EXPANDER GRAPHSEXPANDER GRAPHS

Properties & ApplicationsProperties & Applications

Things to cover !Things to cover ! Definitions Properties

Combinatorial, Spectral properties Constructions

“Explicit” constructions Applications

Networks, Complexity, Coding theory, Sampling, Derandomization

Intuitive DefinitionIntuitive Definition

Intuitively: a graph for which any “small” subset of vertices has a relatively “large” neighborhood.

Conceivably: it allows to build networks with guaranteed access for making connections or routing messages.

Removing random edges (local connection failures) does not reduce the property of an expander by much! Fault-tolerance

Graph Theory VocabularyGraph Theory Vocabulary Neighborhood of a vertex v: Neighborhood of U V: Boundary of U: d-regular graph : every vertex has degree d

Definition 1: a d-regular graph is a (d,c)-expander or has a c-expansion (for some positive c) iff for every subset U V of size at most|V|/2,

}),(:{)( EvuVuv )()( vU

Uv

UUU \)()(

|c|U(U)||

Remark!Remark! Routing a messages from a node A to

another node B in an (d,c) expander graph: At least (1+d)(1+c) nodes at distance 2 from A Further away, (1+d) (1+c)k nodes at distance k from A Continue until having a reachable set of nodes VA that has

more than |V|/2 nodes: the node B may not be in VA

Starting from B, we eventually obtain a set VB that has more that |V|/2.

The sets VA and VB must overlap There is a path of length 2(k+1) from A to B, where

k=logc+1|V|/2 larger c implies shorter path

Simple ResultSimple Result Proposition: For all c > 0 and for all sufficiently

large n, there exists NO (2,c)-expander graph with n vertices.

Proof: without loss of generality, assume that the graph is connected. Consider a connected subset of n/2 vertices. Its boundary is of size 2. Choose the number of vertices in the graph such that c.n/2 > 2!

Isoperimetric ConstantsIsoperimetric Constants Expanding constant = Isoperimetric constant:

The boundary is expressed either in terms of vertices or in terms of edges.

||0,:|\||,|min

|)(|inf)( UVU

UVU

UGh

2

||||0,:

||

|)(|min)(

VUVU

U

UGh

ExamplesExamples Peterson Graph: h(G)=1

ExamplesExamples

Complete Graph Kn of n vertices: If |U|=l then the boundary of U has l(n-l) edges so that h(Kn)=n-[n/2]~n/2

Cycle Cn of n vertices: if |U|=n/2 then the boudary of U has 2 edges, so that h(Cn)4/n

DefinitionDefinition Definition 2: a family (Gn) of finite connected k-

regular graphs is a family of expanders if |Vn| when n and there exists > 0 such that h(Gn) for every n.

Comments: k-regularity assumption included to assure that the number of edges of Gn grows linearly with the number of vertices. Hence a family of complete graphs is a bad example.

Optimization problem: best connectivity from a minimal number of edges.

Spectral PropertiesSpectral Properties Adjacency matrix A: Aij=number of edges

joining vi to vj. It is n-by-n symmetric matrix and it has n real eigenvalues counting multiplicities: 0 … n-1

Proposition 1.1: let G be a k-regular graph of n vertices, then:(1) The largest eigenvalue 0 = k

(2) All eigenvalues i for 1 i n-1 satisfy |i| k

(3) 0 has multiplicity 1 iff G is connected

Spectral PropertiesSpectral Properties Bipartite Graphs: it is possible to paint the

vertices with two colors in such a way no two adjacent vertices have the same color.

Proposition 1.2: let G be a connected, k-regular graph of n vertices. The following are equivalent:

(1) G is bipartite(2) The spectrum of G is symmetric about 0(3) The smallest eigenvalue is n-1 = -k

Spectral Gap of G: k - 1= 0 - 1

Spectral PropertiesSpectral Properties Theorem1.1: Let G be a finite connected k-

regular graph without loops. Then:

Rephrasing the main problem: Give a construction for a family of finite connected k-regular graphs (Gn) such that |Vn| when n and there exists > 0 for which k - 1(Gn) for every n.

