random lifts of graphsnati/papers/rio_09.pdf · 2009. 8. 13. · random graphs come up a lot in...
TRANSCRIPT
![Page 1: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/1.jpg)
Random Lifts of Graphs
Nati Linial
27th Brazilian Math Colloquium, July ’09
Nati Linial Random Lifts of Graphs
![Page 2: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/2.jpg)
Plan of this talk
I A brief introduction to the probabilisticmethod.
I A quick review of expander graphs and theirspectrum.
I Lifts, random lifts and their properties.
I Spectra of random lifts.
Nati Linial Random Lifts of Graphs
![Page 3: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/3.jpg)
What is the probabilistic method?Introduction by example.
TheoremIn every party of 6 people there are either 3 whoknow each other or 3 who are strangers to eachother.
In other words: If you color the edges of K6 (thecomplete graph on 6 vertices) blue and red, younecessarily find a monochromatic (either red or bluetriangle.
Nati Linial Random Lifts of Graphs
![Page 4: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/4.jpg)
What is the probabilistic method?Introduction by example.
TheoremIn every party of 6 people there are either 3 whoknow each other or 3 who are strangers to eachother.
In other words: If you color the edges of K6 (thecomplete graph on 6 vertices) blue and red, younecessarily find a monochromatic (either red or bluetriangle.
Nati Linial Random Lifts of Graphs
![Page 5: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/5.jpg)
The result is tight. There is a red-blue coloring ofthe edges of K5 with no monochromatic triangle.
Nati Linial Random Lifts of Graphs
![Page 6: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/6.jpg)
The result is tight. There is a red-blue coloring ofthe edges of K5 with no monochromatic triangle.
Nati Linial Random Lifts of Graphs
![Page 7: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/7.jpg)
More generally,
Theorem (Ramsey; Erdos-Szekeres)Let N =
(r+s−2r−1
). If you color the edges of KN red
and blue, then you necessarily get a red Kr or a blueKs .
Nati Linial Random Lifts of Graphs
![Page 8: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/8.jpg)
Diagonal Ramsey Numbers
TheoremIn every red-blue coloring of the edges of KN thereis a monochromatic complete subgraph on at least
1
2log2 N vertices.
TheoremEvery N-vertex graph contains either a clique or ananti-clique on at least
1
2log2 N vertices.
Nati Linial Random Lifts of Graphs
![Page 9: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/9.jpg)
Diagonal Ramsey Numbers
TheoremIn every red-blue coloring of the edges of KN thereis a monochromatic complete subgraph on at least
1
2log2 N vertices.
TheoremEvery N-vertex graph contains either a clique or ananti-clique on at least
1
2log2 N vertices.
Nati Linial Random Lifts of Graphs
![Page 10: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/10.jpg)
Diagonal Ramsey Numbers (contd.)
Theorem (Erdos ’49)There are red-blue colorings of KN where nomonochromatic subgraph has more than
2 log2 N vertices.
TheoremThere are N-vertex graphs where no clique oranti-clique has more than
2 log2 N vertices.
Nati Linial Random Lifts of Graphs
![Page 11: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/11.jpg)
Diagonal Ramsey Numbers (contd.)
Theorem (Erdos ’49)There are red-blue colorings of KN where nomonochromatic subgraph has more than
2 log2 N vertices.
TheoremThere are N-vertex graphs where no clique oranti-clique has more than
2 log2 N vertices.
Nati Linial Random Lifts of Graphs
![Page 12: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/12.jpg)
How can you prove such a statement?
One may expect that (like the coloring of K5 we sawbefore) I would show you now a method of coloringthat has no large monochromatic subgraphs.
We do not know how to do this. In fact it is amajor challenge to find such explicit colorings.
Instead, we use the probabilistic method.
Nati Linial Random Lifts of Graphs
![Page 13: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/13.jpg)
How can you prove such a statement?
One may expect that (like the coloring of K5 we sawbefore) I would show you now a method of coloringthat has no large monochromatic subgraphs.
We do not know how to do this. In fact it is amajor challenge to find such explicit colorings.
Instead, we use the probabilistic method.
Nati Linial Random Lifts of Graphs
![Page 14: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/14.jpg)
How can you prove such a statement?
One may expect that (like the coloring of K5 we sawbefore) I would show you now a method of coloringthat has no large monochromatic subgraphs.
We do not know how to do this. In fact it is amajor challenge to find such explicit colorings.
Instead, we use the probabilistic method.
Nati Linial Random Lifts of Graphs
![Page 15: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/15.jpg)
Introducing the probabilistic method
There are 2(N2) ways to color the edges of KN by red
and blue. We think of them as a probability space Ωwith the uniform distribution.
In other words, we give the following recipe forsampling from Ω: For each edge of KN , flip a coin(independently from the rest). If it comes out headscolor the edge red if you get tails, color it blue.
Nati Linial Random Lifts of Graphs
![Page 16: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/16.jpg)
Introducing the probabilistic method
There are 2(N2) ways to color the edges of KN by red
and blue. We think of them as a probability space Ωwith the uniform distribution.
In other words, we give the following recipe forsampling from Ω: For each edge of KN , flip a coin(independently from the rest). If it comes out headscolor the edge red if you get tails, color it blue.
Nati Linial Random Lifts of Graphs
![Page 17: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/17.jpg)
Let us consider an integer r (to be determinedlater) and a random variable B defined on Ω. for agiven coloring C of KN , we define B(C) to be thenumber of sets of r vertices in C all of whose edgesare blue. The expectation of X is:(
N
r
)1
2(r2).
We likewise define a random variable R that countsred subgraphs.
