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Workshop on random graphs 24—26 oct Доклад Кристиана Боргс
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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
Introduction
Given a sequence 𝐺𝑛 of graphs with 𝑉 𝐺𝑛 → ∞,
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
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
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 𝐺𝑛 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 𝐺𝑛−|𝑉 𝐹 |𝑁 𝐹, 𝐺𝑛 ,
where 𝑁 𝐹, 𝐺𝑛 is the # of subgraphs 𝐹’ ⊂ 𝐺𝑛 isomorphic to 𝐹.
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 𝛼𝑖 = 𝑉𝑖 𝑉 𝐺 and edge weights
𝛽𝑖𝑗 =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 𝑆𝑞 𝐺
We say that the set of quotients of a sequence 𝐺𝑛 is convergent if for all 𝑞, the sets 𝑆𝑞(𝐺𝑛) are convergent
in the Hausdorff metric on subsets of ℝ𝑞+𝑞2.
2) Convergence of Quotients (cont.)
Ex. 1: MaxCut
1
𝑉 𝐺 2MaxCut (𝐺) = max 𝛽12 𝐻 ∶ 𝐻 ∈ 𝑆2 𝐺
Ex. 2: MinBisection
1
𝑉 𝐺 2MinBis (𝐺) =
= min 𝛽12 𝐻 ∶ 𝐻 ∈ 𝑆2 𝐺 , 𝛼1 𝐻 =1
2
3) Right Convergence for Dense Graphs (for Physicists)
Soft-core graph: a weighted graph 𝐻 with edge weights
𝛽𝑖𝑗 = 𝛽𝑖𝑗 𝐻 > 0
Given a soft-core graph 𝐻 on 𝑞 nodes, define the
microcanonical homomorphism numbers
hom′ 𝐺,𝐻 =
𝜙:𝑉 𝐺 →𝑉 𝐻
𝜙−1 𝑖 −𝑞−1|𝑉 𝐺 | ≤1
𝛽𝜙 𝑥 𝜙 𝑦 (𝐻)
𝑥𝑦∈𝐸 𝐺
Def: A dense sequence 𝐺𝑛 is called right convergent, if
V Gn−2log hom′(𝐺𝑛, 𝐻) converges for all soft-core
graphs 𝐻
4) Main Theorem for Dense Graphs
Thm [BCLSV]: Let 𝐺𝑛 be a dense sequence of graphs with 𝑉(𝐺𝑛)| → ∞. Then
𝐺𝑛 is right convergent ⇔ the quotients of 𝐺𝑛 are convergent ⇔ 𝐺𝑛 is left convergent
Proof uses three main ingredients: the cut-metric, sampling, and Szemeredi’s Lemma, and establishes that convergence in the cut-metric is also equivalent to the other three notions
5) Left Convergence for Sparse Graphs
From now on, we consider sparse graphs, i.e., sequences
𝐺𝑛 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 𝐺𝑛 is called left convergent if
𝑉 𝐺𝑛−1 hom (𝐹, 𝐺𝑛)
converges for all connected, simple graphs 𝐹
Remark: Using that hom 𝐹, 𝐺 = surj 𝐹, 𝐹′ 𝑁(𝐹′, 𝐺)𝐹′ ,
it is easy to see that left convergence is equivalent to the
convergence of the subgraph counts 𝑉 𝐺𝑛−1 𝑁 (𝐹, 𝐺𝑛)
5) Left Convergence for Sparse Graphs (cont.)
Def: A sequence 𝐺𝑛 is called Benjamini-Schramm convergent (BS-convergent) if for all 𝑅 < ∞, the distribution of the 𝑅-neighborhood around a randomly chosen vertex 𝑥 ∈ 𝑉(𝐺𝑛) is convergent
Lemma: Left convergence is equivalent to Benjamini-Schramm convergence
Rem: The limit of a left convergent sequence 𝐺𝑛 can therefore be expressed as a random, rooted graph (𝑥, 𝐺)
5) Left Convergence for Sparse Graphs (cont.)
