Expander graphs

1
Expander graphs applications and combinatorial
constructions

• Avi Wigderson
• IAS, Princeton

Hoory, Linial, W. 2006 Expander graphs and
applications Bulletin of the AMS.

2
Applications in Math CS
3
Applications of Expanders
In CS

• Derandomization

• Circuit Complexity

• Error Correcting Codes

• Communication Sorting Networks

• Approximate Counting

• Computational Information

• Data Structures

4
Applications of Expanders
In Pure Math

• Topology expanding manifolds Brooks
• - Baum-Connes Conjecture
  Gromov

• Group Theory generating random group elements
  Babai,Lubotzky-Pak

• Measure Theory Ruziewicz Problem Drinfeld,
  Lubotzky-Phillips-Sarnak, F-spaces
  Kalton-Rogers

• Number Theory Thin Sets Ajtai-Iwaniec-Komlos-Pi
  ntz-Szemeredi -Sieve method Bourgain-Gamburd-Sa
  rnak
• - Distribution of integer points on spheres
  Venkatesh

• Graph Theory -

5
Expander graphs Definition and basic properties
6
Expanding Graphs - Properties

• Combinatorial no small cuts, high connectivity

• Probabilistic rapid convergence of random walk

• Algebraic small second eigenvalue

Theorem. Cheeger, Buser, Tanner, Alon-Milman,
Alon, Jerrum-Sinclair, All properties are
equivalent!
7
Expanding Graphs - Properties
S
G(V,E) V vertices, E edges Vn ( ? 8
) d-regular (d fixed)
d
?S Slt n/2 E(S,Sc) gt aSd (what we expect in
a random graph) a constant

• Combinatorial no small cuts, high
  connectivity
• Geometric high isoperimetry

8
Expanding Graphs - Properties
G(V,E) d-regular
v1, v2, v3,, vt, vk1 a random neighbor of
vk vt converges to the uniform distribution in
O(log n) steps (as fast as possible)

• Probabilistic rapid convergence of random walk

9
Expanding Graphs - Properties
G(V,E) V V
AG AG(u,v) normalized adjacency
matrix (random walk matrix)
1 ?1 ?2 ?n -1 ?(G) maxigt1 ?i
max ??AG v?? ??v?? 1, v?u ?(G) ? lt
1 1-?(G) spectral gap
0 (u,v) ? E 1/d (u,v) ? E

• Algebraic small second eigenvalue

10
Expanders - Definition
Undirected, regular (multi)graphs. G is
n,d-graph n vertices, d-regular. Theorem An
n,d-graph G is connected iff ?(G) lt1. Indeed,
if G is (non-bip) connected than ?(G) lt1-
1/dn2 G is n,d, ? -graph ?(G)? ? . G expander
if ? lt1. Definition An infinite family Gi of
ni,d, ?-graphs is an expander family if for all
i ? lt1 .
11
Random walk convergence (algebraic exp. ? fast
mixing)
G(V,E) V V
AG AG(u,v)
1 ?1 ?2 ?n eigenvalues u v1 v2
vn eigenvectors p any probability
distribution on V p(t) (AG)tp distribution
after t steps of a random walk ?p(t) u?1 ? ?n
?? p(t) u ?? ?n ?? (AG)t (pu) ?? ? ?n
(?(G))t Converges in O(log n) steps. Corollary
Diam (G) lt O(log n).
0 (u,v) ? E 1/d (u,v) ? E
12
Expander mixing lemma Alon-Chung(algebraic
exp. ? combinatorial exp.)
G(V,E) V V
AG AG(u,v)
1 ?1 ?2 ?n eigenvalues u v1 v2
vn eigenvectors S? V 1S ?i ?ivi ?1
S/?n T? V 1T ?i ?ivi ?1 T/?n
?1SAG1T ? ?i ?i?i?i ST/n ?igt1
?i?i?I E(S,T)- dST/n ? d?(G)?igt1?i?i ?
d?(G) ?ST ? d?(G)n
0 (u,v) ? E 1/d (u,v) ? E
Expected cut size like a random graph AG
? J/n
13
Existence, explicit construction and basic
parameters
14
Existence of sparse expanders
Theorem Pinsker Most 3-regular graphs are
expanders. Proof The probabilistic method (an
early application). Take a random 3-regular graph
G(V,E) with Vn. Fix a set S? V, S s lt
n/2. PrE(S,Sc) lt .1s lt (s/n)3s Pr G is not
expanding lt ?s ( ) (s/n)3s lt 1/2
n s
Challenge Explicit (small degree) expanders!
15
Explicit construction
G(V,E) V V
AG AG(u,v)
0 (u,v) ? E 1/d (u,v) ? E
Gi(Vi,Ei) infinite sequence of graphs Weakly
explicit AGi can be constructed in poly (Vi)
time Strongly explicit AGi can be constructed
in polylog (Vi) time A poly(n) algorithm,
given i, u ? Vi, lists all v ? Vi s.t. (u,v) ? Ei
16
How small can ?(G) be?
V V
AG
Amplification Consider Gk G is n,d, ? -graph
(with ? lt1) then Gk is n,dk, ?k -graph So
increasing d we can make ?(G) 1/dc for some
cgt0 Theorem Alon-Boppana An infinite family
Gi of ni,d-graphs must have ?(G) gt (2
?(d-1))/d ? 1/?d Proof (of a weaker statement).
Assume d lt n/2. n/d TrAGAGt ?i ?i2 lt
1(n-1)?(G)2 ? 1/?2d lt ?(G)
17
Basic consequences G n,d,?-graph