Nati Linial Random Lifts of Graphs
![Page 18: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/18.jpg)
Let us consider an integer r (to be determinedlater) and a random variable B defined on Ω. for agiven coloring C of KN , we define B(C) to be thenumber of sets of r vertices in C all of whose edgesare blue. The expectation of X is:(
N
r
)1
2(r2).
We likewise define a random variable R that countsred subgraphs.
Nati Linial Random Lifts of Graphs
![Page 19: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/19.jpg)
Note that if B(C) = R(C) = 0, there is nomonochromatic set of r vertices in C, which is justwhat we need.
If the sum of the expectations
E(R) + E(B) < 1.
Then a coloring C exists with no monochromatic setof r vertices. An easy calculation yields that thisholds for
r = 2 log2 N .
Nati Linial Random Lifts of Graphs
![Page 20: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/20.jpg)
Note that if B(C) = R(C) = 0, there is nomonochromatic set of r vertices in C, which is justwhat we need.
If the sum of the expectations
E(R) + E(B) < 1.
Then a coloring C exists with no monochromatic setof r vertices. An easy calculation yields that thisholds for
r = 2 log2 N .
Nati Linial Random Lifts of Graphs
![Page 21: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/21.jpg)
Note that if B(C) = R(C) = 0, there is nomonochromatic set of r vertices in C, which is justwhat we need.
If the sum of the expectations
E(R) + E(B) < 1.
Then a coloring C exists with no monochromatic setof r vertices. An easy calculation yields that thisholds for
r = 2 log2 N .
Nati Linial Random Lifts of Graphs
![Page 22: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/22.jpg)
What has happened here?
Why does the probabilistic method work so well?
In every mathematical field intuition is created fromexamples that we know.But it is hard to analyze large specific examples andthe probabilistic method allows us to bypass thisdifficulty. It serves us as an observational tool,much like the astronomer’s telescope or a biologist’smicroscope.
Nati Linial Random Lifts of Graphs
![Page 23: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/23.jpg)
What has happened here?
Why does the probabilistic method work so well?
In every mathematical field intuition is created fromexamples that we know.
But it is hard to analyze large specific examples andthe probabilistic method allows us to bypass thisdifficulty. It serves us as an observational tool,much like the astronomer’s telescope or a biologist’smicroscope.
Nati Linial Random Lifts of Graphs
![Page 24: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/24.jpg)
What has happened here?
Why does the probabilistic method work so well?
In every mathematical field intuition is created fromexamples that we know.But it is hard to analyze large specific examples andthe probabilistic method allows us to bypass thisdifficulty. It serves us as an observational tool,much like the astronomer’s telescope or a biologist’smicroscope.
Nati Linial Random Lifts of Graphs
![Page 25: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/25.jpg)
Models of random graphs
What we saw is a close relative of the most basicmodel of random graphs, Erdos-Renyi’s G (n, p)model. In this model we start with n vertices. Foreach pair of vertices x , y we decide, independentlyand with probability p, to put an edge between xand y .
There are other important and interesting models ofgraphs. For example, we have known for 30 yearsnow how to sample random d-regular graphs.
Nati Linial Random Lifts of Graphs
![Page 26: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/26.jpg)
Models of random graphs
What we saw is a close relative of the most basicmodel of random graphs, Erdos-Renyi’s G (n, p)model. In this model we start with n vertices. Foreach pair of vertices x , y we decide, independentlyand with probability p, to put an edge between xand y .
There are other important and interesting models ofgraphs. For example, we have known for 30 yearsnow how to sample random d-regular graphs.
Nati Linial Random Lifts of Graphs
![Page 27: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/27.jpg)
What’s there, what’s still needed?
Random graphs come up a lot in (mathematical)statistical mechanics. A major example isPercolation Theory.
Start e.g., from the graph of the d-dimensionallattice. Maintain every edge with probability p, anddiscard it with probability 1− p. (Independentlyover edges).
A typical basic question in this area: What is theprobability that an infinite connected componentremains.
Nati Linial Random Lifts of Graphs
![Page 28: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/28.jpg)
What’s there, what’s still needed?
Random graphs come up a lot in (mathematical)statistical mechanics. A major example isPercolation Theory.
Start e.g., from the graph of the d-dimensionallattice. Maintain every edge with probability p, anddiscard it with probability 1− p. (Independentlyover edges).
A typical basic question in this area: What is theprobability that an infinite connected componentremains.
Nati Linial Random Lifts of Graphs
![Page 29: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/29.jpg)
What’s there, what’s still needed?
Random graphs come up a lot in (mathematical)statistical mechanics. A major example isPercolation Theory.
Start e.g., from the graph of the d-dimensionallattice. Maintain every edge with probability p, anddiscard it with probability 1− p. (Independentlyover edges).
A typical basic question in this area: What is theprobability that an infinite connected componentremains.
Nati Linial Random Lifts of Graphs
![Page 30: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/30.jpg)
You could also seek models to describe natural orartificial phenomena such as the Internet graph orbiological control networks.
Nati Linial Random Lifts of Graphs
![Page 31: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/31.jpg)
In particular, one shortcoming of the G (n, p) modelis the lack of control we have over the graph’sstructure.
We want ”more structured” models of randomgraphs.
Random lifts of graphs, our subject today, are sucha model.
Nati Linial Random Lifts of Graphs
![Page 32: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/32.jpg)
In particular, one shortcoming of the G (n, p) modelis the lack of control we have over the graph’sstructure.
We want ”more structured” models of randomgraphs.
Random lifts of graphs, our subject today, are sucha model.
Nati Linial Random Lifts of Graphs
![Page 33: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/33.jpg)
In particular, one shortcoming of the G (n, p) modelis the lack of control we have over the graph’sstructure.
We want ”more structured” models of randomgraphs.
