an introduction to expander families and ramanujan graphs tony shaheen csu los angeles
TRANSCRIPT
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An introduction to expander families and Ramanujan
graphs
Tony ShaheenCSU Los Angeles
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Before we get started on expander graphs I want to give a definition that we will use in this talk.
A graph is regular if every vertex has the same degree (the number of edges at that vertex).
A 3-regular graph.
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Think of a graph as a communications network.
Now for the motivation behind expander families…
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Two vertices can communicatedirectly with one another iff they are connected by an edge.
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Communication is instantaneousacross edges, but there may bedelays at vertices.
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Edges are expensive.
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Our goal: Let d be a fixed integer with d > 1.
Create an infinite sequence of d-regular graphs
where
1. the graphs are getting bigger and bigger (the number of vertices of goes to infinity asn goes to infinity)
2. each is as good a communications network as possible.
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Questions:
1. How do we measure if a graph is a good communications network?
2. Once we have a measurement, can we find graphs that are optimal withrespect to the measurement?
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Let’s start with the first question.
Questions:
1. How do we measure if a graph is a good communications network?
2. Once we have a measurement, howgood can we make our networks?
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Consider the following graph:
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Let’s look at the set of vertices that we can reach after n steps, starting at the top vertex.
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Here is where we can get to after 1 step.
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Here is where we can get to after 1 step.
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We would like to have many edges going outward from there.
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Here is where we can get to after 2 steps.
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Take-home Message #1:
The expansion constantis one measure of howgood a graph is as acommunications network.
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We want h(X) to be BIG!
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We want h(X) to be BIG!If a graph has small degreebut many vertices, this is noteasy.
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𝐶3 𝐶4 𝐶5 𝐶6
Consider the cycles graphs:
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𝐶3 𝐶4 𝐶5 𝐶6
Consider the cycles graphs:
Each is 2-regular.
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𝐶3 𝐶4 𝐶5 𝐶6
Consider the cycles graphs:
Each is 2-regular.The number of vertices goes to infinity.
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Let S be the bottom half.
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We say that a sequence of regular graphsis an expander family if
• All the graphs have the same degree
• The number of vertices goes to infinity
• There exists a positive lower bound r such that the expansion constant is alwaysat least r.
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We just saw that expander families of degree 2 do not exist.
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We just saw that expander families of degree 2 do not exist.
What is amazing is that if d > 2 then expander families of degree d exist.
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We just saw that expander families of degree 2 do not exist.
What is amazing is that if d > 2 then expander families of degree d exist.
Existence: Pinsker 1973First explicit construction: Margulis 1973
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So far we have looked at the combinatorial way of looking at expander families.
Let’s now look at it from an algebraic viewpoint.
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We form theadjacency matrixof a graph as follows:
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Facts about eigenvalues of a d-regulargraph G:
Facts about the eigenvalues of a d-regular connected graph G with n vertices:
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Facts about eigenvalues of a d-regulargraph G:
● They are all real.
Facts about eigenvalues of a d-regular gra G with n vertices:Facts about the eigenvalues of a d-regular connected graph G with n vertices:
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● They are all real.● The eigenvalues satisfy
… = d
Facts about the eigenvalues of a d-regular connected graph G with n vertices:
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● The second largest eigenvalue
Facts about eigenvalues of a d-regulargraph G:
● They are all real.
(Alon-Dodziuk-Milman-Tanner)
Facts about the eigenvalues of a d-regular connected graph G with n vertices:
● The eigenvalues satisfy
… = d satisfies
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(Alon-Dodziuk-Milman-Tanner)
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(Alon-Dodziuk-Milman-Tanner)
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(Alon-Dodziuk-Milman-Tanner)
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Take-home Message #2:
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The red curve has a horizontal asymptote at 2√𝑑−1
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In other words, is asymptotically
the smallest that can be.
2√𝑑−1𝜆1
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We say that a d-regular graph X is Ramanujan if all the non-trivial eigenvalues of X (the ones that aren’t equal to d or -d) satisfy
𝜆
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Hence, if X is Ramanujan then
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.
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Take-home Message #3:
Ramanujan graphs essentiallyhave the smallest possible 𝜆1
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A family of d-regular Ramanujan graphs is an expander family.
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Shamelessself-promotion!!!
Expander families and Cayley graphs – A beginner’s guide
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