• Theorem. If S,T? V, (u,v) ? E a random edge, then
• Pru ?S and v ?T - ?(S) ?(T) lt ?
• Cor 1 A random neighbor of a random vertex in S
  will land in S with probability lt ?(S) ? ?
• Cor 2 Every set of size gt ?n contains an edge.
• Chromatic number (G) gt 1/?
• Graphs of large girth and chromatic number
• Cor 3 Removing any fraction ? lt ? of the edges
  leaves a connected component of 1-O(?) of the
  vertices.

18
Fault-tolerant computation
19
Infection Processes G n,d,?-graph, ?lt1/4
Cor 4 Every set S of size s lt ?n/2 contains at
most s/2 vertices with gt 2?s neighbors in
S Infection process 1 Adversary infects I0,
I0 ? ?n/4. I0S0, S1, S2, St, are defined
by v ? St1 iff a majority of its neighbors
are in St. Fact St? for t gt log n
infection dies out Proof Si1Si/2
Infection process 2 Adversary picks I0, I1, ,
It? ?n/4. I0R0, R1, R2, Rt, are defined by
Rt St ? It Fact Rt? ?n/2 for all t
infection never spreads
20
Reliable circuits from unreliable components
von Neumann
Given, a circuit C for f of size s Every gate
fails with prob p lt 1/10 Construct C for
C(x)f(x) whp. Possible? With small s?
f
1
V
0
1
V
V
1
0
0
V
V
V
X2
X3
X1
21
Reliable circuits from unreliable components
von Neumann
Given, a circuit C for f of size s Every gate
fails with prob p lt 1/10 Construct C for
C(x)f(x) whp. Possible? With small s? - Add
Identity gates
f
I
V
I
I
V
V
I
I
I
V
V
V
X2
X3
X1
22
Reliable circuits from unreliable components
von Neumann

• Given, a circuit C for f of size s
• Every gate fails with prob p lt 1/10
• Construct C for C(x)f(x) whp.
• Possible? With small s?
• Add Identity gates
• Replicate circuit
• Reduce errors

f
1
23
Reliable circuits from unreliable components von
Neumann, Dobrushin-Ortyukov, Pippenger
Given, a circuit C for f of size s Every gate
fails with prob p lt 1/10 Construct C for
C(x)f(x) whp. Possible? With small s?
Majority expanders of size O(log s)
? Analysis Infection Process 2
f
M
M
M
M
1
V
V
V
V
M
M
M
M
M
M
M
M
V
V
V
V
V
V
V
V
M
M
M
M
M
M
M
M
M
M
M
M
V
V
V
V
V
V
V
V
V
V
V
V
X2
X3
X1
X2
X3
X1
X2
X3
X1
X2
X3
X1
24
Bipartite expanders
25
(unbalanced) Bipartite expanders
G(V,E)
u
w
V
w
V
u
u
w
?S?V S? n/2, NV (S)? (3/2)S
V
H(V,VE) Vn, V(2/3)n,
degree4d ?S?V S? n/2, NV (S)? S ? S
has a perfect matching to V