Random lifts of graphs, our subject today, are sucha model.
Nati Linial Random Lifts of Graphs
![Page 34: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/34.jpg)
A very quick review on expansion in graphs
There are three main perspectives of expansion:
I Combinatorial - isoperimetric inequalities
I Linear Algebraic - spectral gap
I Probabilistic - Rapid convergence of therandom walk (which we do not discuss today)
For (much) more on this: see our survey article withHoory and Wigderson.
Nati Linial Random Lifts of Graphs
![Page 35: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/35.jpg)
The combinatorial definition
A graph G = (V ,E ) is said to be ε-edge-expandingif for every partition of the vertex set V into X andX c = V \ X , where X contains at most a half ofthe vertices, the number of cross edges
e(X ,X c) ≥ ε|X |.
In words: in every cut in G , the number of cutedges is at least proportionate to the size of thesmaller side.
Nati Linial Random Lifts of Graphs
![Page 36: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/36.jpg)
The combinatorial definition
A graph G = (V ,E ) is said to be ε-edge-expandingif for every partition of the vertex set V into X andX c = V \ X , where X contains at most a half ofthe vertices, the number of cross edges
e(X ,X c) ≥ ε|X |.
In words: in every cut in G , the number of cutedges is at least proportionate to the size of thesmaller side.
Nati Linial Random Lifts of Graphs
![Page 37: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/37.jpg)
The combinatorial definition (contd.)
The edge expansion ratio of a graph G = (V ,E ), is
h(G ) = minS⊆V , |S |≤|V |/2
|E (S , S)||S |
.
Nati Linial Random Lifts of Graphs
![Page 38: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/38.jpg)
The linear-algebraic perspective
The Adjacency Matrix of an n-vertex graph G ,denoted A = A(G ), is an n × n matrix whose (u, v)entry is the number of edges in G between vertex uand vertex v . Being real and symmetric, the matrixA has n real eigenvalues which we denote byλ1 ≥ λ2 ≥ · · · ≥ λn.
Nati Linial Random Lifts of Graphs
![Page 39: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/39.jpg)
Simple things that the spectrum of A(G )tells about G
I If G is d-regular, then λ1 = d . In thecorresponding eigenvector all coordinates areequal.
I The graph is connected iff λ1 > λ2. We callλ1 − λ2 the spectral gap.
I The graph is bipartite iff λ1 = −λn.
I χ(G ) ≥ −λ1
λn+ 1.
I A substantial spectral gap implies logarithmicdiameter.
Nati Linial Random Lifts of Graphs
![Page 40: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/40.jpg)
Simple things that the spectrum of A(G )tells about G
I If G is d-regular, then λ1 = d . In thecorresponding eigenvector all coordinates areequal.
I The graph is connected iff λ1 > λ2. We callλ1 − λ2 the spectral gap.
I The graph is bipartite iff λ1 = −λn.
I χ(G ) ≥ −λ1
λn+ 1.
I A substantial spectral gap implies logarithmicdiameter.
Nati Linial Random Lifts of Graphs
![Page 41: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/41.jpg)
Simple things that the spectrum of A(G )tells about G
I If G is d-regular, then λ1 = d . In thecorresponding eigenvector all coordinates areequal.
I The graph is connected iff λ1 > λ2. We callλ1 − λ2 the spectral gap.
I The graph is bipartite iff λ1 = −λn.
I χ(G ) ≥ −λ1
λn+ 1.
I A substantial spectral gap implies logarithmicdiameter.
Nati Linial Random Lifts of Graphs
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Simple things that the spectrum of A(G )tells about G
I If G is d-regular, then λ1 = d . In thecorresponding eigenvector all coordinates areequal.
I The graph is connected iff λ1 > λ2. We callλ1 − λ2 the spectral gap.
I The graph is bipartite iff λ1 = −λn.
I χ(G ) ≥ −λ1
λn+ 1.
I A substantial spectral gap implies logarithmicdiameter.
Nati Linial Random Lifts of Graphs
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Simple things that the spectrum of A(G )tells about G
I If G is d-regular, then λ1 = d . In thecorresponding eigenvector all coordinates areequal.
I The graph is connected iff λ1 > λ2. We callλ1 − λ2 the spectral gap.
I The graph is bipartite iff λ1 = −λn.
I χ(G ) ≥ −λ1
λn+ 1.
I A substantial spectral gap implies logarithmicdiameter.
Nati Linial Random Lifts of Graphs
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Spectrum vs. expansion
TheoremLet G be a d-regular graph with spectrumλ1 ≥ · · · ≥ λn. Then
d − λ2
2≤ h(G ) ≤
√(d + λ2)(d − λ2).
The bounds are tight.
Nati Linial Random Lifts of Graphs
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What’s a ”large” spectral gap?
If expansion is “good” and if a large spectral gapyields large expansion, then it’s natural to ask:
QuestionHow small can λ2 be in a d-regular graph? (i.e.,how large can the spectral gap get)?
Theorem (Alon, Boppana)
λ2 ≥ 2√
d − 1− o(1)
Nati Linial Random Lifts of Graphs
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What’s a ”large” spectral gap?
If expansion is “good” and if a large spectral gapyields large expansion, then it’s natural to ask:
QuestionHow small can λ2 be in a d-regular graph? (i.e.,how large can the spectral gap get)?
Theorem (Alon, Boppana)
λ2 ≥ 2√
d − 1− o(1)
Nati Linial Random Lifts of Graphs
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What’s a ”large” spectral gap?
If expansion is “good” and if a large spectral gapyields large expansion, then it’s natural to ask:
QuestionHow small can λ2 be in a d-regular graph? (i.e.,how large can the spectral gap get)?
