EXPANDER GRAPHS Properties & Applications

Things to cover ! Definitions Properties Combinatorial, Spectral properties Constructions “Explicit” constructions Applications Networks, Complexity, Coding theory, Sampling, Derandomization

Intuitive 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 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,

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 V A that has more than |V|/2 nodes: the node B may not be in V A Starting from B, we eventually obtain a set V B that has more that |V|/2. The sets V A and V B must overlap There is a path of length 2(k+1) from A to B, where k=log c+1 |V|/2 larger c implies shorter path

Simple 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 Constants Expanding constant = Isoperimetric constant: The boundary is expressed either in terms of vertices or in terms of edges.

Examples Peterson Graph: h(G)=1

Examples Complete Graph K n of n vertices: If |U|=l then the boundary of U has l(n-l) edges so that h(K n )=n-[n/2]~n/2 Cycle C n of n vertices: if |U|=n/2 then the boudary of U has 2 edges, so that h(C n )  4/n

Definition Definition 2: a family (G n ) of finite connected k- regular graphs is a family of expanders if |V n |   when n   and there exists  > 0 such that h(G n )   for every n. Comments: k-regularity assumption included to assure that the number of edges of G n 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 Properties Adjacency matrix A: A ij =number of edges joining v i to v j. 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 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 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 (G n ) such that |V n |   when n   and there exists  > 0 for which k -  1 (G n )   for every n. Observation1.1: To have good quality expanders, the spectral gap need to be as large as possible.

Spectral Properties Theorem1.2: Let (G n ) be a family of finite connected k-regular graph with |V n |   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.

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)

Constructibiliy Consider a family of expander graphs (G N ) and assume that N = 2 n for some n, and that the vertices of G N are the 2 n strings of length n. Weak Constructibility: G N is weakly constructible if an explicit representation of it can be given in polynomial time of N. Strong Constructibility: G N is strongly constructible if when given an n-bit long vertex of G N we can construct a list of all its neighbors in polynomial time of n.

Some Explicit Constructions Gabber and Galil: the first construction with an explicitly given constant vertex expansion. Bipartite Graph: V = A  B where |A| = |B| = m 2 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!

Amplification of Expanders Need: some applications need an expansion coefficient that is larger than the one associated with a constructed (G N ). Amplify (G N )  (G N k ) How: add (u,v) such that there exists a path of length exactly k between u and v in G N k. Spectral consequence: M N k = (M N ) k Proposition: if G N is a d-regular graph with expansion coefficient c, then G N k satisfies: (1) It is d k -regular (2) Its expansion coefficient is (1 + c) k - 1 (3) If G N is weakly constructible, so does G N k

An Application of Expanders Problem: Let W be a set of witnesses  {0,1} n of size at least 2 n-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 Expanders Using Expanders: start with a d-regular expander graph G N with expansion c, the construct G l N by choosing l = log(1/  )/log(1 + c) new expansion coefficient ~ 1/ . Select at random a vertex in G l N 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.d l ) = poly(n/  ) Correctness: fails with probability at most .

More 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 Applications Coding theory: asymptotically good error correcting codes based on expanders.