Observation1.1: To have good quality

expanders, the spectral gap need to be as large as possible.

)(2)(2

)(1

1

kkGh

k

Spectral PropertiesSpectral Properties Theorem1.2: Let (Gn) be a family of finite

connected k-regular graph with |Vn| when n . Then:

Observation1.2: the spectral gap cannot be arbitrary large!

Definition: a finite connected k-regular graph G is Ramanujan if for every eigenvalue k,

A family of Ramanujan graphs is an optimal solution from the spectral perspective.

12)(inflim 1 kGnn

12|| k

Some Expanders!Some Expanders! Theorem1.3: For the following values of k, there

exists infinite families of k-regular Ramanujan graphs:

k = p + 1, p an odd prime (Lubotzky-Philips-Sarnak, Margulis)

Algebraic groups, modular forms, Riemann Hypothesis for curves over finite fields.

k = 3 (Chiu) k = q + 1, q is prime power (Morgenstern)

ConstructibiliyConstructibiliy Consider a family of expander graphs (GN) and

assume that N = 2n for some n, and that the vertices of GN are the 2n strings of length n.

Weak Constructibility: GN is weakly constructible if an explicit representation of it can be given in polynomial time of N.

Strong Constructibility: GN is strongly constructible if when given an n-bit long vertex of GN we can construct a list of all its neighbors in polynomial time of n.

Some Explicit ConstructionsSome Explicit Constructions Gabber and Galil: the first construction with an

explicitly given constant vertex expansion. Bipartite Graph: V = A B where |A| = |B| = m2 and

vertices in A and B are indexed by ordered pairs in [m]x[m]. Then,

where the addition is done modulo m.The degree of this graph is 5 and the vertex expansion for a set of size s is where n is the number of vertices.

Reingold, Vadhan, and Wigderson: simple combinatorial construction of constant-degree expander graphs using the zig-zag graph product!

)},1(),,(,1,(),,(),,{(}),({ yyxyyxyxxyxxyxyx

sns )/1)(4/)32((

Amplification of ExpandersAmplification of Expanders Need: some applications need an expansion

coefficient that is larger than the one associated with a constructed (GN). Amplify (GN) (GN

k) How: add (u,v) such that there exists a path of

length exactly k between u and v in GNk.

Spectral consequence: MNk = (MN)k

Proposition: if GN is a d-regular graph with expansion coefficient c, then GN

k satisfies:

(1) It is dk-regular (2) Its expansion coefficient is (1 + c)k - 1 (3) If GN is weakly constructible, so does GN

k

An Application of ExpandersAn Application of Expanders Problem: Let W be a set of witnesses {0,1}n of

size at least 2n-1. Give a randomized algorithm A such that when given < ½ satisfies:

Pr[A outputs an a witness of W] > 1 -

Trivial solution: pick –log() strings independently, each giving a probability of at least ½ to hit W.

Restrictions: running time should be poly(n/), and at most n bits of randomness are allowed to be used.

An Application of ExpandersAn Application of Expanders Using Expanders: start with a d-regular expander

graph GN with expansion c, the construct GlN

by choosing l = log(1/)/log(1 + c) new expansion coefficient ~ 1/.

Select at random a vertex in GlN

Scan the neighbors of v, and output a neighbor in W if such exists, else fail

Remarks: each vertex is represented as a string of n bits, thus only n bits of randomness are required.

Complexity: poly(n.dl) = poly(n/) Correctness: fails with probability at most .

More ApplicationsMore Applications Random walk on expanders: taking an l step

random walk in an expander graph is in a way similar to choosing l vertices at random: Uniform independent sampling with less random bits!

Cryptography: again using random walks on constructive expanders, one can transform any regular weak one-way function (easily inverted on all but a polynomial fraction of the range) into a strong one while preserving security.

Complexity: amplification of success probability of randomized algorithms.

More ApplicationsMore Applications Coding theory: asymptotically good error

correcting codes based on expanders.