1. Convergent Dense Graph Sequences
Jennifer Tour Chayes joint work with
C. Borgs, L. Lovasz, V. Sos, K. Vesztergombi

2. Convergent Dense Graph Sequences
I: Metrics, Sampling & Testing II: Multi-way Cuts & Statistical Physics

3. 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

4. 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)

5. 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?

6. 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 e > 0, 9 k < 1 such that 8 G with |V(G)| > k,
|f(G) – f(G[S])| < e
with probability at least 1 – e, where S ½ V(G) is a uniformly random sample of size k?

7. Preview , f is continuous in the metric , f is testable
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

8. 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

9. 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) = {f: V(F) ! V(G) s.t. f(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

10. 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)| .

11. 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

12. Outline Graph Metrics Cut norm on matrices
Distances between graphs with same number of vertices
Splitting vertices
Cut metric between arbitrary graphs

13. Cut Norm on Matrices kMkW = max |  Mij |
Definition (1999 Frieze, Kannan):
Let M be an n £ n matrix
The cut norm of M is given by
kMkW = max |  Mij |
i 2 S
j 2 T
S,T

14. Distances Between Graphs on Same Number of Vertices
G, G0 weighted graphs on [n] with
common vertex weights a1, …, an s.t. Siai = a
different edge weights bij and bij0
adjacency matrices AG with (AG)ij = ai bijaj
and AG0 with (AG0)ij = ai bij0aj
Define the distance between G and G0:
dW(G, G0) = a-2 k AG - AG0 kW
= a-2 max | S ai aj (bij – bij0)|
S,T ½ [n]
i 2 S
j 2 T

15. Distances between Graphs on Same Number of Vertices
Notice that the distance dW(G, G0) is not invariant under isomorphisms of G and G0
So do an “integer overlay” and define
W(G, G0) = min dW(GI, GI0)
GI,GI0
where the minimum goes over all relabelings G and G0 2 1 3
E.g., G =
G0 =
W(G, G0) = 0 )

16. 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

17. Splitting Vertices ai = Su2n0 Xiu Given: Graph G on [n],
with weights ai, bij
Split i into n0 pieces
(i,1), … , (i,n0)
Split ai into n0 pieces
ai = Su2n0 Xiu i i1 i2 i3 2 1 3

18. Splitting Vertices (continued)
Given: Graph G on [n],
with weights ai, bij
Replace edge ij by
complete bipartite graph
with biu,jv = bij
) new graph G[X] on [nn0]
Fact: t(F,G[X]) = t(F,G) i j i1 i2 i3 j1 j2 j3

19. Cut Metric between Arbitrary Graphs
Given: G graph on [n], weights ai, bij
G0 graph on [n0], weights a0u, b0uv
with Siai = Su a0u
Define our cut metric
dW(G, G0) = min dW(G[X], G0[X]) X
where the minimum goes over all fractional overlays, i.e. all couplings (or joinings) X of ai and a0u, with Xiu ¸ 0, SuXiu = ai and SiXiu = a0u.

20. 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

21. Convergence in Metric
Definition: Let (Gn) be a sequence of simple graphs. We say that (Gn) is convergent in the metric dW if (Gn) is a Cauchy sequence in dW.
Theorem 1 (Convergence in Metric): A sequence of simple graphs (Gn) is left convergent if and only if it is convergent in the metric dW.
I.e., subgraph densities of a sequence of graphs converge , the sequence is Cauchy in the cut metric.

22. 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

23. Szemeredi Lemma Given: simple (unweighted) graph G
disjoint partition P = (V1, ... , Vq) of V(G)
Define the edge density between classes Vi and Vj:
bij = |Vi|-1||Vj|-1 eG(Vi ,Vj)
Define the (weighted) “average graph” GP on V(G) with
nodeweights ax(G)= 1
edgeweights bxy (G) = bij if x 2 Vi and y 2 Vj

24. Szemeredi Lemma (continued)
It turns out that Szemerdi’s Regularity Lemma, in the weak form proved by Frieze and Kannan, describes precisely the dW-distance between a simple graph G and its average over a sufficiently large partition.
Weak Regularity Lemma (1999 Frieze and Kannan): For all e > 0 and all simple graphs G, there exists a partition P = (V1, ... , Vq) of V(G) into q · 41/e2 classes such that
dW(G, GP) < e.

