Expander Graphs: The Unbalanced Case

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 Parvaresh-Vardy Codes [Guruswami,Umans,Vadhan 06]

Symmetric N N  D D G - Undirected N Bipartite Graphs • As a preparation for the unbalanced case we will talk of bipartite expanders. • Can also capture undirected expanders:

N N S, |S| K |(S)|  A |S| (A > 1) D Vertex Expansion Every (not too large) set expands.

N N S, |S| K |(S)|  A |S| (A > 1) D Vertex Expansion • Goal: minimize D(i.e. constant D) • Degree 3 random graphs are expanders [Pin73]

N N S, |S| K |(S)|  A |S| (A > 1) D Vertex Expansion Also: maximize A. • Trivial upper bound: A  D • even A ≲ D-1 • Random graphs: AD-1

N N D 2nd Eigenvalue Expansion • 2nd eigenvalue (in absolute value) of (normalized) adjacency matrix is bounded away from 1 • Can be interpreted in terms of Renyi (l2) entropy

N x’ Expanders Add Entropy N • 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 Prob. dist. X Induced dist. X’ D x

N N D 2nd Eigenvalue Expansion • 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 X X’

N N D 2nd Eigenvalue Expansion • Interestingly, vertex expansion and 2nd-eigenvalue expansion are essentially equivalent for constant degree graphs [Tan84, AM84, Alo86] X X’

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

Unique neighbor of S Non Unique neighbor Property 1: A Very Strong Unique Neighbor Property S, |S| K, |(S)| 0.9 D |S| S • S has 0.8 D |S| unique neighbors ! • We call graphs where every such S has even a single unique neighbor – unique neighbor expanders

Remains a lossless expander even if adversary removes (0.7 D) edges from each vertex. Property 2: Incredibly Fault Tolerant S, |S| K, |(S)| 0.9 D |S|

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 non-constant degree).

N N D Unbalanced Expanders • Many applications need

N D Unbalanced Expanders • Many applications need unbalanced expanders: M

N D Array of Definitions M • Many flavors: • How unbalanced. • Measure of entropy. • Lossless vs. lossy. • Is X’close to full entropy? • Lower vs. upper bound on entropy of X. • … X X’

N D S, |S|= N 0.9 |(S)|  10 D Vertex Expansion Revisited M • Even previously trivial tasks • require D = (log N/log M) • M << N  Farewell constant degree

N S, |S| K D Slightly-Unbalanced Constant-Degree Lossless Expanders M= N |(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/ )))

N D Open: More Unbalanced M • E.g. M=N0.5 and sets of size at most K=N0.2expand. While being greedy: • Unique neighbor expanders • Lossless expanders • Minimal Degree

N D Super-Constant Degree M S, |S| K |(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)

S, |S|≥ K |(S)| > (1-) M D Dispersers [Sipser 88] N M • Bounds: • D ≥ 1/ log(N/K) • DK/M ≥ log 1/ -- must be lossy • Explicit constructions are (comparably) good but still not optimal …

N x’ Increasing Entropy? M • Can Renyi entropy increase ? • |Col(X’)| < |Col(X)|  (essentially) D> min{M0.5, N/M} Prob. dist. X Induced dist. X’ D x

N D Extractors [NZ 93] M ≪N X’ 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  

N T, | e(S,T)/DK - |T|/N | <  S, |S|= K D Equivalently Extractors = Mixing M • Vertex Expansion – Sets on the left have many neighbors. • Mixing Lemma – the neighborhood of S hits any T with roughly the right proportion.

random bits 2-Source Extractors source of biased correlated bits EXT almost uniform output another independent weak source • Recently – lots of attention and results • Randomness Extractors are a special case, where the 2nd source is truly random.

Explicit Constructs. of Extractors Interpretation: extracting an arbitrary constant fraction of entropy • Extractors are highly motivated in applications. As a general rule of thumb: “Anything expanders can do, extractors 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=2kwith D=Poly(LogN / ) Interpretation: extracting all the entropy

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 = (y, f(y), (f h mod E)(y), (f h2mod E)(y), …, (f hm-1mod E)(y)) where E(Y) = irreducible poly of degree nh = 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?

Expander Graphs: The Unbalanced Case