Workshop on random graphs 24—26 oct Доклад Кристиана Боргс

1. Convergence of Sparse Graphs
as a Problem at the Intersection of
Graph Theory, Statistical Physics and
Probability
Christian Borgs
joint work with
J.T. Chayes, D. Gamarnik, J. Kahn and L. Lovasz

2. Introduction


Given a sequence 𝐺 𝑛 of graphs with 𝑉 𝐺 𝑛 → ∞,
what is the “right” notion of convergence?
Answers:
 Extremal Combinatorics: We want subgraph
counts to converge  Left Convergence
 Computer Science: We want MaxCut, MinBisection,
… to converge  Convergence of Quotients
 Statistical Physics, Machine Learning: We want
free energies of graphical models to converge 
Right Convergence

3. Introduction (cont.)



[BCLSV ‘06 – ‘12] Introduced these notions for dense
graphs, and proved they are equivalent
Lots of follow-up work, including the definition of a
limit object [LS ‘06]
This talk: For sequences with bounded degrees
(sparse graphs), we
 show that these notions are not equivalent
 introduce a new notion (Large Deviation
convergence) which implies all others

4. 1) Homorphism Numbers and Left
Convergence (for Combinatorialists)

For two simple graphs 𝐹, 𝐺 , a map 𝜙: 𝑉(𝐹) → 𝑉(𝐺) is
called a homomorphism iff 𝜙 𝐸(𝐹) ⊂ 𝐸 𝐺
Def: A dense sequence of simple graphs 𝐺 𝑛 is left
convergent if the probability that a random map
𝜙: 𝑉 𝐹 → 𝑉 𝐺 is a homomorphism converges for all
simple graphs 𝐹

Remark: Left convergence is equivalent to convergence
of the normalized subgraph counts 𝑉 𝐺 𝑛 −|𝑉 𝐹 | 𝑁 𝐹, 𝐺 𝑛 ,
where 𝑁 𝐹, 𝐺 𝑛 is the # of subgraphs 𝐹’ ⊂ 𝐺 𝑛 isomorphic
to 𝐹.

5. 2) Convergence of Quotients (for
Computer Scientists)


Fix a coloring 𝜙: 𝑉 𝐺 → {1, … , 𝑞} of 𝑉 𝐺 and let 𝑉𝑖 be
the set of vertices of color 𝑖
The quotient 𝐺 ∕ 𝜙 is a weighted graph on {1, … , 𝑞}
with vertex weights 𝛼 𝑖 = 𝑉𝑖
𝑉 𝐺 and edge weights
𝛽 𝑖𝑗 =

1
#
𝑉 𝐺 2
𝑢, 𝑣 ∈ 𝑉𝑖 × 𝑉𝑗 , 𝑢𝑣 ∈ 𝐸(𝐺)
The set of all 𝐺 ∕ 𝜙 for a fixed 𝑞 is called the the set
of 𝑞-quotients of 𝐺, and denoted by 𝑆 𝑞 𝐺
We say that the set of quotients of a sequence 𝐺 𝑛 is
convergent if for all 𝑞, the sets 𝑆 𝑞 (𝐺 𝑛 ) are convergent
2
in the Hausdorff metric on subsets of ℝ 𝑞+𝑞 .

6. 2) Convergence of Quotients (cont.)

Ex. 1: MaxCut
1
MaxCut
𝑉 𝐺 2

(𝐺) = max 𝛽12 𝐻 ∶ 𝐻 ∈ 𝑆2 𝐺
Ex. 2: MinBisection
1
2 MinBis
𝑉 𝐺
= min
(𝐺) =
𝛽12 𝐻 ∶ 𝐻 ∈ 𝑆2 𝐺 , 𝛼1 𝐻 =
1
2

7. 3) Right Convergence for Dense
Graphs (for Physicists)

Soft-core graph: a weighted graph 𝐻 with edge weights
𝛽 𝑖𝑗 = 𝛽 𝑖𝑗 𝐻 > 0

Given a soft-core graph 𝐻 on 𝑞 nodes, define the
microcanonical homomorphism numbers
hom′ 𝐺, 𝐻 =
𝛽𝜙
𝜙:𝑉 𝐺 →𝑉 𝐻
𝜙−1 𝑖 −𝑞 −1 |𝑉 𝐺 | ≤1
𝑥 𝜙 𝑦
(𝐻)
𝑥𝑦∈𝐸 𝐺
Def: A dense sequence 𝐺 𝑛 is called right convergent, if
V Gn −2 log hom′(𝐺 𝑛, 𝐻) converges for all soft-core
graphs 𝐻

