Convergent Dense

Graph Sequences

Jennifer Tour Chayesjoint work with

C. Borgs, L. Lovasz, V. Sos, K. Vesztergombi

Convergent Dense

Graph Sequences

I: Metrics, Sampling & TestingII: Multi-way Cuts & Statistical Physics

Outline of I: Metrics, Sampling & Testing

Introduction: Motivation, Convergence and Testing

Subgraph Densities and Left Convergence Graph Metrics Convergence in Metric Szemeredi Lemma and Sampling Parameter Testing The Limit Object, Metric Convergence &

Testability

Introduction: Motivation Numerous examples of growing graph sequences

– e.g., Internet, WWW, social networks

Want a succinct but faithful representation for

Testing properties – e.g., clustering

Testing algorithms – e.g., for routing, search

Here we deal only with dense graphs

(also have results for bounded-degree graphs)

Introduction: Convergence

Given: Sequence Gn of graphs with |V(Gn)| ! 1

Questions: What is the “right” notion of convergence? Is there a useful metric s.t. Gn convergent

, Gn is Cauchy in the metric? What is the limit object?

Introduction: Testing

Given: a simple graph parameter f, i.e. a real-valued function on simple graphs, invariant under isomorphism

Question: Under what conditions is f testable, i.e. 8 > 0, 9 k < 1 such that 8 G with |V(G)| > k,

|f(G) – f(G[S])| <

with probability at least 1 – , where S ½ V(G) is a uniformly random sample of size k?

Preview

There is a reasonable notion of convergent graph sequences, which turns out to be equivalent to convergence in an appropriate metric, and is closely related to testability.

(Some of the) Main Theorems of Part I:f(Gn) converges 8 convergent graph sequence Gn

, f is continuous in the metric, f is testable, f can be extended to a continuous function on the limit object

Outline

Introduction: Motivation, Convergence and Testing

Subgraph Densities and Left Convergence Graph Metrics Convergence in Metric Szemeredi Lemma and Sampling The Limit Object and Metric Convergence

Subgraph Densities

F, G simple graphs (For Part I, think of F as small and G as large)

Homomorphisms: adjacency preserving maps

Hom(F,G) = {: V(F) ! V(G) s.t. (E(F)) ½ E(G)}

Subgraph densities (1979 Erdos, Lovasz, Spencer):

t(F,G) = |V(G)||V(F)|

|Hom(F,G)|

E.g., t(K3,G) is the triangle density of G

Left Convergence

Our Definition: Gn is said to be (left) convergent if t(F, Gn) converges for all simple F.

Example: Let Gn = Gn,p. Then

t(F, Gn) ! p|E(F)| .

Outline

Introduction: Motivation, Convergence and Testing

Subgraph Densities and Left Convergence Graph Metrics Convergence in Metric Szemeredi Lemma and Sampling The Limit Object and Metric Convergence

Outline

Graph Metrics Cut norm on matrices Distances between graphs with same

number of vertices Splitting vertices Cut metric between arbitrary graphs

Cut Norm on Matrices Definition (1999 Frieze, Kannan):

Let M be an n £ n matrix

The cut norm of M is given by

kMk = max | Mij |i 2 Sj 2 T

S,T

Distances Between Graphs on Same Number of Vertices

G, G0 weighted graphs on [n] with common vertex weights 1, …, n s.t. ii = different edge weights ij and ij

0

adjacency matrices AG with (AG)ij = i ijj

and AG0 with (AG0)ij = i ij0j

Define the distance between G and G0:

d(G, G0) = -2 k AG AG0k

=-2 max | i j (ij – ij0)|

S,T ½ [n] i 2 Sj 2 T

Distances between Graphs on Same Number of Vertices

Notice that the distance d(G, G0) is not invariant under isomorphisms of G and G0

So do an “integer overlay” and define

where the minimum goes over all relabelings G and G0

(G, G0) = min d(GI, GI0)

GI,GI0

E.g., G

=

G0 =

(G, G0) = 0

)

Different Numbers of Vertices

Question: What do we do if G and G0 have different numbers of vertices n n0?

