Exploring Zig-Zag Graph Product for Expander Construction

1. Zig-Zag Expanders Seminar in Theory and Algorithmic ResearchSashka DavisUCSD, April 2005“ Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders”O. Reingold, S. Vadhan, A. Wigderson

2. Talk Outline Introduction: notations, definitions, facts. Zig-Zag graph product: • Overview • Construction • Analysis – Intuition • Analysis

3. Expansion, Expanders For undirected graph G=(V,E) • Vertex expansion parameter is defined as: ε = min |Γ(S)\S| / |S|. S | |S| ≤|V|/2 • G is a good expander if for any S, s.t. |S| ≤|V|/2, then |Γ(S)|≥(1+ε) |S|.

4. Family of Expander Graphs A family of expander graphs {Gi} is a collection of graphs such that for all i: • Gi is d-regular. • |V(Gi)| is strictly increasing. • ε ≥c, for some constant c.

5. Undirected D-regular Graphs Notation: Let G be undirected D-regular, then: • the adjacency matrix is A(G). • the normalized adjacency matrix is M= 1/D A(G). • Spectrum σ(A)={λ0,λ1,…,λn-1}. • λ(G)= λ1. • Each row/column adds up to D • A(G) is (real) symmetric, therefore • A(G) is similar to a diagonal matrix. • σ(A)={λ0,λ1,…,λn-1} are real. • Rⁿ has an orthonormal basis consisting of eigenvectors of A(G). • (D,1n) is an eigenpair.

6. Expansion, Convergence, and λ(G) • G is a good expander then λ(G) is small Cheeger & Buser: (d-λ1)/2D ≤ e ≤ 2√(d-λ1)/D • Random walk on G converges to the uniform distribution rapidly if λ(G) is small. • Proof: (on board) We use Rayleligh-Ritz Theorem λ(G) = max <Mx,x>/<x,x> = max ||Mx||/||x|| x perp. to uniform

7. Talk Outline • Introduction: notations, definitions, facts. Zig-Zag Graph product: • Overview • Construction • Analysis – Intuition • Analysis

8. Zig-Zag Graph Product • Delivers a constant degree family of expanders. • Construction is iterative. • The analysis is algebraic. • Notation: G is (N,D,μ)-graph meaning V(G)=N, G is D-regular and has λ(G) at most μ.

9. Standard Operations • Squaring G: new edge are paths in G of length 2 (N,D,λ)2 = (N,D2,λ2) • Tensoring G (Kronecker product) (N,D,λ)  (N,D,λ) = (N2,D2,λ)

10. Expander Construction Using the Zig-Zag Graph product • Start with a constant-size expander H. • Apply simple operations to Hto construct arbitrarily large expanders. • Main Challenge: prevent the degree from growing. • New Graph Product: compose large graph w/ small graph to obtain a new graph which (roughly) inherits • Size of large graph • Degree of small graph • Expansion from both

11. z The Zig-Zag Graph Product: Theorem 1 Let G1 be (N,D,λ1)-graph and G2 be(D,d,λ2)-graph, then (G1 G2) = (ND, d2, λ1+ λ2+ λ22) Proof: Later. (Big portion of remaining 23 slides...)

12. Talk Outline • Introduction: notations, definitions, facts. Zig-Zag graph product: • Overview • Construction • Analysis – Intuition • Analysis (all the gory details..)

13. z The Construction • Building block: Let H be (D4,D,1/5)-graph • Construct a family{Gi}ofD2-regulargraphs such that • G1=H2 • Gi+1= (Gi)2 H Theorem 2 For every i, Gi is (D4i, D2, 2/5)-graph. Proof: By induction (on the board).

14. z 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 Zig-Zag Graph Product – Construction (by example) Vertices in V(G1 G2) = V(G1) V(G2) G2 u G1 v

15. z Zig-Zag Graph Product – Construction (by example) Vertices in G1 G2 = G1G2 (u,1) (u,2) (v,1) (v,2) (u,3) (v,3)

16. z 1 1 2 3 3 2 Consider ((u,1),0,0) - edge(0,0) incident to vertex (u,1). Edge of G1 G2 = VE2E2 (u,1) 0 (u,2) 1 0 1 1 0 1 (v,1) 2 (v,3) (u,3) 0 1 3 3 0 1 1 0 2 (v,2)

17. 1 1 2 3 3 z Vertex (u,1) and all its neighbors. Edges of G1 G2 (u,1) 0 1 (u,2) 0 1 1 0 0 1 1 0 1 0 1 (v,1) 2 (v,3) 0 (u,3) 1 3 3 0 1 1 0 (v,2) 1 1 (w,1) 0

18. (u,1) 0 (u,2) 1 0 1 1 0 1 1 3 2 (v,1) 2 (v,3) (u,3) 0 1 3 3 3 0 1 1 0 (v,2) z • Connect (u,i) and (v, j) iff  i, j such that • 1. i and i connected in G2 • 2. (u, i ) and (v, j ) correspond to same edge of G1 • 3.j and j connected in G2 Edges of G1 G2 = VE2E2

19. z G = G1 G2 G1 is (N,D,λ1)-graph and G2 is (D,d,λ2 )-graph |V(G)| = |V(G1)||V(G2)| = ND Degree of G = deg(G2)2=d2 Edge set of G: • a step in G2 • a step in G1 • a step in G2 λ(G) ≤ λ1 +λ2 +λ22

20. ANALYSIS of the ZIG-ZAG Graph ProductIntuition

21. z z The Eigenvalue Bound • Need to show: Random step on G1 G2 makes non-uniform probability distributions  closer to uniform. • Random step on G1G2 • 1. random step within “cloud”. • 2. jump between clouds. • 3. random step within new cloud.