8. 4) Main Theorem for Dense Graphs

Thm [BCLSV]: Let 𝐺 𝑛 be a dense sequence of
graphs with 𝑉(𝐺 𝑛)| → ∞. Then
𝐺 𝑛 is right convergent
⇔ the quotients of 𝐺 𝑛 are convergent
⇔ 𝐺 𝑛 is left convergent

Proof uses three main ingredients: the cutmetric, sampling, and Szemeredi’s Lemma, and
establishes that convergence in the cut-metric is
also equivalent to the other three notions

9. 5) Left Convergence for Sparse
Graphs


From now on, we consider sparse graphs, i.e., sequences
𝐺 𝑛 with bounded degrees
Given two simple graphs 𝐹, 𝐺, we denote the number of
homomorphisms from 𝐹 to 𝐺 by hom(𝐹, 𝐺)
Def: A sparse sequence 𝐺 𝑛 is called left convergent if
𝑉 𝐺𝑛
−1
hom(𝐹, 𝐺 𝑛)
converges for all connected, simple graphs 𝐹

Remark: Using that hom 𝐹, 𝐺 = 𝐹′ surj 𝐹, 𝐹 ′ 𝑁(𝐹 ′ , 𝐺),
it is easy to see that left convergence is equivalent to the
convergence of the subgraph counts 𝑉 𝐺 𝑛 −1 𝑁(𝐹, 𝐺 𝑛)

10. 5) Left Convergence for Sparse
Graphs (cont.)



Def: A sequence 𝐺 𝑛 is called Benjamini-Schramm
convergent (BS-convergent) if for all 𝑅 < ∞, the
distribution of the 𝑅-neighborhood around a randomly
chosen vertex 𝑥 ∈ 𝑉(𝐺 𝑛 ) is convergent
Lemma: Left convergence is equivalent to BenjaminiSchramm convergence
Rem: The limit of a left convergent sequence 𝐺 𝑛 can
therefore be expressed as a random, rooted graph
(𝑥, 𝐺)

11. 5) Left Convergence for Sparse
Graphs (cont.)




Ex1: The sequences {1,2, … , 𝑛} 𝑑 and (ℤ/𝑛ℤ) 𝑑 converge
to the rooted graph (0, ℤ 𝑑)
Ex2: Let 𝐺 𝑛,𝑑 be the 𝑑-regular random graph and
𝐵 𝑛,𝑑 be the 𝑑-regular bipartite random graph. Both
are left convergent, and converge to the infinite 𝑑regular tree
Rem1: For sparse graphs, left convergence is a very
local notion
Rem2: Ex2 raises the question whether the topology
defined by left convergence is too coarse

12. 6) Convergence of Quotients for
Sparse Graphs
 Let 𝜙: 𝑉 𝐺 → 1, … , 𝑞 and 𝑉𝑖 be as in the dense
setting
 Define the quotient graph 𝐺 ∕ 𝜙 as the graph with
weights 𝛼 𝑖 = 𝑉𝑖 𝑉 𝐺 and
1
𝛽 𝑖𝑗 =
𝑉 𝐺
#
𝑢, 𝑣 ∈ 𝑉𝑖 × 𝑉𝑗 ,
𝑢𝑣 ∈ 𝐸(𝐺)
and denote the set of all these quotients by 𝑆 𝑞 (𝐺)
We say the quotients of 𝐺 𝑛 are convergent if 𝑆 𝑞 (𝐺)
converges in the Hausdorff metric for all 𝑞

13. 6) Convergence of Quotients for
Sparse Graphs (cont.)
 Q: Does left convergence imply convergence of
quotients?
 Ex: Take 𝐺 𝑛 to be 𝐺 𝑛,𝑑 for odd 𝑛 and 𝐵 𝑛,𝑑 for even
𝑛. For 𝑑 large, we have that
MaxCut 𝐵 𝑛,𝑑 =
MaxCut 𝐺 𝑛,𝑑
≈
𝑑𝑛
2
𝑑𝑛
4
 As a consequence, the 2-quotients of 𝐺 𝑛 are not
convergent. Thus left convergence does NOT imply
convergence of quotients.