Idea: Split each vertex of G into n0 new vertices vertex of G0 into n new vertices

Splitting Vertices

Given: Graph G on [n],

with weights i, ij

Split i into n0 pieces(i,1), … , (i,n0)

Split i into n0 pieces

i = u2n0 Xiu

ii1i2i3

21

3

Splitting Vertices (continued)

Given: Graph G on [n],

with weights i, ij

Replace edge ij bycomplete bipartite graphwith iu,jv = ij

) new graph G[X] on [nn0] Fact: t(F,G[X]) = t(F,G)

i

j

i1 i2 i3

j1 j2 j3

Cut Metric between Arbitrary Graphs

Given: G graph on [n], weights i, ij

G0 graph on [n0], weights 0u,

0uv

with ii = u 0

u

Define our cut metric

(G, G0) = min d(G[X], G0[X])X

where the minimum goes over all fractional overlays, i.e. all couplings (or joinings) X of

i and 0u, with Xiu ¸ 0, uXiu = i and iXiu =

0u.

Outline

Introduction: Motivation, Convergence and Testing

Subgraph Densities and Left Convergence Graph Metrics Convergence in Metric Szemeredi Lemma and Sampling The Limit Object and Metric Convergence

Convergence in Metric

Definition: Let (Gn) be a sequence of simple graphs. We say that (Gn) is convergent in the

metric if (Gn) is a Cauchy sequence in .

Theorem 1 (Convergence in Metric): A sequence of simple graphs (Gn) is left convergent if and only if it is convergent in the metric .

I.e., subgraph densities of a sequence of graphs

converge , the sequence is Cauchy in the cut metric.

Outline

Introduction: Motivation, Convergence and Testing

Subgraph Densities and Left Convergence Graph Metrics Convergence in Metric Szemeredi Lemma and Sampling Parameter Testing The Limit Object and Metric Convergence

Szemeredi Lemma Given:

simple (unweighted) graph G disjoint partition P = (V1, ... , Vq) of V(G)

Define the edge density between classes Vi and Vj:

ij = |Vi|1||Vj|

1 eG(Vi ,Vj)

Define the (weighted) “average graph” GP on V(G) with nodeweights x(G)= 1 edgeweights xy (G) = ij if x 2 Vi and y 2 Vj

Szemeredi Lemma (continued)

It turns out that Szemerdi’s Regularity Lemma, in the weak form proved by Frieze and Kannan, describes precisely the -distance between a simple graph G and its average over a sufficiently large partition.

Weak Regularity Lemma (1999 Frieze and Kannan): For all > 0 and all simple graphs G, there exists a partition P = (V1, ... , Vq) of V(G) into q · 41/2 classes such that

(G, GP) < .

Sampling

Main Technical Lemma: Let k be a positive integer, and let G, G0 be weighted graphs on at least k nodes with nodeweights one and edgeweights in [0,1]. If S is chosen uniformly from all subsets S ½ V of size k, then

|(G[S],G0[S]) (G,G0)| · 10 k1/4

with probability at least 1 – exp(k1/2/8).

related to 2003 Alon, Fernandez de la Vega, Kannan and Karpinski, but with different k dependence and different proofs

Sampling (continued)

Theorem 2 (Closeness of Sample): Let k be a positive integer, and let G be a simple graph on at least k nodes. If S is chosen uniformly from all subsets S ½ V of size k, then

(G,G[S]) · 10 (log2k)1/2

with probability at least 1 – exp(k2/2 log2k).