22. z Analysis, Intuition (cont.) A,C – normalized adjacency matrices of G1,G2 M – normalized adjacency matrix of G • Must show: G1 G2-matrix M shrinks every vector αNDthat is perp. to uniform (Rayeigh-Ritz Thm, for 2-nd eigenvalue). • Decompose α=α||+ α, where α||is probability distribution, where distribution within clouds is uniform, and α is a distribution, where probabilities within cloud are far from uniform.

23. Case I: Non-uniform Distribution • Case I:α very non-uniform (far from uniform) within “clouds” • Step 1 makes α more uniform (by expansion of G2 ). • Steps 2 & 3 cannot make α less uniform.

24. Case II: Uniform Distribution Case II:α uniform within clouds. • Step 1: does not change α. • Step 2: Jump between clouds  random step on G1  Distribution on clouds themselves becomes more uniform (by expansion of G1)

25. Analysis of λ(G) To show that λ(G) ≤ (λ(G1) + λ(G2)+λ(G2)2) suffices to prove that to show that for any αND, perpendicular to 1ND <Mα,α> ≤ (λ(G1) + λ(G2)+λ(G2)2) <α,α > <Mα,α> ≤ (λ1+λ2+λ22)<α,α >

26. z Normalized Adj. Matrix of the Product A,C – normalized adjacency matrices of G1,G2 M – normalized adjacency matrix of G1 G2 • M=ĈÂĈ, where • Ĉ = IN C •  is a permutation matrix (length preserving), where element (u,v) goes to the v-th neighbor of v in G1. • We relate  to A next:

27. Â is.. Given any αND,α=α11, …,α1D,…, αN1, …,αND • For i[N], define: • (α)i D, (α)i=α11, …,α1D distribution within the cloud. • βi=∑j=1,Dαij “distribution” on clouds themselves. • (α)i||= (βi/D) 1D • (α)i= (α)i-(α)i|| • L: ND →N, L(α) = (β1,…, βN)= β N • LÂ(β 1D))= Aβ • Â(β 1D) = Aβ 1D

28. Proof (cont.) • <Mα,α> = <ĈÂĈα,α> = αT ĈÂĈα = (Ĉα)TÂ(Ĉ α) = <ÂĈα, Ĉα> • α= α||+α • Ĉα|| = α|| • Ĉα = Ĉ(α+α||) = α|| +Ĉα

29. Proof (cont.) Ĉα = Ĉ(α + α||) = α|| +Ĉα <Mα,α>=|<ÂĈα,Ĉα>|=<Â (α||+Ĉα),(α||+Ĉα)> = <Âα||,α||>+<Âα||,Ĉα>+<ÂĈα,α||>+<ÂĈα,Ĉα> ≤<Âα||, α||>+||Âα||||||Ĉα||+||ÂĈα|||| α||||+<ÂĈα,Ĉα> =<Âα|| ,α||>+ 2||α||||.||Ĉα|| + ||Ĉα||2

30. Proof (cont.) <Mα,α>≤|<Âα|| ,α||>|+ 2||α||||.||Ĉα|| + ||Ĉα||2 Claim1: ||Ĉα|| ≤ λ(G2)||α|| = λ2||α|| Claim2: <Âα|| ,α||>≤λ1<α|| ,α||>=λ1||α||2 α|| = β  UD Âα|| = Â(β UD) = Aβ UD By expansion of G2 - Aβ≤λ1 β <Âα|| ,α||> = <AβUD ,β  UD > = < λ1 βUD ,β UD > <Âα|| ,α||> ≤λ1<α|| ,α||>= λ1||α||2

31. Proof. <Mα,α>=<ÂĈα,Ĉα> ≤ λ1||α||||2+2λ2||α||||.||α||+ λ22.||α||2 ||α||2=||α + α|||| 2 =||α||2 +||α||||2 <Mα,α> /<α,α> = <Mα,α> / ||α||2 =λ1||α||||2/||α||2 + 2λ2||α||||.||α|| /||α||2 + λ22||α||2 /||α||2 <Mα,α>/<α,α> ≤ λ1+λ2+ λ22 Q.E.D.

32. z The Zig-Zag Graph Product: Theorem 1 Let G1 be (N,D,λ1)-graph and G2 be(D,d,λ2)-graph, then (G1 G2) = (ND, d2, λ1+ λ2+ λ22). Theorem 2 For every i, Gi is (D4i, D2, 2/5)-graph.