-R’enyi random
graph with n=5 and p=0.5
What are the Betti numbers ?
4
1
5
3
2
Simplexes complex with dimension:0 are all the dots
1 are all the lines
2 are all the triangels
RANDOM
CECH & RIPS COMPLEX
The random Cech complex
The random Cech complex C  X n ; r  is a simplicial complex
with vertex set X n , and  a face of C  X n ; r  if xi  B  xi , r   
The random Rips complex
The random Rips complex R  X n ; r  is a simplicial complex with
vertex set X n , and  a face of R  X n ; r  if B  xi , r   B  x j , r   
for every pair xi , x j  
RANDOM
CECH & RIPS COMPLEX
Random geometric graph
Definition: Let f : Rd → R be a probability density function, let
x1, x2, . . ., xn be a sequence of independent and identically
distributed d-dimensional random variables with common density
f, and let Xn = {x1, x2, . . ., xn }.
The geometric random graph G(Xn; r) is the geometric graph with
vertices Xn, and edges between every pair of vertices u, v with
d(u, v) ≤ r.
RANDOM CECH & RIPS COMPLEX EXAMPLE
AND DIFFERENCES
 Let say we are in an instance of random
geometric graph with n=5 and r = 1
4
1
3
2
5
InInCech
Rips configuration the Simplexes are:
ERDOS-R’ENYI RANDOM CLIQUE COMPLEXE
MAIN RESULTS

Theorem on Expectation




If p   n 1/ k and p  o n 1/ k 1 then
lim
E  k 
n 
nk p
 k 1


 2 
1

 k  1!



In particular it is shown that if p  O n1/ k  or p   n1/  2 k 1
for some constant   0, then a.a.s. k  0.

Central limit theorem
If p    n 1/ k  and p  o  n 1/ k 1  then
k  E k
Var   k 
 N  0,1

ERDOS-R’ENYI RANDOM CLIQUE COMPLEXE
MAIN RESULTS
Lower bound
Lower bound
Lower bound
1/ k
p  n1/ k p 0.01
 0.215
n 1/pk n0.1
Upper bound Upper bound
Upper bound
p  n 1/2 k 1  0.215
p  n 1/2 k p1  0.398
n 1/2 k 1  0.517
RANDOM CECH
MAIN RESULTS
& RIPS COMPLEX
There are four main ranges i.e. regimes, with qualitatively
different behavior in each, for different values of r, the
ranges are :

SUBCRITICAL - r  o  n 1/ d 

1/ d
CRITICAL - r    n 

1/ d
1/ d
SUPERCRITICAL - r    n   o  r   n

CONNECTED – r    log n / n 

1/ d

Note – since the results for Cech and Rips complexes are very
similar we will ignore the former.
RANDOM CECH
MAIN RESULTS
& RIPS COMPLEX
- SUBCRITICAL
In the Subcritical regime the simplicial complexes that is
constructed from the random geometric graph G(Xn; r)
intuitively, has many disconnected pieces.
In this regime the writes shows:
Theorem on Expectation and Variance (for Rips
Complexes)

For any d  2, k  1,   0, and r  O n 1/ d 
E  k 
2 k  2 d  2 k 1
 Ck
Var   k 
2 k  2 d  2 k 1
 Ck

n r
n r
as n   where Ck is a constant that depends only on k and the
underlying density function f .
RANDOM CECH
MAIN RESULTS

& RIPS COMPLEX
- SUBCRITICAL
Central limit Theorem


For d  2, k  1,   0, and r  O n 1/ d  this limit holds
k  E  k 
Var   k 
 N  0,1
as n  .
A very interesting outcome from the previous Theorem is
that you can know a.a.s in this regime that:
1 1

If k  
 1 then  k  0
2  d 
1 1

And if k  
 1 then  k  0
2  d 
RANDOM CECH
MAIN RESULTS
& RIPS COMPLEX
- CRITICAL
In the Critical regime the expectation of all the Betti
numbers 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
d
and k  0 fixed, E  k     n
RANDOM CECH
MAIN RESULTS
& RIPS COMPLEX
- SUPERCRITICAL
In the Supercritical regime the writes shows an upper
bound 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)
Let n points taken i.i.d. uniformly from a smoothly bounded convex
body C. Let r   n  , where    as n  , and k  0 is fixed.
1/ d
then

E   k   O  k e  c n
for same c  0

RANDOM CECH
MAIN RESULTS
& RIPS COMPLEX
- CONNECTED
In the Connected regime the graph becomes fully
connected w.h.p for the uniform distribution on a convex
body
In this regime the writes shows:
Theorem on connectivity
For a smoothly bounded convex body C in d , endowed with
1d
a uniform distribution, and fixed k  0, if r    log n n 
then
the random Rips complex R( X n ; r ) is a.a.s. k-connected.


METHODS OF WORK
The main techniques/mode of work to obtain the nice
theorems 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 obtain
the requested theorem
Lets take for Example the Theorem on Expectation for
Erdos-R’enyi random clique complexes :




If p   n 1/ k and p  o n 1/ k 1 then
lim
E  k 
n 
nk p
 k 1


 2 

1
 k  1!
METHODS OF WORK
– FIRST STAGE
The writers uses the following inequality (proven by Allen
Hatcher. In Algebraic topology) :
 fk 1  fk  fk 1  k  fk
Where fi donates the number of i-dimensional simplexes.
In the Erdos-R’enyi case this is simply the number of (k +
1)-cliques in the original graph.
Thus we obtain:
 n 
E  fk   
p
 k  1
 k 1


 2 
 k 1


 2 
n k 1 p

n   k  1 !
METHODS OF WORK
– SECOND STAGE
Now we only need to finish the proof, we know by now
that :
 k 1



k 1  2 

n p

 E  f k  n
 k  1!  E  f k   1  o 1



k
1 k
p


n
k 
E  f k 1  np
 

 
k  2
n
p


 E  f k 1  n
k!
Thus we only need to squeeze the k-Betti number and
obtain the desire result.
SUMMERY
Three types of random generated complexes were
presented
 Theories on expectation and on statistic behavior of their
Betti numbers was given, for each one of the four
regimes (in Rips case)
 And the basic working technique the writers used was
presented


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