26
Concentrators Cn Bassalygo,Pinsker
Cn ?S?V S? n/2, S has a perfect matching
to V
n
V
V
2n/3
27

• Networks
• - Fault-tolerance
• Routing
• Distributed computing
• Sorting

28

• Superconcentrators Valiant
• IOn
• ? k? n SCn
• I ? I
• O ? O
• IOk
• There are k vertex
• disjoint paths I to O

I
How many edges are needed for SCn ? V This
number is a Circuit lower bound for Disc.
Fourier Transform
O
29

• Superconcentrators circuits Valiant
• V E(SCn) is a
• circuit lower bound for
• linear circuits computing
• any matrix A with all
• minors nonsingular
• y Ax
• k? n ?I ? I ?O ? O
• IOk there are k
• Vertex disjoint paths I to O
• Otherwise, by Mengers
• Theorem, ?(k-1)-cut, so
• Rank(AI,O) ltk

X1 X2 X3
Xn
I
?
?
?
?
?
O
?
?
?
?
Y1 Y2 Y3
Yn
30

• Superconcentrators Valiant
• Theorem Valiant
• Linear size SCn
• Proof Pippenger
• IO? n/2
• recursion
• (2) IOgt n/2
• Matching pigeonhole
• Reduce to case (1)
• E(SCn)
• E(SC2n/3)2E(Cn)n
• E(SC2n/3)O(n) O(n)

I
SCn
O
31
Distributed routing Sh,PY,Up,ALM,AC

• n inputs, n outputs, many disjoint paths
• Permutation networks
• Non-blocking network
• Wide-sense non-blocking networks
• on-line routable networks
• .
• Building block G - some expander

Theorem Alon-Capalbo Let G be a sufficiently
strong expander. Given (s1,t1),(s2,t2),(sk,tk),
k lt n/(log n), one can efficiently find (si,ti)
edge-disjoint paths between them, on-line!
32
Sorting networks Ajtai-Komlos-Szemeredi
n inputs (real numbers), n outputs (sorted)
Many sorting algorithms of O(n log n)
comparisons Many sorting networks of O(n log2 n)
comparators
Thm AKS Explicit network with O(n log n)
comparators Proof Extremely sophisticated use
analysis of expanders Cor Monotone Boolean
formula for majority (derandomizing a
probabilistic existence proof of Valiant)
33
Derandomization
34
Deterministic error reduction
Prerror lt 1/3
Bxlt2n/3
Thm Chernoff r1 r2. rk independent (kn
random bits)
Thm AKS r1 r2. rk random path (n O(k)
random bits)
then Prerror Prr1 r2. rk ?Bx
gt k/2 lt exp(-k)
35
Metric embeddings
36
Metric embeddings (into l2)
Def A metric space (X,d) embeds with distortion
? into l2 if ? f X ? l2 such that for all
x,y d(x,y) ? ?? f(x)-f(y) ??
? ? d(x,y) Theorem Bourgain Every
n-point metric space has a O(log n) embedding
into l2 Theorem Linial-London-Rabinovich This
is tight! Let (X,d) be the distance metric of an
n,d-expander G. Proof ?f,(AG-J/n)f ? ? ?(G)
??f??2 ( 2ab a2b2-(a-b)2 ) (1-?(G))Ex,y
(f(x)-f(y))2 ? Exy (f(x)-f(y))2 (Poincare
inequality) (clog n)2