14. 6) Convergence of Quotients for
Sparse Graphs (cont.)
 Q: Does convergence of quotients imply left
convergence?
𝑛
4
 Ex: Take 𝐺 𝑛 to be a union of ⌈ ⌉ 4-cycles for odd 𝑛
and a union of
𝑛
⌈ ⌉
6
MaxCut 𝐺 𝑛 =
6-cycles for even 𝑛. Then
1
2
|𝑉 𝐺 𝑛 |
 More general, it is not hard to show that the 𝑞quotients of 𝐺 𝑛 are convergent. But 𝐺 𝑛 is clearly
not left convergent, so convergence of quotients
does not imply left convergence either.

15. 7) Right Convergence for Sparse
Graphs


Soft-core graph: a weighted graph 𝐻 with edge and vertex
weights 𝛽 𝑖𝑗 𝐻 > 0 and 𝛼 𝑖 𝐻 > 0
Given a simple graph 𝐺 and a soft-core graph 𝐻 , define
hom 𝐺, 𝐻 =
𝛼𝜙
𝜙:𝑉 𝐺 →𝑉 𝐻 𝑥∈𝑉 𝐺
𝑥
(𝐻)
𝛽𝜙
𝑥 𝜙 𝑦
(𝐻)
𝑥𝑦∈𝐸 𝐺
Def: A sparse sequence 𝐺 𝑛 is called right convergent if
ℱ 𝐻 =
1
lim
𝑛→∞ 𝑉 𝐺 𝑛
𝑙𝑜𝑔 hom(𝐺 𝑛, 𝐻)
exists for all soft-core graphs 𝐻.

16. 7) Right Convergence for Sparse
Graphs (cont.)



Lemma: 1,2, … , 𝑛 𝑑 and ℤ 𝑛ℤ 𝑑 are right convergent
Q: Does left convergence imply right convergence?
Ex: Take 𝐺 𝑛 to be 𝐺 𝑛,𝑑 for odd 𝑛 and 𝐵 𝑛,𝑑 for even 𝑛, and
let 𝐻 be the soft-core graph with edge weights
𝛽11 = 𝛽22 = 1


and
𝛽12 = 𝑒.
Then
𝑒 MaxCut(𝐺 𝑛 ) ≤ hom(𝐺 𝑛 , 𝐻) ≤ 2 𝑛 𝑒 MaxCut(𝐺 𝑛 )
We may therefore use our previous results on MaxCut(𝐺 𝑛 )
to show that 𝐺 𝑛 is not right convergent on 𝐻

17. 7) Right Convergence for Sparse
Graphs (cont.)


Q: Does right convergence imply convergence of
quotients?
Ex: Assume 𝐹 𝑛 has MinBisec 𝐹 𝑛 ≥ 𝛿𝑛 and assume (by
compactness) that 𝐹 𝑛 is right convergent. Choose 𝐺 𝑛 = 𝐹 𝑛
if 𝑛 is odd, and 𝐺 𝑛 = 𝐹 𝑛/2 ∪ 𝐹 𝑛/2 if 𝑛 is even. Then
hom 𝐺 𝑛 , 𝐻 = hom 𝐹 𝑛/2 , 𝐻
2
& MinBisec 𝐺 𝑛 = 0
implying that 𝐺 𝑛 is right convergent but that its quotients
are not convergent
Main Thm [BCKL’12] For sequences of bounded maximal
degree, right convergence implies left convergence

18. Proof Idea of Main Theorem
Given a simple graph F and a soft-core graph H define
𝑢 𝐹, 𝐻 =
𝐹 ′ ⊂𝐹
−1
|𝐹F′ |
log hom(𝐹’, 𝐻)
and use inclusion exclusion to conclude that
log hom(𝐺, 𝐻) = 𝐹⊂𝐺 𝑢(𝐹, 𝐻)
By the factorization of hom(𝐺, 𝐻) over connected components,
we get 𝑢(𝐹, 𝐻) = 0 unless 𝐹 is connected. Thus
log hom(𝐺, 𝐻) = 𝐹⊂𝐺 𝑢(𝐹, 𝐻) = 𝐹 𝑁(𝐹, 𝐺)𝑢(𝐹, 𝐻)
where the second sum runs over all (isomorphism classes) of
connected graphs 𝐹.
“As a consequence”
1
𝑁 𝐹, 𝐺 𝑛
lim
log hom 𝐺 𝑛 , 𝐻 = 𝐹 𝑢 𝐹, 𝐻 lim
𝑛→∞ |𝑉 𝐺 𝑛 |
𝑛→∞ |𝑉 𝐺 |
𝑛
Inverting this relation proves that right convergence implies
left convergence

