Post on 19-May-2020
Random Graphs, Ramanujan Graphs
Berkeley Math CircleMay 2012
Alon Amit
Wednesday, May 9, 12
Ramanujan Graphs
Random Graphs>
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?Wednesday, May 9, 12
>Wednesday, May 9, 12
Math isn’t Engineering.
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Engineering: random stuff is useless.
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Math: random objects are amazingly useful and incredibly hard to construct explicitly.
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All Graphs
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Good Graphs
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Graphs we can Make
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“Graph”Wednesday, May 9, 12
“Graph”Wednesday, May 9, 12
“Graph”
NOT
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“Graph”Wednesday, May 9, 12
Vertices
A
FE
D
C
B
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Edges
A
FE
D
C
B
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Edges
A
FE
D
C
B
{A,B,C,D,E, F}Vertices = {{A,B}, {A,E}, {B,C}, {C,D}, {C,F}, {B,F}}Edges =
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What are Graphs good for?
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Lots.
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Computer Science
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Engineering
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Biology
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...and Mathematics.
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Random Graphs
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Ramsey: Among 6 people there must be 3 friends or 3 strangers.
R(3, 3) = 6
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Fact: among 40,000,000,000 people, there must be 20 strangers or 20 mutual
friends.
R(20, 20) <
✓38
19
◆
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However: can you arrange for 1,024 people to avoid 20 strangers and 20
mutual friends?
R(20, 20) > 1024
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Kn
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Can we color the edges of in 2 colors so that there’s no monochromatic ?
K1024
K20
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Yes. Color at Random.
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Tournaments
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No Clear Winner
A
C B
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Can you make a tournament in which for every 2 players there’s someone who beats them all?
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Can we have a tournament in which for every 10 players there’s someone who beats them all?
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Yes. A Random One.
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Expander Graphs
• Sparse: few edges.
• Highly connected: Separating more that 20% of the vertices from the rest requires severing many edges.
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Expander Graphs
• Communication network design.
• Algorithms: derandomization.
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Do Expanders Exist?
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Yes. Random graphs are expanders!
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Explicit Construction?
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Ramanujan Graphs
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Adjacency Matrix0
BBBBBB@
0 1 0 0 1 01 0 1 0 0 10 1 0 1 0 10 0 1 0 1 01 0 0 1 0 00 1 1 0 0 0
1
CCCCCCA
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Graph Spectrum0
BBBBBB@
0 1 0 0 1 01 0 1 0 0 10 1 0 1 0 10 0 1 0 1 01 0 0 1 0 00 1 1 0 0 0
1
CCCCCCA
�1 = 3,�2 = 2.8, . . .
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Graph Spectrum0
BBBBBB@
0 1 0 0 1 01 0 1 0 0 10 1 0 1 0 10 0 1 0 1 01 0 0 1 0 00 1 1 0 0 0
1
CCCCCCA
�1 = 3,�2 = 2.8, . . .
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Ramanujan Graph
• d-regular: every vertex has d neighbors.
• Connected.
• Spectral gap: max
|�i|<d|�i| 2
pd� 1
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Nobody knows if random graphs are Ramanujan.
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The LPS Construction
• Quotient of the tree by a certain discrete subgroup defined via quaternion algebras.
• Proof of Ramanujan property relies on the proof of the Ramanujan Conjecture for certain cusp forms associated with representations of this group.
• (p+1)-regular for prime p.
PGL2(Qp)/PGL2(Zp)
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“Toy” Model
• Define
• Consider the Cayley graph of with generators
• This graph is 4-regular, connected with vertices and “large girth”.
A =
✓1 20 1
◆, B =
✓1 02 1
◆
SL2(q)
A,B,A�1, B�1
q(q2 � 1)
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What you Should Remember
• Random graphs have incredible properties that are hard to achieve explicitly.
• Ramanujan Graphs do achieve many such properties.
• Ramanujan Graphs are crazy hard to construct, and the proof that they “work” is even harder.
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Acknowledgments
• Prof. Alex Lubotzky
• Prof. Nati Linial
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