?
?
?2
All pairs
Neighbors
37
Metric embeddings (into l2)
Def A metric space (X,d) has a coarse embedding
into l2 if ? f X ? l2 and increasing,
unbounded functions ?,?R?R such that for all
x,y ?(d(x,y)) ? ?? f(x)-f(y)
??2 ? ?(d(x,y)) Theorem Gromov There exists
a finitely generated, finitely presented group,
whose Cayley graph metric has no coarse
embedding into l2 Proof Uses an infinite
sequence of Cayley expanders Comment Relevant
to the Novikov Baum-Connes conjectures
38
Nonlinear spectral gaps and metric embeddings
into convex spaces
Poincare inequality G any expander. ? ? ?f V ?
l2 Ex,y ??f(x)-f(y)??2 ? ? Exy ??f(x)-f(y)
??2 ? 1/(1-?(G)) Theorem Matousek G
any expander. ? ?p ?f V ? lp Ex,y
??f(x)-f(y)?? ? ?pExy ??f(x)-f(y) ??
Theorem Lafforgue, Mendel-Naor Construct
explicit G (super-expander) ?K ? ?K ?f V ?
lp Ex,y ??f(x)-f(y)?? ? ?K Exy ??f(x)-f(y)
?? Theorem Kasparov-Yu, Gromov Such
family of constant degree expanders give rise to
a metric space X with no coarse embedding into
any uniformly convex space.
39
Constructions
40
Expansion of Finite Groups

• G finite group, S?G, symmetric. The Cayley graph
• Cay(GS) has x?sx for all x?G, s?S.
• Cay(Cn -1,1) Cay(F2n
  e1,e2,,en)
• ?(G) ? 1-1/n2 ?(G) ? 1-1/n
• Basic Q for which G,S is Cay(GS) expanding ?

41
Algebraic explicit constructions
Margulis,Gaber-Galil,Alon-Milman,Lubotzky-Philips
-Sarnak,Nikolov,Kassabov,..
Theorem. LPS Cay(A,S) is an expander
family. Proof The mother group
approach Appeals to a property of SL2(Z)
Selbergs 3/16 thm Strongly explicit Say that
we need n bits to describe a matrix M in SL2(p)
. Vexp(n) Computing the 4 neighbors of M
requires poly(n) time!
42
Algebraic Constructions (cont.)
Very explicit -- computing neighbourhoods in
logspace
Gives optimal results Gn family of
n,d-graphs -- Theorem. AB
d?(Gn) ? 2? (d-1) --Theorem. LPS,M Explicit
d?(Gn) ? 2? (d-1)
(Ramanujan graphs)
Recent results -- Theorem KLN All finite
simple groups expand. -- Theorem H,BG SL2(p)
expands with most generators. -- Theorem BGT
same for all Chevalley groups
43
Zigzag graph product Combinatorial construction
of expanders
44
Explicit Constructions (Combinatorial)-Zigzag
Product Reingold-Vadhan-W
G an n, m, ?-graph. H an m, d, ?-graph.
Combinatorial construction of expanders.
45
Proof of the zigzag theorem
Proof Information theoretic view of
expanders. When is G an expander? Random walk is
entropy boost! p, p distributions on V before
and after a random step. DefG is an expander
iff whenever Ent(v) ltlt log n, Ent(v) gt Ent(v)
? (?gt0) Ent0(p) log supp(p) Ent1(p)
Shannons ent. Ent2(p) log ??p??2
v
v
46
Proof of the zigzag thm Reingold-Vadhan-W
(u,d)
(v,b)
(u,c)
(v,a)
Want Ent(u,d) gt Ent(v,a) ? (assuming
Ent(v,a) ltlt log nm) Case 1 Ent(av) ltlt log m
? Ent(bv) gt Ent(av) ? ?
Ent(u,d) ? Ent(v,b) ? Ent(v,a) ? Case 2
Ent(av) log m ? Ent(bv) log m ? Ent(v) ltlt
log n ? Ent(u) ? Ent(v) ? ?
Ent(cu) ltlt log m ? Ent(u,d) ?
Ent(u,c) ?
Mutually exclusive?
Linear Algebra!
47
Iterative Construction of Expanders
G an n,m,?-graph. H an m,d,? -graph.
The construction
Start with a constant size H a d4,d,1/4-graph.