19. Summary so Far
(local)
L-Convergence
++
x
(local & global)
R-Convergence
++
Convergence of Quotients
(global)

20. 8) Large Deviation Convergence
 Convergence of Quotients: convergence of the sets
𝑆 𝑞 𝐺 𝑛 = {𝐺 𝑛 𝜙 ∣ 𝜙: 𝑉 𝐺 → 1, … , 𝑞 } ⊂ 0, 𝐷
𝑞 2 +𝑞
 Large Deviation Convergence [BCG ‘12]: choose
𝜙: 𝑉 𝐺 → 1, … , 𝑞 uniformly at random, and study the
𝑞2 +𝑞
random variable 𝐹𝑞 𝐺 = 𝐺 𝜙 ∈ 0, 𝐷
Def: 𝐺 𝑛 is large deviation (LD) convergent ⇔ for all 𝑞,
𝐹𝑞 𝐺 𝑛 obeys a LD-Principle with suitable rate function 𝐼 𝑞
 Informally:
Pr 𝐹𝑞 𝐺 𝑛 = 𝐹 ≈ 𝑒 −𝐼 𝑞(𝐹)|𝑉
𝐺𝑛 |

21. 8) Large Deviation Convergence
(cont.)
 Def: 𝐹𝑞 𝐺 𝑛 obeys a LD-Principle ⇔ ∃ rate function 𝐼 𝑞
s.th.
Pr 𝐹𝑞 𝐺 𝑛 ∈ 𝐴 ≈ sup 𝑒 −𝐼 𝑞(𝐹)|𝑉
𝐹∈𝐴
𝐺𝑛 |

22. 8) Large Deviation Convergence
(cont.)
 Def: 𝐹𝑞 𝐺 𝑛 obeys a LD-Principle ⇔ ∃ rate function 𝐼 𝑞
s.th.
log Pr 𝐹𝑞 𝐺 𝑛 ∈ 𝐴
− inf 𝐼 𝑞 (𝐹) = lim
𝐹∈𝐴
|𝑉 𝐺 𝑛 |

23. 8) Large Deviation Convergence
(cont.)
 Def: 𝐹𝑞 𝐺 𝑛 obeys a LD-Principle ⇔ ∃ rate function 𝐼 𝑞
s.th.
log Pr 𝐹𝑞 𝐺 𝑛 ∈ 𝐴
≤ lim
𝑉 𝐺𝑛
− inf0 𝐼 𝑞 𝐹
𝐹∈𝐴
log Pr 𝐹𝑞 𝐺 𝑛 ∈ 𝐴
≤ lim
≤ − inf 𝐼 𝑞 𝐹
𝑉 𝐺𝑛
𝐹∈𝐴
 Lemma: 1,2, … , 𝑛
𝑑
and ℤ 𝑛ℤ
𝑑
are LD-convergent

24. 8) Large Deviation Convergence
(cont.)

Thm: If 𝐺 𝑛 is LD-convergent, then 𝐺 𝑛 is right convergent
In fact, if 𝐻 is a soft-core graph with 𝑉 𝐻
= 𝑞 , then
ℱ 𝐻 = sup {log 𝑊 𝐻 𝐹 + log 𝑞 − 𝐼 𝑞 𝐹 }
𝐹
where
𝑊𝐻 𝐹 =
𝛼𝑖 𝐻
𝑖
𝛼𝑖 𝐹
𝛽 𝑖𝑗 𝐻
𝛽 𝑖𝑗 (𝐹)
𝑖𝑗
 So in the limiting free energy ℱ 𝐻 , the sequence 𝐺 𝑛 only
appears via 𝐼 𝑞 , and the “target graph” 𝐻 only appears via
𝑊𝐻

25. Summary
LD
Conv.
Left
Conv.
x
Conv. of
Quotients
Right
Conv.