Theorem (Alon, Boppana)
λ2 ≥ 2√
d − 1− o(1)
Nati Linial Random Lifts of Graphs
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The meaning of the number 2√
d − 1
A good approach to extremal problems is to comeup with a candidate for an ideal example, and showthat there are no better instances.
What, then, is the ideal expander? A goodcandidate is the infinite d-regular tree. Using (alittle) spectral theory it is possible to define aspectrum for infinite graphs. It turns out that thespectrum of the d-regular infinite tree spans theinterval
(−2√
d − 1, 2√
d − 1)
Nati Linial Random Lifts of Graphs
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The meaning of the number 2√
d − 1
A good approach to extremal problems is to comeup with a candidate for an ideal example, and showthat there are no better instances.
What, then, is the ideal expander? A goodcandidate is the infinite d-regular tree. Using (alittle) spectral theory it is possible to define aspectrum for infinite graphs.
It turns out that thespectrum of the d-regular infinite tree spans theinterval
(−2√
d − 1, 2√
d − 1)
Nati Linial Random Lifts of Graphs
![Page 50: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/50.jpg)
The meaning of the number 2√
d − 1
A good approach to extremal problems is to comeup with a candidate for an ideal example, and showthat there are no better instances.
What, then, is the ideal expander? A goodcandidate is the infinite d-regular tree. Using (alittle) spectral theory it is possible to define aspectrum for infinite graphs. It turns out that thespectrum of the d-regular infinite tree spans theinterval
(−2√
d − 1, 2√
d − 1)
Nati Linial Random Lifts of Graphs
![Page 51: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/51.jpg)
The meaning of the number 2√
d − 1
A good approach to extremal problems is to comeup with a candidate for an ideal example, and showthat there are no better instances.
What, then, is the ideal expander? A goodcandidate is the infinite d-regular tree. Using (alittle) spectral theory it is possible to define aspectrum for infinite graphs. It turns out that thespectrum of the d-regular infinite tree spans theinterval
(−2√
d − 1, 2√
d − 1)
Nati Linial Random Lifts of Graphs
![Page 52: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/52.jpg)
Some questions
How tight is this bound?
ProblemAre there d-regular graphs with second eigenvalue
λ2 ≤ 2√
d − 1 ?
When such graphs exist, they are called RamanujanGraphs.
Nati Linial Random Lifts of Graphs
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What is the typical behavior?
ProblemHow likely is a (large) random d-regular graph to beRamanujan?
Nati Linial Random Lifts of Graphs
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What is currently known aboutRamanujan Graphs?
Margulis; Lubotzky-Phillips-Sarnak; Morgenstern:d-regular Ramanujan Graphs exist whend − 1 is a prime power. The constructionis easy, but the proof uses a lot of heavymathematical machinery.
Friedman: If you are willing to settle forλ2 ≤ 2
√d − 1+ε, they exist. Moreover,
almost every d-regular graph satisfiesthis condition.
Nati Linial Random Lifts of Graphs
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What is currently known aboutRamanujan Graphs?
Margulis; Lubotzky-Phillips-Sarnak; Morgenstern:d-regular Ramanujan Graphs exist whend − 1 is a prime power. The constructionis easy, but the proof uses a lot of heavymathematical machinery.
Friedman: If you are willing to settle forλ2 ≤ 2
√d − 1+ε, they exist. Moreover,
almost every d-regular graph satisfiesthis condition.
Nati Linial Random Lifts of Graphs
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The distribution of the second eigenvalue
3.46 3.465
n=400000
n=100000
n=40000n=10000
2*sqrt(3)
Nati Linial Random Lifts of Graphs
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Some open problems on RamanujanGraphs
I Are there arbitrarily large d-regular RamanujanGraphs (i.e. λ2 ≤ 2
√d − 1) for every d ≥ 3?
The first unknown case is d = 7.
I Can we find combinatorial/probabilisticmethods to construct graphs with large spectralgap (or even Ramanujan)? As we’ll see randomlifts of graphs (Bilu-L.) yield graphs with
λ2 ≤ O(√
d log3/2 d).
Nati Linial Random Lifts of Graphs
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Some open problems on RamanujanGraphs
I Are there arbitrarily large d-regular RamanujanGraphs (i.e. λ2 ≤ 2
√d − 1) for every d ≥ 3?
The first unknown case is d = 7.
I Can we find combinatorial/probabilisticmethods to construct graphs with large spectralgap (or even Ramanujan)? As we’ll see randomlifts of graphs (Bilu-L.) yield graphs with
λ2 ≤ O(√
d log3/2 d).
Nati Linial Random Lifts of Graphs
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Covers and lifts - the abstract approach
DefinitionA map ϕ : V (H)→ V (G ) where G ,H are graphs isa covering map if for every x ∈ V (H), the neighborset ΓH(x) is mapped 1 : 1 onto ΓG (ϕ(x)).
This is a special case of a fundamental conceptfrom topology. Recall that a graph is aone-dimensional simplicial complex, so coveringmaps can be defined and studied for graphs.
Nati Linial Random Lifts of Graphs
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Covers and lifts - the abstract approach
DefinitionA map ϕ : V (H)→ V (G ) where G ,H are graphs isa covering map if for every x ∈ V (H), the neighborset ΓH(x) is mapped 1 : 1 onto ΓG (ϕ(x)).
This is a special case of a fundamental conceptfrom topology. Recall that a graph is aone-dimensional simplicial complex, so coveringmaps can be defined and studied for graphs.
Nati Linial Random Lifts of Graphs
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A little terminology
When there is a covering map from H to G , we saythat H is a lift of G .
We also call G the base graph
Convention: We will always assume that the basegraph is connected. This creates no loss ingenerality.