25. 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
|dW(G[S],G0[S]) - dW(G,G0)| · 10 k-1/4
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

26. 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
dW(G,G[S]) · 10 (log2k)-1/2
with probability at least 1 – exp(-k2/2 log2k).

27. 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

28. “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

29. 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

30. The Limit Object ) 9 limit object G1
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 dW.
) 9 limit object G1
Question: Is there a useful representation of this limit object?
Answer: Yes – the graphon.

31. 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)

32. Graphons as Limit Objects
For |V(F)| = k, define
[0,1]k
ij 2 E(F)
t(F,W) = s dx1… dxk P W(xi, xj)
Theorem (2005 Lovasz and B. Szegedy):
(Gn) is left convergent , 9 graphon W s.t. t(F, Gn) ! t(F,W).

33. Graphons and Metric Convergence
Definition (Frieze and Kannan): The cut norm of a graphon W is given by
kWkW = 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 - WGnkW ! 0.

34. 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.

35. Summary of Part I , f is continuous in the cut metric
(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

36. 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

37. 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

38. Homomorphisms into (Small) Weighted Graphs
Recall that in Part I, we considered
Hom(F,G) = {f: V(F) ! V(G) s.t. f(E(F)) ½ E(G)}
with F small and simple, and G large
Now instead consider Hom(G,H) with G large and
H small and weighted, with vertex weights ai = ai(H) and edge weights bij = bij(H), so that
f:V(G)!V(H) x2V(G) xy2E(G)
|Hom(G,H)| = S P af(x) P bf(x) f(y)
This is a weighted count of the number of colorings

39. Example: Ising Magnet V(H) = {-1,+1} af = ehf , bff’ = eJff’
|Hom(G,H)| = S e-Energy(f)
f:V(G)! {-1,+1}
with
Energy(f) = - J S fxfy - h S fx
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 ].

40. 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

41. 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 ai = ai(H) > 0 and
edge weights bij = bij(H) > 0.

42. Ground State Energy J = ( ) Let bij(H) = eJij
Then the ground state energy of model H on graph G is
E(G,J) = |V(G)|-2 min S Jf(x)f(y)
f:V(G)!V(H) xy2E(G)
Example: Maxcut Density 1 e
H =
E(G,J) = |V(G)|-2 Maxcut (G)
J = ( ) 1

43. Naïve Right Convergence and Ground State Energies
Lemma: If bij(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 V-2 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.

44. 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

45. Incomplete Equivalences
Left Convergence
Part I
Part I
Metric Convergence
f(Gn) is Cauchy 8 testable f
E(G,J) is continuous in cut metric
Cut densities are testable parameters OR
E(G,J) is Cauchy 8 J
Quotients Cauchy in Hausdorff Metric
Part II
(previous Lemma)
Naïve Right Convergence

46. 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

47. Microcanonical Ensemble
Given q color classes with fraction ai in color class i: a = (a1, … , aq) with ai ≥ 0 and Sai = 1
Define the microcanonical homomorphism number:
f:V(G)!V(H) x2V(G) xy2E(G)
||f-1(i)|-ai|V(G)||≤1
|Homa(G,H)| = S P af(x) P bf(x) f(y)
and microcanonical ground state energy:
Ea (G,J) = |V(G)|-2 min S Jf(x)f(y)
f:V(G)!V(H) xy2E(G)
||f-1(i)|-ai|V(G)||≤1

48. Examples
Max/Min Bisection
J = ( ) ±1
Densest Subgraph
J = ( ) 1

49. 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 ai = ai(H) > 0 and
edge weights bij = bij(H) > 0.
This is equivalent to convergence of all microcanonical ground state energies.

50. 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

51. Complete Equivalences = Summary
Left Convergence
Part I
Part I
Metric Convergence
f(Gn) is Cauchy 8 testable f
Ea (G,J) is continuous in cut metric
Lots of Work
Cut densities are testable parameters OR
Ea(G,J) is Cauchy 8 J
Quotients Cauchy in Hausdorff Metric
Part II
(previous Lemma)
Right Convergence

52. 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)

53. THE END

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