EXPANDER GRAPHS

1. EXPANDER GRAPHS Properties & Applications

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

3. 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

4. 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,

5. 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

6. 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!

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

8. Examples • Peterson Graph: h(G)=1

9. Examples • 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

10. Definition • Definition 2: a family (Gn) of finite connectedk-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.

11. Spectral 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

12. 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

13. 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 (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.

14. Spectral 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.

15. 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)

16. Constructibiliy • 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.

17. Some 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: simplecombinatorial construction of constant-degree expander graphs using the zig-zag graph product!

18. Amplification of Expanders • Need: some applications need an expansion coefficient that is larger than the one associated with a constructed (GN). Amplify (GN)  (GNk) • 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 GNk satisfies: (1) It is dk-regular (2) Its expansion coefficient is (1 + c)k - 1 (3) If GN is weakly constructible, so does GNk

19. An 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.

20. An Application of Expanders • Using Expanders: start with a d-regular expander graph GN with expansion c, the construct GlNby 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 .

21. 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.

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