Explicit nearRamanujan graphs of every degree Ryan ODonnell

Explicit near-Ramanujan graphs of every degree Ryan O’Donnell Carnegie Mellon Sidhanth Mohanty (Berkeley) Pedro Paredes (Carnegie Mellon)

Overture

d-regular, n-vertex graph G, with adj. matrix A Fix d, n, and any vertex v. Let Teven �∞. v The number of closed walks v is length T v T d Θ(______). Trace Method: ρ(A) = max |λi(A)| = the base of this exponential

If G is directed (so A is not symmetric) you look at AT/2 (A*)T/2… …closed walks where the first T/2 steps are forward, last T/2 steps are backward

d-regular graph G The number of closed walks e length T/2 e T (d− 1) is Θ( ). Look at nonbacktracking walks. Let B be the associated adj. matrix (rows/cols indexed by directed edges) Trace Method ⇒ ρ(B) = d− 1

d-regular infinite tree The number of closed nonbacktracking walks e e length T/2 T/2 (d− 1) is Θ( ). ∴ ρ(B∞) = Key: The closed walks must reduplicate each edge. Image credit: Hoory−Linial−Wigderson

“But I care about A, not B!” Ihara−Bass Formula: β is an eigenvalue of B For a d-regular graph G, � is an eigenvalue of A (plus B has m−n eigs of ± 1) finite d-reg graph max eigenvalue: infinite d-reg tree max eigenvalue: d− 1 d

On to the main topic

d-regular, n-vertex graph G, with adj. matrix A Write A’s eigenvalues as d = λ 1 ≥ λ 2 ≥ ··· ≥ λn λ = max |λj| j≠ 1 Smaller λ = better expander

Smaller λ = better expander “Gn is an expander (family)” � λ ≤ . 99 d

Smaller λ = better expander “Gn is an expander (family)” � λ ≤ [Alon−Boppana’ 86]: λ≥ − on(1) E. g. : There are only finitely many 10 -regular graphs with λ ≤ 5. 999. (1−Ωn(1)) d

Q: Are there infinitely many 10 -regular 10 6? graphs Gn with λ ≤ 6? λ≥ Answer 1: [Iha’ 66, LPS’ 88, Mar’ 88] [Morgenstern’ 94] Yes, if 10− 1=9 is a prime. Ramanujan Graph: λ ≤ Yes, if d− 1 is a prime power. They’re explicit: Gn computable in deterministic poly(n) time. In fact, strongly explicit: G 2ⁿ computable in deterministic in poly(n) time.

λ≥ Answer 2: [MSS’ 15] [Cohen’ 16] Q: Are there infinitely many 10 -regular 7 ? graphs Gn with λ ≤ 6? Yes if you only want λ 2 ≤ . Their graphs are bipartite, hence have λn = −d. These graphs can be made explicit.

λ≥ Answer 3: [Friedman’ 08, Bordenave’ 17] Our result: Q: Are there infinitely many 10 -regular 7 ? graphs Gn with λ ≤ 6? ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w. h. p. We get explicit such Gn.

Explicit near-Ramanujan graphs of every degree Ryan O’Donnell Carnegie Mellon Sidhanth Mohanty (Berkeley) Pedro Paredes (Carnegie Mellon)

λ≥ Answer 3: [Friedman’ 08, Bordenave’ 17] Our result: Q: Are there infinitely many 10 -regular 7 ? graphs Gn with λ ≤ 6? ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w. h. p. We get explicit such Gn. E. g. : In deterministic poly(n) time, we can construct an n-vertex, 101 -regular graph G with λ ≤ 20. 000001.

“Probabilistically Strongly Explicit” seed s∈{0, 1}O(n) det. poly(n)-time computable [Bilu−Linial’ 06] circuit Cn Cn implements the adjacency list of G 2ⁿ a 2 n-vertex, d-regular graph W. h. p. over choice of s, we have λ ≤ +ϵ (Implies explicit: replace n with log n; try all seeds; check. )

More prior work Zig-Zag Product: λ≤ , strongly explicit λ≤ , probabilistically strongly explicit [RVW’ 02, BT’ 11] Iterated lifts: [BL’ 06] Add matchings to LPS/Margulis: [CM’ 08] λ≤ (assuming Cramér’s Conjecture) , strongly explicit

About our work

So you want to derandomize this… ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w. h. p. [Friedman’ 08] [Bordenave’ 17] 100 pages 30 pages I understand about 70% of it. With 10% understanding, you’ll see that O(log n)-wise independent permutations derandomize it, in n. O(log n) time.

So you want to derandomize this… ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w. h. p. With 10% understanding, you’ll see that O(log n)-wise independent permutations derandomize it, in n. O(log n) time.

