Conjectures 3-Regular Graphs Refs On the probability that random graphs are Ramanujan Steven J Miller, Anthony Sabelli (Brown University) Tim Novikoff (Cornell University) Slides and paper available at http://www.math.brown.edu/∼sjmiller Expanders and Ramanujan Graphs: Construction and Applications AMS National Meeting, San Diego, January 9, 2008. 1
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On the probability that random graphs are Ramanujan · Conjectures 3-Regular Graphs Refs Conjectures Conjectures The distribution of (G) converges to the = 1 Tracy-Widom distribution
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Conjectures 3-Regular Graphs Refs
On the probability that random graphs areRamanujan
Steven J Miller, Anthony Sabelli (Brown University)Tim Novikoff (Cornell University)
Slides and paper available athttp://www.math.brown.edu/∼sjmiller
Expanders and Ramanujan Graphs:Construction and Applications
AMS National Meeting, San Diego, January 9, 2008.
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Conjectures 3-Regular Graphs Refs
Conjectures
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Conjectures 3-Regular Graphs Refs
Expanders and Eigenvalues
Expanding Constanth(G) := inf
{|∂U|
min(|U|,|V\U|): U ⊂ V , |U| > 0
}
{Gm} family of expanders if ∃ε with h(Gm) ≥ ε and|Gm| → ∞.
Cheeger-Buser Inequalitiesd−λ2(G)
2 ≤ h(G) ≤ 2√
2d(d − λ2(G))
Applications: sparse (|E | grows at most linearly with|V |), highly connected.� communication network theory:
(Alon-Boppana, Burger, Serre) {Gm} family of finiteconnected d -regular graphs, limm→∞ |Gm| = ∞:
lim infm→∞
λ2(Gm) ≥ 2√
d − 1
As |G| → ∞, for d ≥ 3 and any ε > 0, “most"d -regular graphs G have
λ2(G) ≤ 2√
d − 1 + ε
(conjectured by Alon, proved for many families byFriedman).
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Conjectures 3-Regular Graphs Refs
Questions
For a family of d -regular graphs:
What is the distribution of λ2?
What percent of the graphs are Ramanujan?
λ(G) = max (λ+(G), λ−(G)), where λ±(G) are largestnon-trivial positive (negative) eigenvalues. If bipartiteλ−(G) = −λ+(G). If connected λ2(G) = λ+(G).
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Conjectures 3-Regular Graphs Refs
Families Investigated (N even)
CIN,d : d -regular connected graphs generatedby choosing d perfect matchings.
SCIN,d : subset of CIN,d that are simple.
CBN,d : d -regular connected bipartite graphsgenerated by choosing d permutations.
SCBN,d : subset of CBN,d that are simple.
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Conjectures 3-Regular Graphs Refs
Tracy-Widom Distribution
Limiting distribution of the normalized largest eigenvaluesfor ensembles of matrices: GOE (β = 1), GUE (β = 2),GSE (β = 4)
ApplicationsLength of largest increasing subsequence of randompermutations.Largest principle component of covariances matrices.Young tableaux, random tilings, queuing theory,superconductors....
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Tracy-Widom Plots
Plots of the three Tracy-Widom distributions: f1(s) is red,f2(s) is blue and f4(s) is green.
-6 -4 -2 2 4
0.1
0.2
0.3
0.4
0.5
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Conjectures 3-Regular Graphs Refs
Tracy-Widom Distributions
Parameters for the Tracy-Widom distributions. Fβ is thecumulative distribution function for fβ, and Fβ(µβ) is themass of fβ to the left of its mean.
Mean µ Std Dev σ Fβ(µβ)TW(β = 1) -1.21 1.268 0.5197TW(β = 2) -1.77 0.902 0.5150TW(β = 4) -2.31 0.720 0.5111Std Normal 0.00 1.000 0.5000
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Conjectures 3-Regular Graphs Refs
Normalized Tracy-Widom Plots
Plots normalized to have mean 0 and variance 1: f norm1 (s)
is red, f norm2 (s) is blue, f norm
4 (s) is green, standard normal isblack.
-4 -2 2 4
0.1
0.2
0.3
0.4
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Conjectures 3-Regular Graphs Refs
Conjectures
ConjecturesThe distribution of λ±(G) converges to the β = 1Tracy-Widom distribution as N → ∞ in all studiedfamilies.For non-bipartite families, λ±(G) are independent.The percent of the graphs that are Ramanujanapproaches 52% as N → ∞ (resp., 27%) in bipartite(resp., non-bipartite) families.
Evidence weaker for CBN,d (d -regular connected bipartitegraphs, not necessarily simple).
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Conjectures 3-Regular Graphs Refs
Distribution of λ+(G)
Distribution of λ+(G) for 1000 graphs randomly chosenfrom CIN,3 for various N (vertical line is 2
Well-modeled by Tracy-Widom with β = 1.Means approach 2
√d − 1 according to power law.
Variance approach 0 according to power law.Comparing the exponents of the power laws, see thenumber of standard deviations that 2
√d − 1 falls to
the right of the mean goes to 0 as N → ∞.λ±(G) appear independent in non-bipartite families.As N → ∞ the probability that a graph is Ramanujanis the mass of the Tracy-Widom distribution to the leftof its mean (52%) if bipartite (27% otherwise).
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Conjectures 3-Regular Graphs Refs
Power law exponents of means and standard deviations
Means: µFN,d ≈ 2√
d − 1 − cµ,N,dNm(FN,d )
Standard Deviations: σFN,d ≈ cσ,N,dNs(FN,d )
Thus 2√
d − 1 ≈ µFN,d +cµ,N,dcσ,N,d
Nm(FN,d )−s(FN,d )σFN,d
Ramanujan Threshold
As N → ∞, if m(FN,d) < s(FN,d) then 2√
d − 1 falls zerostandard deviations to the right of the mean.
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3-Regular Graphs
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Experiments: Comparisons with Tracy-Widom Distribution
Each set is 1000 random 3-regular graphs from CIN,3normalized to have mean 0 and variance 1.
19 degrees of freedom, critical values 30.1435(α = .05) and 36.1908 (α = .01).
Only showing subset of data.
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Conjectures 3-Regular Graphs Refs
χ2-Tets of λ+(G) for CIN,3 versus Tracy-Widom Distributions
Experiment: Mass to the left of the mean for λ+(G)
Each set of 1000 3-regular graphs.mass to the left of the mean of the Tracy-Widomdistributions:� 0.519652 (β = 1)� 0.515016 (β = 2)� 0.511072 (β = 4)� 0.500000 (standard normal).two-sided z-test: critical thresholds: 1.96 (for α = .05)and 2.575 (for α = .01).
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Conjectures 3-Regular Graphs Refs
Experiment: Mass to the left of the mean for CIN,3
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