Spectra of Graphons - Iowa State Universityorion.math.iastate.edu/butler/2017/spring/x95/...Spectra...

Spectra of Graphons:some spectral results from

Laszlo Lovasz’s textbook Large Networks and Graph Limits

Alexander W. N. Riasanovsky

[email protected] Graph Theory (MATH 595)

at Iowa State University

April 19, 2017

Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 1 / 22

Motivation

study large networks

neurosciencethe Internet, social networks

utilize analytic tools in graph theory

continuity, compactness, measure theory, integration, variationalcalculusquasirandomness, Szemeredi’s regularity lemma

compare graphs of different orders to each other

graphs which change slightly over time (Internet, social media)graphs with the same basic shape but vastly different orders


Motivation


neuroscience

the Internet, social networks






Motivation








Motivation








Motivation




continuity, compactness, measure theory, integration, variationalcalculus

quasirandomness, Szemeredi’s regularity lemma




Motivation








Motivation








Motivation






graphs which change slightly over time (Internet, social media)

graphs with the same basic shape but vastly different orders


Motivation








Pixel Pictures of Graphs

0 1 1 11 0 1 01 1 0 01 0 0 0

Table: The Paw graph.

Pixel Picture

The pixel picture WG is the {0, 1}-valued step function found by fittingthe adjacency matrix of G in the unit square.



0 1 1 11 0 1 01 1 0 01 0 0 0


Pixel Picture




0 1 1 11 0 1 01 1 0 01 0 0 0


Pixel Picture




0

1 1 1

1

0 1 0

1

1 0 0

1

0 0 0


Pixel Picture




0 1

1 1

1 0

1 0

1 1

0 0

1 0

0 0


Pixel Picture




0 1 1

1

1 0 1

0

1 1 0

0

1 0 0

0


Pixel Picture




0 1 1 11 0 1 01 1 0 01 0 0 0


Pixel Picture




0 1 1 11 0 1 01 1 0 01 0 0 0


Pixel Picture




0 1 1 11 0 1 01 1 0 01 0 0 0


Pixel Picture



Two Sequences of Random Graphs

Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 21 vertices.
























Table: The only probable limits in growing sequences of Erdos-Renyi randomgraphs (p = 1/2) and uniform attachment graphs.


Cut Distance of Graphs

Labeled Cut Distance of Graphs

Given two graphs G and H with vertex sets [n],

the labeled graph cutdistance D�(G ,H) ∈ [0, 1] of G and H is given by

D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣∣1TS (AG − AH)1Tn2

∣∣∣∣∣ .Cut Norm of a Matrix [[Frieze, Kannan]’99]

For A some n × n matrix,

||A||� := maxS,T⊆[n]

∣∣∣∣∣∣∑

i∈S ,j∈T

aijn2

∣∣∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS A1Tn2

∣∣∣∣ .

For A = AG , this is different from Cheeger’s constant hG .




Given two graphs G and H with vertex sets [n], the labeled graph cutdistance D�(G ,H) ∈ [0, 1] of G and H is given by

D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2

∣∣∣∣ .Cut Norm of a Matrix [[Frieze, Kannan]’99]


||A||� := maxS,T⊆[n]

∣∣∣∣∣∣∑

i∈S ,j∈T

aijn2

∣∣∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS A1Tn2

∣∣∣∣ .






D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣

= maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2



||A||� := maxS,T⊆[n]

∣∣∣∣∣∣∑

i∈S ,j∈T

aijn2

∣∣∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS A1Tn2

∣∣∣∣ .






D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2

∣∣∣∣ .

Cut Norm of a Matrix [[Frieze, Kannan]’99]


||A||� := maxS,T⊆[n]

∣∣∣∣∣∣∑

i∈S ,j∈T

aijn2

∣∣∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS A1Tn2

∣∣∣∣ .






D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2



||A||� := maxS,T⊆[n]

∣∣∣∣∣∣∑

i∈S ,j∈T

aijn2

∣∣∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS A1Tn2

∣∣∣∣ .






