Betti numbers of random simplicial complexes MATTHEW KAHLE & ELIZABETH MECKE

1. Betti numbers of random simplicial complexes MATTHEW KAHLE & ELIZABETH MECKE Presented by Ariel Szapiro

2. 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 …

3. 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.

4. 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))

5. 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 What are the Betti numbers ? 1 3 2 4 5 Simplexes complex with dimension: 0 are all the dots 1 are all the lines 2 are all the triangels

6. random Cech & Rips complex The random Cech complex The random Rips complex

7. 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.

8. random Cech & Rips complex example and differences • Let say we are in an instance of random geometricgraph with n=5 and r = 1 1 3 2 4 5 In Cech configuration the Simplexes are: In Rips configuration the Simplexes are:

9. erdos-r’enyi random clique complexemain results • Theorem on Expectation • Central limit theorem

10. erdos-r’enyi random clique complexemain results

11. random cech& Rips complexmain results • There 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 – • Note – since the results for Cechand Rips complexes are very similar we will ignore the former.

12. random cech& rips complexmain results - subcritical • In the Subcritical regime the simplicial complexes that is constructed 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)

13. random cech& rips complexmain results - subcritical • Central limit Theorem A very interesting outcome from the previous Theorem is that you can know a.a.s in this regime that:

14. random cech& rips complexmain results - critical • In the Criticalregime 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)

15. random cech& Rips complexmain results - supercritical • In the Supercriticalregime 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)

16. random cech& rips complexmain results - connected • In the Connectedregime 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

17. 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 :

18. 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:

19. 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.

20. 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