Sampling (continued)

Key Elements of the Proof of Theorem 2 Use an easy sampling argument to prove the

theorem for the special case in which V(G) can be decomposed into only a few large sets V1, …, Vq, with constant weights for an edge between any given pair Vi and Vj

Use the weak regularity lemma to approximate an arbitrary graph by the simple special case above

Use the previous (main technical) lemma to show that this approximation induces only a small error

“Proof” of Theorem 1 Recall Theorem 1: Convergence from the left

(i.e., subgraph convergence) , convergence in cut metric

Idea of Proof of Theorem 1 By the triangle inequality, Theorem 2 implies that two

graphs G and G0 (possibly with n n0) are close in metric only if their samples are close

But knowledge of all subgraph frequencies is more or less equivalent to knowledge of all sampling probabilities

t(F,G) ¼ Prob(G[S] = F)where S is uniform among all sets of size |V(F)|

“thus” (lots of work), convergence of subgraph frequencies , convergence in metric

Outline

Introduction: Motivation, Convergence and Testing

Subgraph Densities and Left Convergence Graph Metrics Convergence in Metric Szemeredi Lemma and Sampling The Limit Object, Metric Convergence &

Testability

The Limit Object

Recall Theorem 1 (Convergence in Metric): A sequence of simple graphs (Gn) is left convergent if and only if it is

convergent in the metric .

) 9 limit object G1

Question: Is there a useful representation of this limit object?

Answer: Yes – the graphon.

Graphons

Definition: A function W: [0,1]2 ! R is called a graphon if W is measurable W(x,y) = W(y,x) kWk1 < 1

Example: Step functions G graph on n vertices WG(x,y) = Idxnedyne 2 E(G)

Graphons as Limit Objects

For |V(F)| = k, define

[0,1]k ij 2 E(F)t(F,W) = s dx1

… dxk W(xi, xj)

Theorem (2005 Lovasz and B. Szegedy): (Gn) is left convergent , 9 graphon W s.t. t(F, Gn) ! t(F,W).

Graphons and Metric Convergence Definition (Frieze and Kannan): The cut

norm of a graphon W is given by

kWk = sup | s W(x,y) |S,T½[0,1] S£T

Theorem 10:

(Gn) is left convergent , 9 graphon W and a relabeling of (Gn) s.t.

kW WGnk ! 0.

Summary of Part I

There is a reasonable notion of convergent graph sequences – convergence of subgraph densities, which turns out to be equivalent to convergence in an appropriate (and useful) metric – the cut metric, and is closely related to sampling and testability.

Summary of Part I

(Some of the) Main Theorems:f(Gn) converges 8 convergent graph sequence Gn

, f is continuous in the cut metric

, f is a testable graph parameter

, f can be extended to a continuous function on the limit object

Part II: Multi-way Cuts & Statistical Physics In Part I, we learned that a graph

sequence Gn can be probed “from the left” by studying the densities of subgraphs occurring in it

In Part II, we will learn that Gn can be probed “from the right” by studying generalized colorings (or multi-way cuts or statistical mechanical models) on it, giving us a dual notion of convergence

Outline of II: Multi-way Cuts & Statistical Physics

Homomorphisms into (Small) Weighted Graphs

Naïve Right Convergence and Ground State Energies

Incomplete Equivalences Microcanonical Ensemble and Right

Convergence Complete Equivalences

Homomorphisms into (Small) Weighted Graphs

Recall that in Part I, we consideredHom(F,G) = {: V(F) ! V(G) s.t. (E(F)) ½ E(G)}with F small and simple, and G large

Now instead consider Hom(G,H) with G large andH small and weighted, with vertex weights i = i(H) and edge weights ij = ij(H), so that

:V(G)!V(H) x2V(G) xy2E(G)|Hom(G,H)| = (x)

(x)(y) This is a weighted count of the number of colorings

Example: Ising Magnet V(H) = {-1,+1} = eh, ’ = eJ’

|Hom(G,H)| = eEnergy()

:V(G)! {-1,+1}

with

Energy() = J xy h x xy2E(G) x2V(G)

Note that this is not the conventional normalization of the energy for a dense graph, so. |Hom(G,H)| exp[|V(G)|2 ].

Outline of II: Multi-way Cuts & Statistical Physics

Homomorphisms into (Small) Weighted Graphs

Naïve Right Convergence and Ground State Energies

Incomplete Equivalences Microcanonical Ensemble and Right

Convergence Complete Equivalences

Naïve Right Convergence Let r(G,H) = |V(G)|2 log |Hom(G,H)| Note that r(G,H) is not the free energy Our Definition:

Gn is said to be naïvely right convergent if r(Gn,H) converges for all soft-core weighted H, i.e. all H with

vertex weights i = i(H) > 0 and

edge weights ij = ij(H) > 0.