D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2



||A||� := maxS,T⊆[n]

∣∣∣∣∣∣∑

i∈S ,j∈T

aijn2

∣∣∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS A1Tn2

∣∣∣∣ .For A = AG , this is different from Cheeger’s constant hG .Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 5 / 22

Example of Cut Distance of Graphs

AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example

If G has edges 12 and 34 and H has edges 13 and 24,

thenD�(G ,H) = ||AG − AH || = 1/8. Note also that ||AG ||� = ||AH ||� = 1/4.



AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example





AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example





AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example





AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example





AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example

If G has edges 12 and 34 and H has edges 13 and 24, thenD�(G ,H) = ||AG − AH || = 1/8.

Note also that ||AG ||� = ||AH ||� = 1/4.



AG − AH

=

1 −1

1 −1

−1 1−1 1

=

1 −1

1 −1

−1 1−1 1

Example

If G has edges 12 and 34 and H has edges 13 and 24, thenD�(G ,H) = ||AG − AH || = 1/8. Note also that ||AG ||� = ||AH ||� = 1/4.





D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2

∣∣∣∣ .Quasirandomness

Pretend that H is a p = 1/2 ER random graph: replace “eH(S ,T )” with“|S ||T |/2”, and “AH” with “1

2J”, the all-12 matrix. Then D�(G , 1/2) < εimplies that G is ε-quasirandom.





D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2


Pretend that H is a p = 1/2 ER random graph:

replace “eH(S ,T )” with“|S ||T |/2”, and “AH” with “1






D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2


Pretend that H is a p = 1/2 ER random graph: replace “eH(S ,T )” with“|S ||T |/2”

, and “AH” with “12J”, the all-12 matrix. Then D�(G , 1/2) < ε

implies that G is ε-quasirandom.





D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2



2J”, the all-12 matrix.

Then D�(G , 1/2) < εimplies that G is ε-quasirandom.





D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2








D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2

∣∣∣∣ .Cut Distance of Graphs

Given two graphs G and H with vertex sets [n], the (unlabeled) graph cutdistance d�(G ,H) of G and H is given by

d�(G ,H) := minσ∈Sn{D�(G ,Hσ)} .





D�(G ,H) := maxS ,T⊆[n]

∣∣∣∣eG (S ,T )− eH(S ,T )

n2

∣∣∣∣ = maxS ,T⊆[n]

∣∣∣∣1TS (AG − AH)1Tn2

∣∣∣∣ .Cut Distance of Graphs

Given two graphs G and H with vertex sets [n], the (unlabeled) graph cutdistance d�(G ,H) of G and H is given by

d�(G ,H) := minσ∈Sn{D�(G ,Hσ)} .


Cut Distance and Graphons

Definition

If W : [0, 1]2 → [−1, 1] is symmetric and integrable, it is a signed graphon.If 0 ≤W ≤ 1, it is a graphon. For any signed graphon, let its cut norm

||W ||� of W be ||W ||� := supS,T⊆[0,1]

∣∣∣∫S×T W (x , y)dxdy∣∣∣ . Also,

D�(W ,X ) := ||W − X ||� is the labeled cut distance.

− =

Table: WG −WH with G and H from before. Here, ||WG ||� = ||WH ||� = 1/4.Also, D�(WG ,WH) = ||WG −WH ||� = 1/8.


Cut Distance of Graphs and Graphons

Cut Distances

For any graphons W ,X : [0, 1]2 → [0, 1] W ,X and graphs G ,H,

therespective cut distances are:

δ�(W ,X ) := minσ∈S [0,1]

D�(W ,X σ) and δ�(G ,H) := δ�(WG ,WH).

σ−→

Table: If G = e12 + e34 and H = e13 + e24

and σ swaps [1/4, 1/2]↔ [1/2, 3/4],then W σ

G = WH . So δ�(G1,G2) = 0.



Cut Distances

For any graphons W ,X : [0, 1]2 → [0, 1] W ,X and graphs G ,H, therespective cut distances are:

δ�(W ,X ) := minσ∈S [0,1]


σ−→



G = WH . So δ�(G1,G2) = 0.



Cut Distances


δ�(W ,X ) := minσ∈S [0,1]


σ−→



G = WH . So δ�(G1,G2) = 0.



Cut Distances


δ�(W ,X ) := minσ∈S [0,1]

D�(W ,X σ)

and δ�(G ,H) := δ�(WG ,WH).

σ−→



G = WH . So δ�(G1,G2) = 0.



