Expander Graphs The Unbalanced Case Omer Reingold The

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]

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 [Pin 73]

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

Expanders Add Entropy N N Prob. dist. X x 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 2 ndeigenvalue expansion are essentially equivalent for constant degree graphs [Tan 84, AM 84, Alo 86]

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 – 2 nd Eigenvalue • Celebrated sequence of algebraic constructions [Mar 73, GG 80, JM 85, LPS 86, AGM 87, Mar 88, Mor 94, . . . ]. • Optimal 2 nd eigenvalue (Ramanujan graphs) • “Combinatorial” constructions: [Ajt 87, RVW 00, BL 04]. • Open: Combinatorial constructions of strongly explicit Ramanujan (or almost Ramanujan) graphs. • Getting “close”: [Ben-Aroya, Ta-Shma 08]

Explicit constructions – Vertex Expansion • Optimal 2 nd eigenvalue expansion does not imply optimal vertex expansion • Exist Ramanujan graphs with vertex expansion D/2 [Kah 95]. • 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 [AC 02] • For the directed case (expansion only from left side), lossless expanders are known [CRVW 02]. 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 Constant. Degree Lossless Expanders N S, |S| K M= N D | (S)| (1 - ) D |S| CRVW 02: 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=N 0. 5 and sets of size at most K=N 0. 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 [GUV 06]: D=Poly(Log. N), M=Poly(KD) (w. some tradeoff). • Open: M=O(KD) (known w. D=Quasi. Poly(Log. N)) • Open: D= O(Log. N)

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{M 0. 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 max. T | Pr[X T] - Pr[Y T] | = ½ ||X-Y||1

Equivalently Extractors = Mixing N M T, S, |S|= K 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 2 nd 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 [LRVW 03, GUV 06]: D=Poly(Log. N / ), M=2 k(1 - ) • Selected open problem: M=2 k with D=Poly(Log. N / )

A Word About Techniques • Research on randomness extractors was invigorated with the discovery of a beautiful and surprising connection to pseudorandom generators [Tre 99]. • This further led to discoveries of connections between extractors and error correcting codes [Tre 99, RRV 99, TZ 01, TZS 01, SU 01]. • In particular, [GUV 06] relies on Parvaresh-Vardy list-decodable codes

[GUV 06] - 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)) 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?