BETTI NUMBERS OF RANDOM SIMPLICIAL COMPLEXES MATTHEW KAHLE & ELIZABETH MECKE

Presented by Ariel Szapiro

INTRODUCTION : BETTI NUMBERSInformally, 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 NUMBERSSimilarity 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 COMPLEXEErdos-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 complexThe 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 COMPLEXEEXAMPLE Let say we are in an instance of Erdos-R’enyi random

graph with n=5 and p=0.5

1

32

4

5

Simplexes complex with dimension:

0 are all the dots1 are all the lines2 are all the triangels

What are the Betti numbers ?

RANDOM CECH & RIPS COMPLEX

The random Cech complex ; is a simplicial complexwith vertex set , and a face of ; if ,

i

n

n n x i

C X rX 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 rX X r B x r B x r

x

The random Rips complex

The random Cech complex

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

1

3 2

4

5

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

ERDOS-R’ENYI RANDOM CLIQUE COMPLEXEMAIN RESULTS Theorem on Expectation

1/ 1/ 1

0,1

k k

k k

k

If p n and p o n then

EVar

N

Central limit theorem

1/ 1/ 1

12

1lim1 !

k k

kkn

k

If p n and p o n then

Ek

n 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

ERDOS-R’ENYI RANDOM CLIQUE COMPLEXEMAIN RESULTS

1/

Lower bound 0.01kp n 1/2 1

Upper bound 0.215kp n 1/

Lower bound 0.1kp n 1/2 1

Upper bound 0.398kp n 1/

Lower bound 0.215kp n 1/2 1

Upper bound 0.517kp n

RANDOM CECH & RIPS COMPLEXMAIN RESULTSThere are four main ranges i.e. regimes, with qualitatively different behavior in each, for different values of r, the ranges are :

SUBCRITICAL -

CRITICAL -

SUPERCRITICAL -

CONNECTED –

1/dr o n

1/dr n

1/ 1/d dr n o r n

1/log / dr n n

Note – since the results for Cech and Rips complexes are very similar we will ignore the former.

RANDOM CECH & RIPS COMPLEXMAIN RESULTS - SUBCRITICALIn 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)

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 kk kd k d kk k

k

d k r O n

E VarC C

n r n rn C k

f

RANDOM CECH & RIPS COMPLEXMAIN RESULTS - 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 1If 1 then 02

1 1And if 1 then 02

k

k

kd

kd

RANDOM CECH & RIPS COMPLEXMAIN RESULTS - CRITICALIn 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 and 0 fixed, dkk E n

RANDOM CECH & RIPS COMPLEXMAIN RESULTS - SUPERCRITICALIn 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 growthof 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 , where as , and k 0 is fixed.

then

dr n n

0

k ckE O e n

for same c

RANDOM CECH & RIPS COMPLEXMAIN RESULTS - CONNECTEDIn 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

1For a smoothly bounded convex body in , endowed witha uniform distribution, and fixed 0, if log thenthe random Rips complex ( ; ) is a.a.s. k-connected.

d

d

n

Ck r n n

R X r

METHODS OF WORKThe 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 :

1/ 1/ 1

12

1lim1 !

k k

kkn

k

If p n and p o n then

Ek

n p

METHODS OF WORK – FIRST STAGE The writers uses the following inequality (proven by Allen Hatcher. In Algebraic topology) :

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:

1 1k k k k kf f f f

11 21

2

1 1 !

kk k

k n

n n pE f pk k

METHODS OF WORK – SECOND STAGE Now we only need to finish the proof, we know by now that :

Thus we only need to squeeze the k-Betti number and obtain the desire result.

1

121

12

1

1 ! 1 1

!

k

kk

k nk

kk p nkk

k n

n pE fE fk

oE f np

n pE fk

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