Nati Linial Random Lifts of Graphs
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A little terminology
When there is a covering map from H to G , we saythat H is a lift of G .
We also call G the base graph
Convention: We will always assume that the basegraph is connected. This creates no loss ingenerality.
Nati Linial Random Lifts of Graphs
![Page 63: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/63.jpg)
A little terminology
When there is a covering map from H to G , we saythat H is a lift of G .
We also call G the base graph
Convention: We will always assume that the basegraph is connected. This creates no loss ingenerality.
Nati Linial Random Lifts of Graphs
![Page 64: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/64.jpg)
An example - The 3-cube is a 2-lift of K4
Figure: The 3-dimensional cube is a 2-lift of K4
Nati Linial Random Lifts of Graphs
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The icosahedron is a 2-lift of K6
!!
" "" "##
Figure: The icosahedron graph is a 2-lift ofK6
Nati Linial Random Lifts of Graphs
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Making this definition more concrete
We see in the previous examples that the coveringmap ϕ is 2 : 1.
I The 3-cube is a 2-lift of K4.
I The graph of the icosahedron is a 2-lift of K6.
In general, if G is a connected graph, then everycovering map ϕ : V (H)→ V (G ) is n : 1 for someinteger n (easy).
Nati Linial Random Lifts of Graphs
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Fold numbers etc.
I We call n the fold number of ϕ.
I We say that H is an n-lift of G , or an n-coverof G .
I The set of those graphs that are n-lifts of G isdenoted by Ln(G ).
Nati Linial Random Lifts of Graphs
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Fold numbers etc.
I We call n the fold number of ϕ.
I We say that H is an n-lift of G , or an n-coverof G .
I The set of those graphs that are n-lifts of G isdenoted by Ln(G ).
Nati Linial Random Lifts of Graphs
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Fold numbers etc.
I We call n the fold number of ϕ.
I We say that H is an n-lift of G , or an n-coverof G .
I The set of those graphs that are n-lifts of G isdenoted by Ln(G ).
Nati Linial Random Lifts of Graphs
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A direct, constructive perspective
I Every H ∈ Ln(G ), has vertex setV (H) = V (G )× [n].
I We call the set Fx = x × [n] the fiber over x .
I For every edge e = xy ∈ E (G ) we have toselect some perfect matching between thefibers Fx and Fy , i.e., a permutationπ = πe ∈ Sn and connect (x , i) with (y , π(i))for i = 1, . . . , n.
I This set of edges is denoted by Fe , the fiberover e.
Nati Linial Random Lifts of Graphs
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A direct, constructive perspective
I Every H ∈ Ln(G ), has vertex setV (H) = V (G )× [n].
I We call the set Fx = x × [n] the fiber over x .
I For every edge e = xy ∈ E (G ) we have toselect some perfect matching between thefibers Fx and Fy , i.e., a permutationπ = πe ∈ Sn and connect (x , i) with (y , π(i))for i = 1, . . . , n.
I This set of edges is denoted by Fe , the fiberover e.
Nati Linial Random Lifts of Graphs
![Page 72: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/72.jpg)
A direct, constructive perspective
I Every H ∈ Ln(G ), has vertex setV (H) = V (G )× [n].
I We call the set Fx = x × [n] the fiber over x .
I For every edge e = xy ∈ E (G ) we have toselect some perfect matching between thefibers Fx and Fy , i.e., a permutationπ = πe ∈ Sn and connect (x , i) with (y , π(i))for i = 1, . . . , n.
I This set of edges is denoted by Fe , the fiberover e.
Nati Linial Random Lifts of Graphs
![Page 73: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/73.jpg)
A direct, constructive perspective
I Every H ∈ Ln(G ), has vertex setV (H) = V (G )× [n].
I We call the set Fx = x × [n] the fiber over x .
I For every edge e = xy ∈ E (G ) we have toselect some perfect matching between thefibers Fx and Fy , i.e., a permutationπ = πe ∈ Sn and connect (x , i) with (y , π(i))for i = 1, . . . , n.
I This set of edges is denoted by Fe , the fiberover e.
Nati Linial Random Lifts of Graphs
![Page 74: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/74.jpg)
u v
FvFu
Figure 1: Lifting an edge
1
Nati Linial Random Lifts of Graphs
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Random lifts of graphs
I When the permutations πe are selected atrandom, we call the resulting graph a randomn-lift of G.
I They can be used in essentially every way thattraditional random graphs are employed:
I To construct graphs with certain desirableproperties. In our case, to achieve large spectralgaps.
I To model various phenomena.I To study their typical properties.
Nati Linial Random Lifts of Graphs
![Page 76: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/76.jpg)
Random lifts of graphs
I When the permutations πe are selected atrandom, we call the resulting graph a randomn-lift of G.
I They can be used in essentially every way thattraditional random graphs are employed:
I To construct graphs with certain desirableproperties. In our case, to achieve large spectralgaps.
I To model various phenomena.I To study their typical properties.
Nati Linial Random Lifts of Graphs
![Page 77: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/77.jpg)
Random lifts of graphs
I When the permutations πe are selected atrandom, we call the resulting graph a randomn-lift of G.
I They can be used in essentially every way thattraditional random graphs are employed:
I To construct graphs with certain desirableproperties. In our case, to achieve large spectralgaps.
I To model various phenomena.I To study their typical properties.
Nati Linial Random Lifts of Graphs
![Page 78: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/78.jpg)
Random lifts of graphs
I When the permutations πe are selected atrandom, we call the resulting graph a randomn-lift of G.
I They can be used in essentially every way thattraditional random graphs are employed:
I To construct graphs with certain desirableproperties. In our case, to achieve large spectralgaps.
I To model various phenomena.
I To study their typical properties.