Cut Distances


δ�(W ,X ) := minσ∈S [0,1]


σ−→



G = WH . So δ�(G1,G2) = 0.



Cut Distances


δ�(W ,X ) := minσ∈S [0,1]


σ−→



G = WH . So δ�(G1,G2) = 0.



Cut Distances


δ�(W ,X ) := minσ∈S [0,1]


σ−→

Table: If G = e12 + e34 and H = e13 + e24 and σ swaps [1/4, 1/2]↔ [1/2, 3/4],

then W σG = WH . So δ�(G1,G2) = 0.



Cut Distances


δ�(W ,X ) := minσ∈S [0,1]


σ−→

Table: If G = e12 + e34 and H = e13 + e24 and σ swaps [1/4, 1/2]↔ [1/2, 3/4],then W σ

G = WH . So δ�(G1,G2) = 0.


Cut Distance Examples

Table: Complete graphsand J, the all-onesgraphon:δ�(WKn , J) = 1/n→ 0.

Table: Completebipartite graphs:δ�(K2,Kdn/2e,bn/2c)→0.

Question

How do the spectra compare?


Cut Distance Examples

Table: Complete graphsand J, the all-onesgraphon:δ�(WKn , J) = 1/n→ 0.

Table: Completebipartite graphs:δ�(K2,Kdn/2e,bn/2c)→0.

Question

How do the spectra compare?


Revisiting Convergence

























Table: The only probable limits in growing sequences of Erdos-Renyi randomgraphs (p = 1/2) and uniform attachment graphs.


Density and Completeness

TheoremLet W be the metric space ofgraphons modded out by δ� = 0.Then W is compact and the subsetG ⊆ W of graph graphons (those ofthe form WG ) is dense.

In other words...

1 Convergent sequences ofgraphs are graphons

2 Graphons are convergentsequences of graphs

3 Any sequence of graph(on)shas a limit graphon

4 extremal constructions haveone or more limts




In other words...








In other words...








In other words...








In other words...






AG versus TW

AG

Given |V (G )| = n and v ∈ Rn

(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW

Given a graphon W and f a“nice” f : [0, 1]→ R

(TW f )(x) =

∫ 1

0W (x , y)f (y)dy

Formally, TW : nice→ nice


AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn

0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW

Given a graphon W

and f a“nice” f : [0, 1]→ R

(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy

Formally,

TW : nice→ nice


AG versus TW

AG


(AGv)k =n∑

i=1

akivi

AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0

πξχρ

=

π + ξ

TW


(TW f )(x) =

∫ 1

0W (x , y)f (y)dy



Spectral Decomposition of TW

Theorem

If W is a graphon, then its eigenvalues can be ordered as1 ≥ |λ1| ≥ |λ2| ≥ · · · so that λk → 0 as k →∞ and

TW ∼∞∑k=1

λk fk(x)fk(y) treating TW : L2[0, 1]→ L2[0, 1]

where the inner product∫ 10 fk(x)fl(y) = 1 if k = l and 0 otherwise.

Example

If W (x , y) = xy , λ1 = 1/4 andf1(x) = 2x and all other λk = 0.



Theorem


TW ∼∞∑k=1



Example

If W (x , y) = xy ,

λ1 = 1/4 andf1(x) = 2x and all other λk = 0.



Theorem


TW ∼∞∑k=1



Example

If W (x , y) = xy , λ1 = 1/4 andf1(x) = 2x

and all other λk = 0.



Theorem


TW ∼∞∑k=1



Example

If W (x , y) = xy , λ1 = 1/4 andf1(x) = 2x and all other λk = 0.


Exercise: From spec(G ) to spec(W )

To turn a spectral decomposition of AG into one of TW ...

1 Write AG out spectrally.

2 Replace the standard basis of Rn with weighted inticator functions.(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )

3 Justify that there are no more eigenvalues. (Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)











2 Replace the standard basis of Rn with weighted inticator functions.

(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )













3 Justify that there are no more eigenvalues.

(Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)








Spectral Convergence

Table: WKn → J asn→∞.

Table:WKdn/2e,bn/2c →WK2 asn→∞.

Spectra?

spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and

spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and

spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?

spec(Kn) = {n − 1,−1, . . . ,−1} and

spec(J) = {1, 0, . . . , 0} and


spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?



spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?

spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} andspec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and

spec(WK2) = {1/2, 0, . . . , 0,−1/2}.