• G1 H 2

Theorem. RVW Gk is a d4k, d2, ½-graph.
Proof Gk2 is a d 4k,d 4, ¼-graph.
H is a d 4, d, ¼-graph.
Gk1 is a d 4(k1), d 2, ½-graph.
48
Consequences of the zigzag product

• Isoperimetric inequalities beating e-value
  bounds
• Reingold-Vadhan-W, Capalbo-Reingold-Vadhan-W
• Connection with semi-direct product in groups
• Alon-Lubotzky-W
• - New expanding Cayley graphs for non-simple
  groups
• Meshulam-W Iterated group algebras
• Rozenman-Shalev-W Iterated wreath products
• SLL Escaping every maze deterministically
  Reingold 05
• Super-expanders Mendel-Naor
• Monotone expanders Dvir-W

49
SLL escaping mazes and navigating unknown
terrains
50
Getting out of mazes / Navigating unknown
terrains (without map memory)
Only a local view (logspace)
nvertex maze/graph
Mars 2006
Crete 1000BC
Thm Reingold 05 SLL A deterministic walk,
computable in Logspace, will visit every
vertex. Uses ZigZag expanders
Thm Aleliunas-Karp-Lipton-Lovasz-Rackoff 80 A
random walk will visit every vertex in n2 steps
(with probability gt99 )
51
Expander from any connected graph
ReingoldAnalogy with the iterative construction
G an n,m,?-graph. H an m,d,? -graph.
G an n,m, 1-?-graph. H an m,d,1/4 -graph.
nm,d2, 1-?/2-graph.
The construction
Fix a constant size H a d4,d,1/4-graph.
H a d10,d,1/4-graph.

• G1 H 2

• G1 G

Theorem. RVW Gk is a d4k, d2, ½-graph
Theorem R G1 is n, d2, 1-1/n3 ?
Proof Gk2 is a d 4k,d 4, ¼-graph.
H is a d 4, d, ¼-graph.
Gclog n is nO(1), d2, ½
Gk1 is a d 4(k1), d 2, ½-graph.
52
Undirected connectivity in Logspace R

• Algorithm
• Input GG1 an n,d2-graph
• Compute Gclog n
• Try all paths of length clog n from vertex 1.
• Correctness
• Gi1 is connected iff Gi is
• If G is connected than it is an n,d2,
  1-1/n3-graph
• G1 connected ? Gclog n has diameter lt clog n
• Space bound
• Gi? Gi1 in constant space (squaring zigzag
  are local)
• - Gclog n from G1 requires O(log n) space

53
Zigzag graph product Semi-direct group
product Expansion in non-simple groups
54
Semi-direct Product of groups
A, B groups. B acts on A as automorphisms.
Let ab denote the action of b on a.
Definition. A ? B has elements (a,b) a?A,
b?B. group mult
(a,b ) (a,b) (aab , bb)
Connection semi-direct product is a special case
of zigzag Assume ltTgt B, ltSgt A , S sB (S
is a single B-orbit)
Proof By inspection (a,b)(1,t) (a,bt)
(Step in a cloud)
(a,b)(s,1)
(asb,b) (Step between clouds)
Theorem ALW Expansion is not a group property
Theorem MW,RSW Iterative construction of Cayley
expanders
55
Example
z
AF2m, the vector space, Se1, e2,, em ,
the unit vectors
BZm, the cyclic group, T?1, shift by 1
B acts on A by shifting coordinates. Se1B.
G Cay(A,S), H Cay(B,T), and
56
Is expansion a group property?

• Lubotzky-Weiss93 Is there a group G, and two
  generating subsets S1,S2O(1) such that
• Cay(GS1) expands but Cay(GS2) doesnt ?
• (call such G schizophrenic)
• nonEx1 Cn - no S expands
• nonEx2 SL2(p)-every S expandsBreuillard-Gamburd
  09
• Alon-Lubotzky-W01 SL2(p)?(F2)p1 schizophrenic
  