J = ( )0 1

01

1 1eH =

Ground State Energy

Let ij(H) = eJij Then the ground state energy of model

H on graph G isE(G,J) = |V(G)|-2 min J(x)(y)

:V(G)!V(H) xy2E(G)

Example: Maxcut Density

E(G,J) = |V(G)|-2 Maxcut (G)

Naïve Right Convergence and Ground State Energies

Lemma: If ij(H) = eJij , then

r(G,H) = E(G,J) + O(|V(G)|-1 ) Again note that r(G,H) is not the free energy

– to leading order, it is just the ground state energy. The entropy has been wiped out by the V2 normalization.

So naïve right convergence of Gn is the same as convergence of the ground state energy for all soft-core models H on Gn.

Outline of II: Multi-way Cuts & Statistical Physics

Homomorphisms into (Small) Weighted Graphs

Naïve Right Convergence and Ground State Energies

Incomplete Equivalences Microcanonical Ensemble and Right

Convergence Complete Equivalences

Incomplete Equivalences

Metric Convergenc

e

f(Gn) is

Cauchy 8 testable f

Naïve Right Convergenc

e

Quotients Cauchy in

Hausdorff Metric

E(G,J) is Cauchy 8 J

Left Convergenc

ePart I Part I

Part II(previous Lemma)

Cut densities are testable

parameters

E(G,J) is continuous in

cut metricOR

Outline of II: Multi-way Cuts & Statistical Physics

Homomorphisms into (Small) Weighted Graphs

Naïve Right Convergence and Ground State Energies

Incomplete Equivalences Microcanonical Ensemble and Right

Convergence Complete Equivalences

Microcanonical Ensemble

Given q color classes with fraction aiincolor class i:

a = (a1, … , aq) with ai≥ 0 and ai= 1 Define the microcanonical homomorphism number:

:V(G)!V(H) x2V(G) xy2E(G)||1(i)|ai|V(G)||≤1

|Homa(G,H)| = (x)

(x)(y)and microcanonical ground state energy:

Ea (G,J) = |V(G)|-2 min J(x)(y) :V(G)!V(H) xy2E(G)||1(i)|-ai|V(G)||≤1

Examples Max/Min Bisection

J = ( )0 ±10±1

Densest Subgraph

J = ( )1 00 0

Right Convergence

Let ra(G,H) = |V(G)|2 log |Homa (G,H)|

Definition: Gn is said to be right convergent if ra(Gn,H) converges for all a and all soft-core weighted H, i.e. all H with

vertex weights i = i(H) > 0 and

edge weights ij = ij(H) > 0.

This is equivalent to convergence of all microcanonical ground state energies.

Outline of II: Multi-way Cuts & Statistical Physics

Homomorphisms into (Small) Weighted Graphs

Naïve Right Convergence and Ground State Energies

Incomplete Equivalences Microcanonical Ensemble and Right

Convergence Complete Equivalences

Complete Equivalences= Summary

Metric Convergenc

e

f(Gn) is

Cauchy 8 testable f

Right Convergenc

e

Quotients Cauchy in

Hausdorff Metric

Ea(G,J) is

Cauchy 8 J

Left Convergenc

ePart I Part I

Part II(previous Lemma)

Cut densities are testable

parameters

Ea (G,J) is continuous in

cut metricOR

Lots of Work

Lots of Work

Summary There is a reasonable notion of convergent graph

sequences – convergence of subgraph densities – which turns out to be equivalent to convergence in an appropriate (and useful) metric – the cut metric, and is closely related to sampling and testability

Convergence of subgraph densities is also equivalent to the dual notion of convergence of all microcanonical ground state energies of all soft-core models (and also to convergence of quotients, moding out by Szemeredi partitions, in the natural Hausdorff metric)

THE ENDTHE END

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