Spectra?

spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} andspec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} andspec(WK2) = {1/2, 0, . . . , 0,−1/2}.


Spectral Convergence Theorem

“Padded” Bi-Infinite Spectra

Pad every spectrum with an infinite number of “0”s in the middle.

Viewspec as a two sequences: nonnegative and nonpositive.

Example

spec(Petersen) = {3, 1, 1, 1, 1, 1, 0, . . . , 0, − 2,−2,−2,−2}

Theorem [Borgs, Chayes, Lovasz, Sos, Vesztergombi ’12]

Suppose Gn →W in δ�. Then as k →∞

{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}




Pad every spectrum with an infinite number of “0”s in the middle. Viewspec as a two sequences: nonnegative and nonpositive.

Example

spec(Petersen) = {3, 1, 1, 1, 1, 1, 0, . . . , 0, − 2,−2,−2,−2}



{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}





Examplespec(Petersen) =

{3, 1, 1, 1, 1, 1, 0, . . . , 0, − 2,−2,−2,−2}



{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}





Examplespec(Petersen) = {3, 1, 1, 1, 1, 1,

0, . . . , 0, − 2,−2,−2,−2}



{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}





Examplespec(Petersen) = {3, 1, 1, 1, 1, 1,

0, . . . , 0,

− 2,−2,−2,−2}



{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}





Examplespec(Petersen) = {3, 1, 1, 1, 1, 1, 0, . . . , 0, − 2,−2,−2,−2}



{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}








{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}







Suppose Gn →W in δ�.

Then as k →∞

{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}








{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}








{

λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}

↓ ↓ ↓ ↓ ↓

{

λ1(W )

,

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}








{ λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )|

}↓

↓ ↓ ↓ ↓

{ λ1(W ),

λ2(W )

, · · · ,

λ−2(W ) λ−1(W )

}








{ λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| ,

· · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )| }

↓

↓ ↓ ↓

↓{ λ1(W ),

λ2(W )

, · · · ,

λ−2(W )

λ−1(W ) }








{ λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| , · · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )| }

↓ ↓

↓ ↓

↓{ λ1(W ), λ2(W ), · · · ,

λ−2(W )

λ−1(W ) }








{ λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| , · · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )| }

↓ ↓

↓

↓ ↓{ λ1(W ), λ2(W ), · · · , λ−2(W ) λ−1(W ) }








{ λ1(Gk )|V (Gk )| ,

λ2(Gk )|V (Gk )| , · · · ,

λ−2(Gk )|V (Gk )| ,

λ−1(Gk )|V (Gk )| }

↓ ↓ ↓ ↓ ↓{ λ1(W ), λ2(W ), · · · , λ−2(W ) λ−1(W ) }


The End


Applications

Triangle Removal Lemma

For all ε > 0, there exists some ε′ > 0 so that if G a graph on n verticeshas at most ε′n3 triangles, then there exists some triangle-free G ′ ⊆ Gwith e(G )− e(G ′) ≤ εn2.

Quasirandomness [Chung, Graham, Wilson ’89]

If G1,G2, . . . are graphs where Gn is εn-quasirandom (εn as small aspossible) and |Gn| → ∞, then∑

k

λk(Gn)2 → 1/2 and∑k

λk(Gn)4 → 1/16.

if and only if εn →∞.


More Applications

Many Strong Szemeredi Regularity Lemmas for Graphs and Graphons

All large graphs (all graphons) may be approxmated in cut distance witharbitrary pre-specified precision by a random graph (graphon) which isgiven by an equipartition.

“Disguises” of Graphons

The following models are cryptomorphic (i.e., the same information):

1 a graphon, up to weak isomorphism

2 a multiplicative, normalized simple graph parameter that isnonnegative on signed graphs

3 a consistent and local graph model

4 a local random countable graph model

5 a point in the completion of the space of finite graphs with the cutdistance


References

A. Frieze, R. Kannan (1999)

Quick approximation to matrices and applications

Combinatorica 19(3), 175 – 220.

Lovasz (2012)

Large networks and graph limits

Providence: American Mathematical Society 60.


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