1. Presented by Alon Levin 29.10.2006
Expanders
Presented by Alon Levin

2. Expander - Intuitive Definition
Expander graph is an undirected graph:
Without “bottlenecks”
With high connectivity
With a “large” minimal cut
With “large” edge expansion

3. Expanders and their applications
Computational Complexity Theory
Economical Robust Networks (Computer Network, Phone Network)
Construction of Hash Functions
Error Correcting Codes
Extractors
Pseudorandom Generators
Sorting Networks

4. Expander – Definition via Edge Expansion
The edge expansion of a graph G=(V,E) is :
Theorem:
Gn are d-regular.
{ Gn } is an explicit expander family.

5. Expanders – Definition via Spectral Gap
We shall discuss undirected d-regular graphs from now and on
We shall adopt the notion of A = A(G) as G’s adjacency matrix
Since A is a real symmetric nn matrix it has n real eigenvalues:
We will denote = max( |2|, |n|)
The spectral gap is defined as d- .

6. Expanders – Definition via Spectral Gap
A d-regular graph G is an (n, d, )-expander if
 = max { |i|:1<i≤n } = max { |2(G)|, |n(G)| } and d>
If G is an (n, d, )-expander then
2(G)/2d ≤ d- ≤ 2(G)
 large expansion ~ large spectral gap!
* We will prove the inequality right after we show a rather helpful characterization of 2(G)

7. Reminder – Euclidean Scalar Product and Norm
Cauchy–Schwarz inequality :
Proof:

8. Reminder – Triangle Inequality
The triangle inequality:
Proof:

9. Rayleigh Quotient
The second largest eigenvalue of a real symmetric matrix can be computed this way:
Proof: Let be an orthonormal basis of A’s eigenvectors, where the ith vector corresponds to eigenvalue i(A).
“≤”: If we let then we would get |2| and
|n|, so the maximum is at least .

10. Rayleigh Quotient (Proof continues):
“≥”: Denote an arbitrary x as and since
 a1=0. Now and since that:
=max{|i|:1<i≤n}

11. Spectral gap and edge expansion
Now we are ready to prove the following lemma:
if G is an (n, d, )-expander then d- ≤ 2(G)
Proof: G=(V,E), |V|=n. Let SV, |S|≤n/2. We shall set a vector x similar to S’s indicator, but so that <x, >=0 and so by the Rayleigh quotient we have
So, we will define:

12. Spectral gap and edge expansion
As we can easily see, x is orthogonal to since
We can also notice the equalities: ;

13. Spectral gap and edge expansion
By A’s definition, , and since A is symmetric: X1 - Xi xn

14. Spectral gap and edge expansion
Edges originating in S
Cut in half due to edges double count
Cut edges

15. Spectral gap and edge expansion
Which implies
since by our assumption , because

16. Since it’s a multigraph (edge set is a multiset)
Regular Graph Union
Here we shall prove a simple lemma:
If G is a d-regular graph over the vertex set V, and H is a
d’-regular graph over the same vertices, then
G’ = GH = (V,E(G)  E(H)) is a d+d’-regular graph such that (G’)  (G) + (H)
Proof: Take x s.t.
Then,
Rayleigh
quotient
AG’=AG+ AH
Since it’s a multigraph (edge set is a multiset)

17. The Final Expander Lemma
Let G=(V,E) be an (n, d, )-expander and FE.
Then the probability that a random walk which starts on an edge from F will pass on an edge from F on it’s tth step is
bounded by
Motivation: Showing that a random walk on a good expander (large spectral gap) behaves similarly to independent choice of random edges F

18. The Final Expander Lemma
Proof: x is a vertices distribution vector s.t.
xv=Pr[Our walk starts at vertex v]
The probability to reach vertex u at certain point is
, thus
, where x is the current distribution and x’ is
the new one. By the definition of A=A(G) we can write
x’=Ax/d. We denote Ã=A/d, so x’=Ãx.
After i steps, the distribution is x’=Ãix.
Pr[u is reached from v]=1/d
Since G is d-regular

19. The Final Expander Lemma
P is the probability that an edge from F will be traversed on the tth step.
Let yw be the number of edges from F incident on w divided by d.
Then,

20. The Final Expander Lemma
Calculation of initial x: first step is picking an edge from F and then one of it’s vertices, so for vertex v it’s
dyv is the number of edges from F incident on v

21. The Final Expander Lemma
G is d-regular, thus each row in à sums to one.
If is the uniform distribution on G, i.e , then Since x is a probability function, it can be decomposed into , where
since x and are probability functions:
Then,

22. The Final Expander Lemma
When G is d-regular, each row in à sums to one, and thus if then
A more intuitive way of seeing this, other than algebra, is considering a random walk with uniform distribution on a d-regular graph:

23. The Final Expander Lemma
Linearity of scalar product
Hence,

24. The Final Expander Lemma
Cauchy-Schwartz inequality
(Ã)=(A)/d +
Rayleigh quotient t-1 times

25. The Final Expander Lemma
Since xi are positive,
Maximum is achieved when all edges incident to v are in F, and in that case , so:

26. The Lubotzky-Phillips-Sarnak Expander
Take a prime p, let V=Zp{}. Define 0-1=  and connect every vertex x to:
x+1
x-1
It’s a 3-regular graph with <3 

27. The Margulis/Gaber-Galil Expanders
Take V=ZnZn, so |V|=n2. Given v=(x,y)V connect it to the following vertices:
(x+2y,y)
(x,2x+y)
(x,2x+y+1)
(x+2y+1,y)
(all operations are done modulo n)
This is an 8-regular graph with =52<8, so it’s spectral gap is about 0.93

28. The End
Questions?