Nati Linial Random Lifts of Graphs
![Page 79: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/79.jpg)
Random lifts of graphs
I When the permutations πe are selected atrandom, we call the resulting graph a randomn-lift of G.
I They can be used in essentially every way thattraditional random graphs are employed:
I To construct graphs with certain desirableproperties. In our case, to achieve large spectralgaps.
I To model various phenomena.I To study their typical properties.
Nati Linial Random Lifts of Graphs
![Page 80: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/80.jpg)
A few more general properties of lifts
I Vertex degrees are maintained. If x has dneighbors, then so do all the vertices in thefiber of x . In particular, a lift of a d-regulargraph is d-regular.
I The cycle Cn is a lift of Cm iff m|n.
I The d-regular tree covers every d-regulargraph. This is the universal cover of ad-regular graph. Every connected base graphhas a universal cover which is an infinite tree.
Nati Linial Random Lifts of Graphs
![Page 81: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/81.jpg)
A few more general properties of lifts
I Vertex degrees are maintained. If x has dneighbors, then so do all the vertices in thefiber of x . In particular, a lift of a d-regulargraph is d-regular.
I The cycle Cn is a lift of Cm iff m|n.
I The d-regular tree covers every d-regulargraph. This is the universal cover of ad-regular graph. Every connected base graphhas a universal cover which is an infinite tree.
Nati Linial Random Lifts of Graphs
![Page 82: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/82.jpg)
A few more general properties of lifts
I Vertex degrees are maintained. If x has dneighbors, then so do all the vertices in thefiber of x . In particular, a lift of a d-regulargraph is d-regular.
I The cycle Cn is a lift of Cm iff m|n.
I The d-regular tree covers every d-regulargraph. This is the universal cover of ad-regular graph. Every connected base graphhas a universal cover which is an infinite tree.
Nati Linial Random Lifts of Graphs
![Page 83: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/83.jpg)
Old vs. New Eigenvalues
Here is an easy observation:
The lifted graph inherits every eigenvalue of thebase graph.
Namely, if H is a lift of G , then every eigenvalue ofG is also an eigenvalue of H
(Pf: Pullback, i.e., takeany eigenfunction f of G , and assign the value f (x)to every vertex in the fiber of x . It is easily verifiedthat this is an eigenfunction of H with the sameeigenvalue as f in G ).
Nati Linial Random Lifts of Graphs
![Page 84: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/84.jpg)
Old vs. New Eigenvalues
Here is an easy observation:
The lifted graph inherits every eigenvalue of thebase graph.
Namely, if H is a lift of G , then every eigenvalue ofG is also an eigenvalue of H(Pf: Pullback, i.e., takeany eigenfunction f of G , and assign the value f (x)to every vertex in the fiber of x . It is easily verifiedthat this is an eigenfunction of H with the sameeigenvalue as f in G ).
Nati Linial Random Lifts of Graphs
![Page 85: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/85.jpg)
Old vs. New Eigenvalues (contd.)
These are called the old eigenvalues of H . If G isgiven, the old eigenvalues appear in every lift, andwe can only hope to control the values of the neweigenvalues.
This suggests the following approach to theconstruction of d-regular Ramanujan Graphs byrepeated lifts:
I Start from a small d-regular Ramanujan Graph(e.g. Kd+1).
I In every step apply a lift to the previous graphwhile keeping all new eigenvalues in the interval[−2√
d − 1, 2√
d − 1]
Nati Linial Random Lifts of Graphs
![Page 86: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/86.jpg)
Old vs. New Eigenvalues (contd.)
These are called the old eigenvalues of H . If G isgiven, the old eigenvalues appear in every lift, andwe can only hope to control the values of the neweigenvalues.
This suggests the following approach to theconstruction of d-regular Ramanujan Graphs byrepeated lifts:
I Start from a small d-regular Ramanujan Graph(e.g. Kd+1).
I In every step apply a lift to the previous graphwhile keeping all new eigenvalues in the interval[−2√
d − 1, 2√
d − 1]
Nati Linial Random Lifts of Graphs
![Page 87: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/87.jpg)
Old vs. New Eigenvalues (contd.)
These are called the old eigenvalues of H . If G isgiven, the old eigenvalues appear in every lift, andwe can only hope to control the values of the neweigenvalues.
This suggests the following approach to theconstruction of d-regular Ramanujan Graphs byrepeated lifts:
I Start from a small d-regular Ramanujan Graph(e.g. Kd+1).
I In every step apply a lift to the previous graphwhile keeping all new eigenvalues in the interval[−2√
d − 1, 2√
d − 1]
Nati Linial Random Lifts of Graphs
![Page 88: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/88.jpg)
How to think about 2-lifts
In an n-lift of a graph G we associate with everyedge e = xy of G a permutation πe ∈ Sn which tellsus how to connect the n vertices in the fiber Fx withthe n vertices of Fy .
But in S2 the only permutations are the identity idand the switch σ = (12). So a 2-lift of G isspecified by deciding, for every edge e whether πe
equals id or σ.
Alternatively, we sign the edges of G where +1stands for id and −1 for σ.
Nati Linial Random Lifts of Graphs
![Page 89: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/89.jpg)
How to think about 2-lifts
In an n-lift of a graph G we associate with everyedge e = xy of G a permutation πe ∈ Sn which tellsus how to connect the n vertices in the fiber Fx withthe n vertices of Fy .
But in S2 the only permutations are the identity idand the switch σ = (12). So a 2-lift of G isspecified by deciding, for every edge e whether πe
equals id or σ.
Alternatively, we sign the edges of G where +1stands for id and −1 for σ.
