Expander Graphs: The
Unbalanced Case
Omer Reingold
The Weizmann Institute
What's in This Talk?
• Expander Graphs – an array of definitions.
• Focus on most established notions, and open
problems on explicit constructions. Mainly
in the unbalanced case since this is
– What applications often require
– Where constructions are very far from
optimal
• Will flash one construction (no details) Unbalanced expanders based on ParvareshVardy Codes [Guruswami,Umans,Vadhan 06]
Bipartite Graphs
• As a preparation for the unbalanced case
we will talk of bipartite expanders.
• Can also capture undirected expanders:
G - Undirected
N
D
Symmetric
N

N
D
Vertex Expansion
N
S, |S| K
N
D
|(S)|  A |S|
(A > 1)
Every (not too large) set expands.
Vertex Expansion
N
S, |S| K
N
D
|(S)|  A |S|
(A > 1)
•Goal: minimize D (i.e. constant D)
• Degree 3 random graphs are
expanders [Pin73]
Vertex Expansion
N
S, |S| K
N
D
|(S)|  A |S|
(A > 1)
Also: maximize A.
• Trivial upper bound: A  D
– even A ≲ D-1
• Random graphs: AD-1
nd
2
Eigenvalue Expansion
N
N
D
• 2nd eigenvalue (in absolute value) of (normalized)
adjacency matrix is bounded away from 1
• Can be interpreted in terms of Renyi (l2) entropy
Expanders Add Entropy
N
Prob. dist. X
x
N
D
Induced dist. X’
x’
• Vertex expansion:
|Support(X’)|  A |Support(X)|
• Some applications rely on “less naïve”
measures of entropy.
• Col(X) = Pr[X(1)=X(2)] = ||X||2
nd
2
Eigenvalue Expansion
N
X
N
D
X’
• Col(X’) –1/N  2 (Col(X) –1/N)
• Renyi entropy (log 1/Col(X)) increases as long as:
 < 1 and Col(X) is not too small
nd
2
Eigenvalue Expansion
N
X
N
D
X’
• Interestingly, vertex expansion and 2ndeigenvalue expansion are essentially
equivalent for constant degree graphs
[Tan84, AM84, Alo86]
Explicit Constructions
Applications need explicit constructions:
• Weakly explicit: easy to build the
entire graph (in time poly N).
• Strongly explicit:
– Given vertex name x and edge label i
easy to find the ith neighbor of x
(in time poly log N).
Explicit constructions – 2nd Eigenvalue
• Celebrated sequence of algebraic
constructions [Mar73,
GG80,JM85,LPS86,AGM87,Mar88,Mor94,...].
• Optimal 2nd eigenvalue (Ramanujan graphs)
• “Combinatorial” constructions: [Ajt87,
RVW00, BL04].
• Open: Combinatorial constructions of
strongly explicit Ramanujan (or almost
Ramanujan) graphs.
• Getting “close”: [Ben-Aroya,Ta-Shma 08]
Explicit constructions – Vertex
Expansion
• Optimal 2nd eigenvalue expansion does not
imply optimal vertex expansion
• Exist Ramanujan graphs with vertex
expansion  D/2 [Kah95].
• Lossless Expander – Expansion > (1-) D
• Why should we care?
– Limitation of previous techniques
– Many applications
Property 1: A Very Strong
Unique Neighbor Property
S, |S| K, |(S)|  0.9 D |S|
S
Unique neighbor of S
Non Unique neighbor
S has  0.8 D |S| unique neighbors !
• We call graphs where every such S has even a
single unique neighbor – unique neighbor expanders
Property 2: Incredibly Fault
Tolerant
S, |S| K, |(S)|  0.9 D |S|
Remains a lossless expander even if adversary
removes (0.7 D) edges from each vertex.
Explicit constructions – Vertex
Expansion
• Open: lossless expanders for the
undirected case.
– Unique neighbor expanders are known [AC02]
• For the directed case (expansion only
from left side), lossless expanders are
known [CRVW02]. Expansion D-O(D).
• Open: expansion D-O(1) (even with nonconstant degree).
