Betti numbers of random simplicial complexes
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Transcript Betti numbers of random simplicial complexes
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 n1/ k or p n1/ 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 n1/ 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