Nati Linial Random Lifts of Graphs
![Page 90: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/90.jpg)
How to think about 2-lifts
In an n-lift of a graph G we associate with everyedge e = xy of G a permutation πe ∈ Sn which tellsus how to connect the n vertices in the fiber Fx withthe n vertices of Fy .
But in S2 the only permutations are the identity idand the switch σ = (12). So a 2-lift of G isspecified by deciding, for every edge e whether πe
equals id or σ.
Alternatively, we sign the edges of G where +1stands for id and −1 for σ.
Nati Linial Random Lifts of Graphs
![Page 91: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/91.jpg)
Signing and spectra
A signing is a symmetric matrix in which some ofthe entries in the adjacency matrix of G arechanged from +1 to −1.
We think of a signing in two equivalent ways: A wayof specifying a 2-lift of G ,and a real symmetricmatrix with entries 0, 1,−1.
An easy but useful observation:
PropositionThe new eigenvalues of a 2-lift of G are theeigenvalues of the corresponding signing matrix.
Nati Linial Random Lifts of Graphs
![Page 92: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/92.jpg)
Signing and spectra
A signing is a symmetric matrix in which some ofthe entries in the adjacency matrix of G arechanged from +1 to −1.
We think of a signing in two equivalent ways: A wayof specifying a 2-lift of G ,
and a real symmetricmatrix with entries 0, 1,−1.
An easy but useful observation:
PropositionThe new eigenvalues of a 2-lift of G are theeigenvalues of the corresponding signing matrix.
Nati Linial Random Lifts of Graphs
![Page 93: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/93.jpg)
Signing and spectra
A signing is a symmetric matrix in which some ofthe entries in the adjacency matrix of G arechanged from +1 to −1.
We think of a signing in two equivalent ways: A wayof specifying a 2-lift of G ,and a real symmetricmatrix with entries 0, 1,−1.
An easy but useful observation:
PropositionThe new eigenvalues of a 2-lift of G are theeigenvalues of the corresponding signing matrix.
Nati Linial Random Lifts of Graphs
![Page 94: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/94.jpg)
Signing and spectra
A signing is a symmetric matrix in which some ofthe entries in the adjacency matrix of G arechanged from +1 to −1.
We think of a signing in two equivalent ways: A wayof specifying a 2-lift of G ,and a real symmetricmatrix with entries 0, 1,−1.
An easy but useful observation:
PropositionThe new eigenvalues of a 2-lift of G are theeigenvalues of the corresponding signing matrix.
Nati Linial Random Lifts of Graphs
![Page 95: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/95.jpg)
Signing and spectra
A signing is a symmetric matrix in which some ofthe entries in the adjacency matrix of G arechanged from +1 to −1.
We think of a signing in two equivalent ways: A wayof specifying a 2-lift of G ,and a real symmetricmatrix with entries 0, 1,−1.
An easy but useful observation:
PropositionThe new eigenvalues of a 2-lift of G are theeigenvalues of the corresponding signing matrix.
Nati Linial Random Lifts of Graphs
![Page 96: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/96.jpg)
Recall: The spectral radius of a matrix is the largestabsolute value of an eigenvalue.
The above approach to the construction ofRamanujan Graphs can be stated as follows:
ConjectureEvery d-regular Ramanujan Graph has a signingwith spectral radius ≤ 2
√d − 1.
Nati Linial Random Lifts of Graphs
![Page 97: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/97.jpg)
Recall: The spectral radius of a matrix is the largestabsolute value of an eigenvalue.
The above approach to the construction ofRamanujan Graphs can be stated as follows:
ConjectureEvery d-regular Ramanujan Graph has a signingwith spectral radius ≤ 2
√d − 1.
Nati Linial Random Lifts of Graphs
![Page 98: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/98.jpg)
Recall: The spectral radius of a matrix is the largestabsolute value of an eigenvalue.
The above approach to the construction ofRamanujan Graphs can be stated as follows:
ConjectureEvery d-regular Ramanujan Graph has a signingwith spectral radius ≤ 2
√d − 1.
Nati Linial Random Lifts of Graphs
![Page 99: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/99.jpg)
The signing conjecture
But it seems that something much stronger is true
ConjectureEvery d-regular graph G has a signing with spectralradius ≤ 2
√d − 1.
This conjecture, if true, is tight.
Nati Linial Random Lifts of Graphs
![Page 100: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/100.jpg)
The signing conjecture
But it seems that something much stronger is true
ConjectureEvery d-regular graph G has a signing with spectralradius ≤ 2
√d − 1.
This conjecture, if true, is tight.
Nati Linial Random Lifts of Graphs
![Page 101: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/101.jpg)
The signing conjecture
But it seems that something much stronger is true
ConjectureEvery d-regular graph G has a signing with spectralradius ≤ 2
√d − 1.
This conjecture, if true, is tight.
Nati Linial Random Lifts of Graphs
![Page 102: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/102.jpg)
What is known
Theorem (Yonatan Bilu + L.)By repeated application of 2-lifts it is possible toexplicitly construct d-regular graphs (d ≥ 3) whosesecond eigenvalue
λ2 ≤ O(√
d log3/2 d)
Nati Linial Random Lifts of Graphs
![Page 103: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/103.jpg)
A highlight of the proof
The most unexpected part of the proof is a converseof the so-called Expander Mixing Lemma.
Our new lemma says that λ2 is controlled by theextent to which G is pseudo-random.
What’s involved is the graph’s discrepancy, i.e. themaximum of
e(A,B)− dn |A||B |√
|A||B |
Nati Linial Random Lifts of Graphs
![Page 104: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/104.jpg)
A highlight of the proof
The most unexpected part of the proof is a converseof the so-called Expander Mixing Lemma.
Our new lemma says that λ2 is controlled by theextent to which G is pseudo-random.
