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

2. Overture

3. 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

4. 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

5. (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)

6. (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

7. “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:

8. On to the main topic

9. 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

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

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

12. 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.

13. 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.

14. 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.

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

16. 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 ≤

17. “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.)

18. 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)

19. About our work

20. 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.

21. 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.

22. 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

23. 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]

24. 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

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

26. 2-lifts = edge-signings

27. 2-lifts = edge-signings+ + + + + + + + + + + +

28. 2-lifts = edge-signings+ + + + + + + + + + + +

29. 2-lifts = edge-signings+ + + + + + + + + + + +

30. 2-lifts = edge-signings+ + + + + + + + + + + +

31. 2-lifts = edge-signings+ + + + + + + + + + − +

32. 2-lifts = edge-signings+ + + + + + + + + + − +

33. 2-lifts = edge-signings+ + + + + + + + + + − +

34. 2-lifts = edge-signings+ + + + + + + + − + − +

35. 2-lifts = edge-signings+ + + + + + + + − + − +

36. 2-lifts = edge-signings+ + + + + + + + − + − +
n-vertex
2n-vertex

37. 2-lifts = edge-signings+ + + + + + + + − + − +
d-regular
d-regular

38. 2-lifts = edge-signings+ + + + + + + + − + − +
(similar to “girth ≥ L”)

39. 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±)

40. 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]

41. 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.

42. 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.

43. 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

44. 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.

45. 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

46. v

47. 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)

48. 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

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

50. 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.

51. 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.

52. “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:

53. 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.

54. (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

55. 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…

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

57. The End - Thanks!