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

Overture

d T d-regular, n-vertex graph G, with adj. matrix A Fix d, n, and any vertex v. Let T ⟶ ∞. even The number of closed walks v v v length T d T is 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−1)T d-regular graph G The number of closed walks e e is Θ( ). length T/2 length T/2 (d−1)T is Θ( ). Trace Method ⇒ ρ(B) = d−1 Look at nonbacktracking walks. Let B be the associated adj. matrix (rows/cols indexed by directed edges)

(d−1)T/2 d-regular infinite tree The number of closed nonbacktracking walks e e length T/2 length T/2 (d−1)T/2 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: For a d-regular graph G, is an eigenvalue of A β is an eigenvalue of B ⟺ (plus B has m−n eigs of ±1) finite d-reg graph max eigenvalue: d−1 d Well, it’s quite annoying to count closed usual walks in the infinite tree, but the nonbacktracking ones were trivial. IB: Proof is easy, I promise you. infinite d-reg tree max eigenvalue:

On to the main topic

d-regular, n-vertex graph G, with adj. matrix A Write A’s eigenvalues as d = λ1 ≥ λ2 ≥ ··· ≥ λn λ = max |λj| j≠1 Smaller λ = better expander

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

λ ≥ Smaller λ = better expander “Gn is an expander (family)” ⟺ λ ≤ [Alon−Boppana’86]: − on(1) λ ≥ E.g.: There are only finitely many 10-regular graphs with λ ≤ 5.999.

λ ≥ Q: Are there infinitely many 10-regular graphs Gn with λ ≤ 6? 10 Yes, if 10−1=9 is a prime. Answer 1: [Iha’66,LPS’88,Mar’88] Ramanujan Graph: λ ≤ [Morgenstern’94] Yes, if d−1 is a prime power. They’re explicit: Gn computable in deterministic poly(n) time. In fact, strongly explicit: G2ⁿ computable in deterministic in poly(n) time.

Q: Are there infinitely many 10-regular graphs Gn with λ ≤ 6? 7 λ ≥ ? Answer 2: [MSS’15] Yes if you only want λ2 ≤ . Their graphs are bipartite, hence have λn = −d. [Cohen’16] So things are super-great when d-1 is a prime power. Having small lambda_2 is good for some expander consequences. Not expander mixing lemma, though. These graphs can be made explicit.

λ ≥ Q: Are there infinitely many 10-regular graphs Gn with λ ≤ 6? 7 ? Answer 3: [Friedman’08, Bordenave’17] ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w.h.p. Our result: We get explicit such Gn. Friedman/Bordenave: obviously, not explicit.

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

λ ≥ Q: Are there infinitely many 10-regular graphs Gn with λ ≤ 6? 7 ? Answer 3: [Friedman’08, Bordenave’17] ∀ ϵ > 0, a random d-regular graph Gn has λ ≤ + ϵ w.h.p. Our result: 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” [Bilu−Linial’06] det. poly(n)-time computable seed s∈{0,1}O(n) circuit Cn Cn implements the adjacency list of G2ⁿ a 2n-vertex, d-regular graph So there are polylog-time *navigable* near-Ramanujan graph, and they can be constructed whp. Useful perhaps for Ramanujan-based cryptographic hash functions. -- Despite this, I will henceforth only talk about achieving “explicit”; please henceforth forget about this. 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: [RVW’02,BT’11] λ ≤ , strongly explicit Iterated lifts: [BL’06] λ ≤ , probabilistically strongly explicit Add matchings to LPS/Margulis: [CM’08] λ ≤ , strongly explicit (assuming Cramér’s Conjecture)

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 nO(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 nO(log n) time.

and also the “no bicycles” property 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 d-regular Gn0 with λ ≤ + ϵ and also the “no bicycles” property

Our Main Technical Theorem Given any d-regular G0 satisfying , a random 2-lift G2 has λ ≤ max{ λ(G0), + ϵ } w.h.p. Facts: 1. G2 is d-regular and |V(G2)| = 2·|V(G0)| Compare MSS and BL 2. G2 also satisfies 3. Easily derandomizable in poly(n) time using almost-(log n)-wise independent strings. [NN93]

weakly derandomizing Bordenave in P derand 2-lift in P derand 2-lift in P derand 2-lift in P • • • d-regular λ ≤ + ϵ vertices d-regular λ ≤ + ϵ 2n0 vertices d-regular λ ≤ + ϵ 4n0 vertices d-regular λ ≤ + ϵ n vertices

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 2n-vertex

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

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

2-lifts = edge-signings + + + contains d for sure, constructed to have all other |eigs| ≤ d+ ϵ + + + hopefully has all |eigs| ≤ + ϵd + + − + − + Fact: eigs(G2) = eigs(G0) ∪ eigs(G±)

Our Main Technical Theorem Given any d-regular G0 satisfying , a random 2-lift G2 has λ ≤ max{ λ(G0), + ϵ } w.h.p. Facts: 1. G2 is d-regular and |V(G2)| = 2·|V(G0)| Compare MSS and BL 2. G2 also satisfies 3. Easily derandomizable in poly(n) time using almost-(log n)-wise independent strings. [NN93]

Our Main Technical Theorem Given any d-regular G0 satisfying , a random 2-lift G2 has λ ≤ max{ λ(G0), + ϵ } w.h.p. Need to show: Compare MSS and BL a random edge-signing G± has ρ(G±) ≤ + ϵ w.h.p.

Our Actual Main Technical Theorem Given any d-regular G0 satisfying , Compare MSS and BL a random edge-signing G± has ρ(G±) ≤ + ϵ w.h.p.

Our Actual Main Technical Theorem Given any d-regular G0 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. Compare MSS and BL

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] Sometimes – e.g. 2XOR – you might actually *want* random edge-signs.

Our Main Technical Theorem Given any d-regular n-vertex G satisfying , a random edge-signing G± has ρ(G±) ≤ + ϵ w.h.p. What is ?! Define L = log log n · poly log 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±) ≤ + ϵ w.h.p. What is ?! Define L = log log n · poly log 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 derand 2-lift in P derand 2-lift in P • • • d-regular λ ≤ + ϵ vertices d-regular λ ≤ + ϵ 2n0 vertices d-regular λ ≤ + ϵ 4n0 vertices d-regular λ ≤ + ϵ n vertices

weakly derandomizing Bordenave derand 2-lift derand 2-lift derand 2-lift • • • at radius ≈ vertices 2n0 vertices 4n0 vertices n vertices

Our Main Technical Theorem Let G be d-regular, n-vertex, every log log n-nbhd has ≤ 1 cycle. Then a random edge-signing G± has ρ(G±) ≤ + ϵ w.h.p. If I had about 20 minutes, I’d tell you the proof. As it stands, I can just give you some hints.

Our Main Technical Theorem Let G be d-regular, n-vertex, every log 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*: For a d-regular graph G, is an eigenvalue of A β is an eigenvalue of B ⟺ finite graph spectral radius: d−1 d infinite graph spectral radius:

Our Main Technical Theorem Let G be d-regular, n-vertex, every log 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−1)T/2 d-regular infinite tree The number of closed walks e e length T/2 length T/2 (d−1)T/2 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 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!