Unbalanced Expanders
• Many applications need
N
N
D
Unbalanced Expanders
• Many applications need unbalanced
expanders:
N
M
D
Array of Definitions
N
• Many flavors:
X
M
D
– How unbalanced.
– Measure of entropy.
– Lossless vs. lossy.
– Is X’ close to full entropy?
– Lower vs. upper bound on entropy of X.
–…
X’
Vertex Expansion Revisited
N
S, |S|= N 0.9
M
D
|(S)|  10 D
• Even previously trivial tasks
require D = (log N/log M)
• M << N  Farewell constant degree
Slightly-Unbalanced ConstantDegree Lossless Expanders
M= N
N
S, |S| K
D
|(S)| (1-) D |S|
CRVW02:
0<, 1 constants  D constant & K= (N)
In case someone asks:
K= ( M/D) & D= poly(1/ , log (1/ ))
(fully explicit: D= quasipoly(1/ , log (1/ )))
Open: More Unbalanced
N
M
D
• E.g. M=N0.5 and sets of size at most
K=N0.2 expand. While being greedy:
• Unique neighbor expanders
• Lossless expanders
• Minimal Degree
Super-Constant Degree
N
S, |S| K
M
D
|(S)|  (1-)D |S|
• State of the art [GUV06]: D=Poly(LogN),
M=Poly(KD) (w. some tradeoff).
• Open: M=O(KD)
(known w. D=QuasiPoly(LogN))
• Open: D= O(LogN)
Dispersers [Sipser 88]
N
S, |S|≥ K
M
D
|(S)| >
(1-) M
• Bounds:
• D ≥ 1/ log(N/K)
• DK/M ≥ log 1/ -- must be lossy
• Explicit constructions are (comparably)
good but still not optimal …
Increasing Entropy?
N
Prob. dist. X
x
M
D
Induced dist. X’
x’
•Can Renyi entropy increase ?
• |Col(X’)| < |Col(X)|  (essentially)
D> min{M0.5, N/M}
Extractors [NZ 93]
N
X
M≪N
D
X’
• (k,)-extractor if
Min-entropy(X)  k  X’ -close to uniform
• Min-entropy(X)  k if x, Pr[x]  2-k
• X and Y are -close if
maxT | Pr[XT] - Pr[YT] | = ½ ||X-Y||1  
Equivalently Extractors = Mixing
N
S, |S|= K
M
T,
D
| e(S,T)/DK - |T|/N | < 
• Vertex Expansion – Sets on the left have
many neighbors.
• Mixing Lemma – the neighborhood of S hits
any T with roughly the right proportion.
2-Source Extractors
source of biased correlated bits
another independent weak source
EXT
almost uniform output
random bits
• Recently – lots of attention and results
• Randomness Extractors are a special case,
where the 2nd source is truly random.
Explicit Constructs. of Extractors
• Extractors are highly motivated in
Interpretation: extracting an
applications.
As
a
general
rule
of
thumb:
arbitrary constant
fraction
Interpretation:
extracting
all of
“Anythingentropy
expanders can do, extractors
the entropy
can do better” …
• Lots of progress. Still very far from
optimal. Best in one direction [LRVW03,
GUV06]: D=Poly(LogN / ), M=2k(1-)
• Selected open problem: M=2k with
D=Poly(LogN / )
A Word About Techniques
• Research on randomness extractors
was invigorated with the discovery of
a beautiful and surprising connection
to pseudorandom generators [Tre99].
• This further led to discoveries of
connections between extractors and
error correcting codes [Tre99, RRV99,
TZ01, TZS01, SU01].
• In particular, [GUV06] relies on
Parvaresh-Vardy list-decodable codes
[GUV06] - Basic Construction
• Left vertex f  Fqn (poly. of degree· n-1 over Fq)
• Edge Label y  F
• Right vertices = Fqm+1
y’th neighbor of f =
2
m-1
h
h
h
(y, f(y), (f mod E)(y), (f mod E)(y), …, (f
mod E)(y))
where E(Y) = irreducible poly of degree n
h = a parameter
Thm: This is a (K,A) expander with K=hm, A = q-hnm.
Conclusions
• Many interesting variants of
expander graphs
• Constructions in general – very far
from optimal
• Any clean and useful algebraic
characterization?