• Kassabov05 Symn
  schizophrenic

57
Expansion in Near-Abelian Groups
G group. GG commutator subgroup of G GG
lt xyx-1y-1 x,y ?G gt G G0 gt G1gt gt Gk
Gk1 Gi1GiGi G is k-step solvable
if Gk1. Abelian groups are 1-step
solvable Lubotzky-Weiss93 If G is k-step
solvable, Cay(GS) expanding, then S
O(log(k)G) Meshulam-W04 There exists k-step
solvable Gk, Sk O(log(k/2)Gk), and
Cay(GkSk) expanding.
loglog.log k times
58
Near-constant degree expanders for near Abelian
groups Meshulam-W04
Iterate G G ? FqG Start with G1 Z2 Get
G1 , G2,, Gk , Gk1gtexp
(Gk) S1 , S2,, Sk , ltSk gt Gk
Sk1ltpoly (Sk)
- Sk ? O(log(k/2)Gk) deg approaching
constant
- Cay(Gk, Sk) expanding
Theorem/Conjecture Cay(G,S) expands, then G has
at most exp(d) irreducible reps of dimension d.
59
Expanders from iterated wreath products
Rozenman-Shalev-W04
d fixed. Gk Aut(Tdk) Odd automorphisms of a
depth-k uniform d-ary tree Iterative Gk1 Gk
? Ad Thm Gk expands with explicit O(1)
generators Ingredients zigzag, equations in
perfect groupsNikolov, correlated random walks
expand,
60
Beating eigenvalue expansion
61
Beating e-value expansion WZ, RVW
In the following a is a large constant.
Task Construct an n,d-graph s.t. every two
sets of size n/a are connected by an edge.
Minimize d
Ramanujan graphs d?(a2)
Random graphs dO(a log a)
Zig-zag graphs RVW dO(a(log a)O(1))
Uses zig-zag product on extractors!
Applications Sorting selection in rounds,
Superconcentrators,
62
Lossless expanders Capalbo-Reingold-Vadhan-W
Task Construct an n,d-graph in which every
set S, Sltltn/d has gt cS neighbors. Max c
(vertex expansion)
Upper bound c?d
Ramanujan graphs Kahale c ? d/2
Random graphs c ? (1-?)d
Lossless
Zig-zag graphs CRVW c ? (1-?)d Lossless
Use zig-zag product on conductors!
Extends to unbalanced bipartite graphs.
Applications (where the factor of 2
matters) Data structures, Network routing,
Error-correcting codes
63
Error correcting codes
64
Error Correcting Codes Shannon, Hamming
C 0,1k ? 0,1n CIm(C) Rate (C)
k/n Dist (C) min dH(C(x),C(y)) C good if
Rate (C) ?(1), Dist (C) ?(n) Theorem
Shannon 48 Good codes exist (prob.
method) Challenge Find good, explicit, efficient
codes. - Many explicit algebraic constructions
Hamming, BCH, Reed-Solomon, Reed-muller,
Goppa, - Combinatorial constructions Gallager,
Tanner, Luby-Mitzenmacher-Shokrollahi-Spielman,
Sipser-Spielman.. Thm Spielman good,
explicit, O(n) encoding decoding Inspiration
Superconcentrator construction!
65
Graph-based Codes Gallager60s
C 0,1k ? 0,1n CIm(C) Rate (C)
k/n Dist (C) min dH(C(x),C(y)) C good if
Rate (C) ?(1), Dist (C) ?(n)
0 0 0 0 0
0 Pz

G
1 1 0 1 0
0 1 1 z
z?C iff Pz0 C is a linear
code LDPC Low Density Parity Check (G has
constant degree)
Trivial Rate (C) ? k/n , Encoding time
O(n2)
G lossless ? Dist (C) ?(n), Decoding time
O(n)
66
Decoding
Thm CRVW Can explicitly construct graphs
kn/2, bottom deg 10, ?B?n, B? n/200,
?(B) ? 9B
0 0 1 0 1 1 Pw

1 1 1 0 1 0 1 1
w
Decoding algorithm Sipser-Spielman while Pw?0
flip all wi with i ? FLIP i ?(i) has
more 1s than 0s
B corrupted positions (B ? n/200) B set
of corrupted positions after flip
Claim SS B ? B/2 Proof B \ FLIP ?
B/4, FLIP \ B ? B/4
67
Rapidly mixing Markov chains Uniform
generation and enumeration problems Expansion
of (exponential-size) graphs which arise
naturally (as opposed to specially designed)
68
Volumes of convex bodies
Given (implicitly) a convex body K in Rd (d
large!) (e.g. by a set of linear
inequalities) Estimate volume (K) Comment
Computing volume(K) exactly is P-complete