So you want to derandomize this… ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w. h. p. ∴ in deterministic poly(n) time, can construct dregular Gn 0 with λ ≤ +ϵ and also the “no bicycles” property

Our Main Technical Theorem Given any d-regular G 0 satisfying a random 2 -lift G 2 has λ ≤ max{ λ(G 0), Facts: , + ϵ } w. h. p. 1. G 2 is d-regular and |V(G 2)| = 2·|V(G 0)| 2. G 2 also satisfies 3. Easily derandomizable in poly(n) time using almost-(log n)-wise independent strings. [NN 93]

weakly derandomizing Bordenave in P derand 2 -lift in P • • • d-regular vertices 2 n 0 vertices 4 n 0 vertices n vertices λ≤ +ϵ

End of derandomization On to random 2 -lifts of fixed graphs

2 -lifts = edge-signings

2 -lifts = edge-signings + + +

2 -lifts = edge-signings + + +

2 -lifts = edge-signings + + +

2 -lifts = edge-signings + + +

2 -lifts = edge-signings + + + + + − +

2 -lifts = edge-signings + + + + + − +

2 -lifts = edge-signings + + + + + − +

2 -lifts = edge-signings + + + + − +

2 -lifts = edge-signings + + + + − +

2 -lifts = edge-signings + + + + − + n-vertex 2 n-vertex

2 -lifts = edge-signings + + + + − + d-regular

2 -lifts = edge-signings + + + + − + (similar to “girth ≥ L”)

2 -lifts = edge-signings G 0, ± + + − G 2 + + contains d for sure, + constructed to have all other |eigs| d+ ϵ + ≤ + hopefully has all |eigs| ≤ − + Fact: eigs(G 2) = eigs(G 0) ∪ eigs(G±) + ϵd

Our Main Technical Theorem Given any d-regular G 0 satisfying a random 2 -lift G 2 has λ ≤ max{ λ(G 0), Facts: , + ϵ } w. h. p. 1. G 2 is d-regular and |V(G 2)| = 2·|V(G 0)| 2. G 2 also satisfies 3. Easily derandomizable in poly(n) time using almost-(log n)-wise independent strings. [NN 93]

Our Main Technical Theorem Given any d-regular G 0 satisfying a random 2 -lift G 2 has λ ≤ max{ λ(G 0), , + ϵ } w. h. p. Need to show: a random edge-signing G± has ρ(G±) ≤ + ϵ w. h. p.

Our Actual Main Technical Theorem Given any d-regular G 0 satisfying a random edge-signing G± has ρ(G±) ≤ , + ϵ w. h. p.

Our Actual Main Technical Theorem Given any d-regular G 0 satisfying a random edge-signing G± has ρ(G±) ≤ , + ϵ w. h. p. Remark: Easy to show a random d-regular G super-satisfies w. h. p.

Hence our theorem implies: A random d-regular G with random edge-signs has ρ(G) ≤ + ϵ w. h. p. • [Bordenave’ 17] would have proven this had he been asked • Implicit in [DMOSS’ 19, MOP’ 19] • If you want to see proof, I humbly suggest our paper • That will give you about 70% understanding of [Bordenave’ 17]

Our Main Technical Theorem Given any d-regular n-vertex G satisfying a random edge-signing G± has ρ(G±) ≤ What is ? ! Define L = log n , + ϵ w. h. p. · poly log log n = ∀ v, the L-neighborhood of v has ≤ 1 cycle

v

Our Main Technical Theorem Given any d-regular n-vertex G satisfying a random edge-signing G± has ρ(G±) ≤ What is ? ! Define L = log n , + ϵ w. h. p. · poly log log n = ∀ v, the L-neighborhood of v has ≤ 1 cycle Ex: Random G has this w. h. p. , even for any L = o(log n)

weakly derandomizing Bordenave in P derand 2 -lift in P • • • d-regular vertices 2 n 0 vertices 4 n 0 vertices n vertices λ≤ +ϵ

weakly derandomizing Bordenave derand 2 -lift • • • at radius ≈ vertices 2 n 0 vertices 4 n 0 vertices n vertices

Our Main Technical Theorem Let G be d-regular, n-vertex, every log n-nbhd has ≤ 1 cycle. Then a random edge-signing G± has ρ(G±) ≤ + ϵ w. h. p.

Our Main Technical Theorem Let G be d-regular, n-vertex, every log n-nbhd has ≤ 1 cycle. Then a random edge-signing G± has ρ(G±) ≤ + ϵ w. h. p. Hints: • Use Ihara−Bass; suffices to show ρ(B±) ≤ + ϵ w. h. p.

“But I care about A, not B!” Ihara−Bass Formula*: β is an eigenvalue of B finite graph spectral radius: infinite graph spectral radius: d− 1 For a d-regular graph G, � is an eigenvalue of A d

Our Main Technical Theorem Let G be d-regular, n-vertex, every log n-nbhd has ≤ 1 cycle. Then a random edge-signing G± has ρ(G±) ≤ + ϵ w. h. p. Hints: • Use Ihara−Bass; suffices to show ρ(B±) ≤ + ϵ w. h. p. • Use Trace Method in expectation over the random edge-signs. Closed walks must use each edge an even number of times, or else they count 0 in expectation.

d-regular infinite tree The number of closed walks e length T/2 e T/2 (d− 1) is Θ( ). ∴ ρ(B∞) = Key: The closed walks must reduplicate each edge. Image credit: Hoory−Linial−Wigderson

Our Main Technical Theorem Let G be d-regular, n-vertex, every log n-nbhd has ≤ 1 cycle. Then a random edge-signing G± has ρ(G±) ≤ + ϵ w. h. p. Hints: • Use Ihara−Bass; suffices to show ρ(B±) ≤ + ϵ w. h. p. • Use Trace Method in expectation over the random edge-signs. Closed walks must use each edge an even number of times. • In a length-T walk, ≤ T/2 steps go to ‘new’ vertices… (d− 1)T/2 G has cycles; the other T/2 steps need not go straight home. But ⇒ sufficiently ‘tree-like’, so counting works out okay…

Conclusion We proved: For every d, ϵ, there are explicit graphs Gn with λ ≤ + ϵ. Obvious open question: Explicit near-Ramanujan quantum expanders?

The End - Thanks!