What’s involved is the graph’s discrepancy, i.e. themaximum of
e(A,B)− dn |A||B |√
|A||B |
Nati Linial Random Lifts of Graphs
![Page 105: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/105.jpg)
A highlight of the proof
The most unexpected part of the proof is a converseof the so-called Expander Mixing Lemma.
Our new lemma says that λ2 is controlled by theextent to which G is pseudo-random.
What’s involved is the graph’s discrepancy, i.e. themaximum of
e(A,B)− dn |A||B |√
|A||B |
Nati Linial Random Lifts of Graphs
![Page 106: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/106.jpg)
A few more things about random lifts
A matching M in a graph G is a collection ofdisjoint edges. If the edges in M meet every vertexin G , we say that M is a perfect matching=PM.The defect of G is the number of vertices missed bythe largest matching in G . (So the existence of aPM is the same as zero defect).
Question: Given a base graph G and a large eveninteger n, how likely is an n-lift of G to contain aperfect matching?
Nati Linial Random Lifts of Graphs
![Page 107: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/107.jpg)
A few more things about random lifts
A matching M in a graph G is a collection ofdisjoint edges. If the edges in M meet every vertexin G , we say that M is a perfect matching=PM.The defect of G is the number of vertices missed bythe largest matching in G . (So the existence of aPM is the same as zero defect).
Question: Given a base graph G and a large eveninteger n, how likely is an n-lift of G to contain aperfect matching?
Nati Linial Random Lifts of Graphs
![Page 108: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/108.jpg)
Theorem (L. + Rozenman)Let G be a base graph and let H be a random 2nlift of G . For every base graph exactly one of thefollowing four situations occurs:
I Every 2n-lift H of G contains a PM.
I Every H must have defect ≥ αn for someconstant α > 0.
I The probability that H has a PM is 1− o(1)(but not 1).
I Almost surely H has defect Θ(log n).
Nati Linial Random Lifts of Graphs
![Page 109: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/109.jpg)
Theorem (L. + Rozenman)Let G be a base graph and let H be a random 2nlift of G . For every base graph exactly one of thefollowing four situations occurs:
I Every 2n-lift H of G contains a PM.
I Every H must have defect ≥ αn for someconstant α > 0.
I The probability that H has a PM is 1− o(1)(but not 1).
I Almost surely H has defect Θ(log n).
Nati Linial Random Lifts of Graphs
![Page 110: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/110.jpg)
Theorem (L. + Rozenman)Let G be a base graph and let H be a random 2nlift of G . For every base graph exactly one of thefollowing four situations occurs:
I Every 2n-lift H of G contains a PM.
I Every H must have defect ≥ αn for someconstant α > 0.
I The probability that H has a PM is 1− o(1)(but not 1).
I Almost surely H has defect Θ(log n).
Nati Linial Random Lifts of Graphs
![Page 111: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/111.jpg)
Theorem (L. + Rozenman)Let G be a base graph and let H be a random 2nlift of G . For every base graph exactly one of thefollowing four situations occurs:
I Every 2n-lift H of G contains a PM.
I Every H must have defect ≥ αn for someconstant α > 0.
I The probability that H has a PM is 1− o(1)(but not 1).
I Almost surely H has defect Θ(log n).
Nati Linial Random Lifts of Graphs
![Page 112: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/112.jpg)
Theorem (L. + Rozenman)Let G be a base graph and let H be a random 2nlift of G . For every base graph exactly one of thefollowing four situations occurs:
I Every 2n-lift H of G contains a PM.
I Every H must have defect ≥ αn for someconstant α > 0.
I The probability that H has a PM is 1− o(1)(but not 1).
I Almost surely H has defect Θ(log n).
Nati Linial Random Lifts of Graphs
![Page 113: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/113.jpg)
What else?
There are theorems about the typical behavior
I The typical degree of connectivity of a lift of G .
I The chromatic numbers of typical lifts.
I The typical distribution of new eigenvalues.
I ... and more ....
Nati Linial Random Lifts of Graphs
![Page 114: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/114.jpg)
What else?
There are theorems about the typical behavior
I The typical degree of connectivity of a lift of G .
I The chromatic numbers of typical lifts.
I The typical distribution of new eigenvalues.
I ... and more ....
Nati Linial Random Lifts of Graphs
![Page 115: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/115.jpg)
What else?
There are theorems about the typical behavior
I The typical degree of connectivity of a lift of G .
I The chromatic numbers of typical lifts.
I The typical distribution of new eigenvalues.
I ... and more ....
Nati Linial Random Lifts of Graphs
![Page 116: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/116.jpg)
What else?
There are theorems about the typical behavior
I The typical degree of connectivity of a lift of G .
I The chromatic numbers of typical lifts.
I The typical distribution of new eigenvalues.
I ... and more ....
Nati Linial Random Lifts of Graphs
![Page 117: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/117.jpg)
... and much more that we do not know...
Open Problem
I Is there a zero-one law for Hamiltonian cycles?
I What is the typical chromatic number of ann-lift of K5? Is it 3 or 4, perhaps each withpositive probability?
Nati Linial Random Lifts of Graphs
![Page 118: Random Lifts of Graphsnati/PAPERS/rio_09.pdf · 2009. 8. 13. · Random graphs come up a lot in (mathematical) statistical mechanics. A major example is Percolation Theory. Start](https://reader036.vdocuments.us/reader036/viewer/2022071015/5fce48ca5c3499666862acb9/html5/thumbnails/118.jpg)
... and much more that we do not know...
Open Problem
I Is there a zero-one law for Hamiltonian cycles?
I What is the typical chromatic number of ann-lift of K5? Is it 3 or 4, perhaps each withpositive probability?
Nati Linial Random Lifts of Graphs