• Algorithm Dyer-Frieze-Kannan 91
• Approx counting ? random sampling
• Random walk inside K.
• Rapidly mixing Markov chain.
• Techniques Spectral gap ?
• isoperimetric inequality

K
69
Statistical mechanics

• Example the dimer problem
• Count of domino configurations?
• Given G, count the number of
• perfect matchings
  G
• Glauber Dynamics (MCMC sampling the Gibbs
  distribution)
• Construct HG on configurations, with edges
  representing local changes (e.g. rotate adjacent
  parallel dominos).
• Then run a random walk for sufficiently long
  time.
• TheoremsJerrum-Sinclair poly(n) time
  convergence for
• near-uniform perfect matching in dense random
  graphs
• sampling Gibbs dist in ferromagnetic Ising model
• Techniques coupling, conductance,

approximately
v ? HG
70
Generating random group elements

• Given Sg1,g2,gd generators of a group G (of
  size n)
• Find a near-uniform element x of G (?g?G
  Prxg lt ?/n )
• Straight-line program (SLP) Babai-Szemeredi
• x1,x2,xt where every xi is either
• A generator gj or gj-1
• xkxm or xm-1 for m,k lt i
• TheoremBS ?G,S,g ? an SLP for g of length
  tO(log2 n)
• TheoremBabai ?G,S ? a near-uniform SLP of
  tO(log5 n)
• TheoremCooperman,Dixon,Green . SLP of
  tO(log2 n)
• Proof Cube(z1,z2,zt) subwords of
  zt-1z2-1z1-1 z1z2zt )
• x1,x2,xr with xk1 ?R Cube(x1,x2,xk), with
  rO(log n)

Techniques local expansion, Arithmetic
combinatorics
71
Extensions of expanders
72
Dimension Expanders

• Expander
• Permutations ?1, ?2, , ?k n ? n are an
  expander if for every subset S?n, S lt n/2
• ? i?k s.t. TiS?S lt (1-?)S
• Dimension expander Barak-Impagliazzo-Shpilka-W01
  
• Linear operators T1,T2, ,Tk Fn ? Fn are
  (n,F)-dimension expander if subspace V?Fn with
  dim(V) lt n/2
• ? i?k s.t. dim(TiV?V) lt (1-?) dim(V)
• Fact kO(1) random Tis suffice for every F,n.
• Lubotzky-Zelmanov04 Construction for FC
• What about finite fields?

73
Monotone Expanders

• f n ? n partial monotone map
• xlty and f(x),f(y) defined, then f(x)ltf(y).
• f1,f2, ,fk n ? n are a k-monotone expander
• if fi partial monotone and the (undirected) graph
  on n with edges (x,fi(x)) for all x,i, is an
  expander.
• Dvir-Shpilka k-monotone exp ? 2k-dimension exp
  ?F,d
• Explicit (log n)-monotone expander
• Dvir-W09 Explicit (logn)-monotone expander
  (zig-zag)
• Bourgain09 Explicit O(1)-monotone expander
• Dvir-W09 Existence ? Explicit reduction
• Open Prove that O(1)-mon exp exist!

74
Real Monotone Expanders Bourgain09

• Explicitly constructs
• f1,f2, ,fk 0,1 ? 0,1 continuous, Lipshitz,
  monotone maps, such that for every S ? 0,1 with
  ?(S)lt ½, there exists i?k such that ?(S?f2(S))
  lt (1-?) ?(S)
• Monotone expanders on n by discretization
• M( )?SL2(R), x?R, let fM(x) (axb)/(cxd)
• Take ?-net of such Mis in an ?-ball around I.
• Proof
• Expansion in SU(2) Bourgain-Gamburd
• Tits alternative Bruillard

a b c d
75
Open Problems
76
Open Problems

• Characterize Cayley expanders
• Construct Lossless expanders
• Construct
• Rate concentrators
• of constant
• left degree