Ex1: The sequences {1,2, … , 𝑛}𝑑 and (ℤ/𝑛ℤ)𝑑 converge
to the rooted graph (0, ℤ𝑑)
Ex2: Let 𝐺𝑛,𝑑 be the 𝑑-regular random graph and
𝐵𝑛,𝑑 be the 𝑑-regular bipartite random graph. Both
are left convergent, and converge to the infinite 𝑑-regular 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
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 𝛼𝑖 = 𝑉𝑖 𝑉 𝐺 and
𝛽𝑖𝑗 =1
𝑉 𝐺 # 𝑢, 𝑣 ∈ 𝑉𝑖 × 𝑉𝑗 , 𝑢𝑣 ∈ 𝐸(𝐺)
and denote the set of all these quotients by 𝑆𝑞(𝐺)
We say the quotients of 𝐺𝑛 are convergent if 𝑆𝑞(𝐺)
converges in the Hausdorff metric for all 𝑞
6) Convergence of Quotients for Sparse Graphs (cont.)
Q: Does left convergence imply convergence of quotients?
Ex: Take 𝐺𝑛 to be 𝐺𝑛,𝑑 for odd 𝑛 and 𝐵𝑛,𝑑 for even
𝑛. For 𝑑 large, we have that
MaxCut 𝐵𝑛,𝑑 =𝑑𝑛
2
MaxCut 𝐺𝑛,𝑑 ≈𝑑𝑛
4
As a consequence, the 2-quotients of 𝐺𝑛 are not convergent. Thus left convergence does NOT imply convergence of quotients.
6) Convergence of Quotients for Sparse Graphs (cont.)
Q: Does convergence of quotients imply left convergence?
Ex: Take 𝐺𝑛 to be a union of ⌈𝑛
4⌉ 4-cycles for odd 𝑛
and a union of ⌈𝑛
6⌉ 6-cycles for even 𝑛. Then
MaxCut 𝐺𝑛 =1
2 |𝑉 𝐺𝑛 |
More general, it is not hard to show that the 𝑞-quotients of 𝐺𝑛 are convergent. But 𝐺𝑛 is clearly
not left convergent, so convergence of quotients does not imply left convergence either.
7) Right Convergence for Sparse Graphs
Soft-core graph: a weighted graph 𝐻 with edge and vertex
weights 𝛽𝑖𝑗 𝐻 > 0 and 𝛼𝑖 𝐻 > 0
Given a simple graph 𝐺 and a soft-core graph 𝐻, define
hom 𝐺,𝐻 = 𝛼𝜙 𝑥 (𝐻)
𝑥∈𝑉 𝐺𝜙:𝑉 𝐺 →𝑉 𝐻
𝛽𝜙 𝑥 𝜙 𝑦 (𝐻)
𝑥𝑦∈𝐸 𝐺
Def: A sparse sequence 𝐺𝑛 is called right convergent if
ℱ 𝐻 = lim𝑛→∞
1
𝑉 𝐺𝑛 𝑙𝑜𝑔 hom (𝐺𝑛, 𝐻)
exists for all soft-core graphs 𝐻.
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 𝐺𝑛 to be 𝐺𝑛,𝑑 for odd 𝑛 and 𝐵𝑛,𝑑 for even 𝑛, and
let 𝐻 be the soft-core graph with edge weights
𝛽11 = 𝛽22 = 1 and 𝛽12 = 𝑒.
Then
𝑒MaxCut(𝐺𝑛) ≤ hom (𝐺𝑛, 𝐻) ≤ 2𝑛𝑒MaxCut(𝐺𝑛)
We may therefore use our previous results on MaxCut(𝐺𝑛)
to show that 𝐺𝑛 is not right convergent on 𝐻
7) Right Convergence for Sparse Graphs (cont.)
Q: Does right convergence imply convergence of quotients?
Ex: Assume 𝐹𝑛 has MinBisec 𝐹𝑛 ≥ 𝛿𝑛 and assume (by compactness) that 𝐹𝑛 is right convergent. Choose 𝐺𝑛 = 𝐹𝑛 if 𝑛 is odd, and 𝐺𝑛 = 𝐹𝑛/2 ∪ 𝐹𝑛/2 if 𝑛 is even. Then
hom 𝐺𝑛, 𝐻 = hom 𝐹𝑛/2, 𝐻2 & MinBisec 𝐺𝑛 = 0
implying that 𝐺𝑛 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
Proof Idea of Main Theorem
Given a simple graph F and a soft-core graph H define
𝑢 𝐹,𝐻 = 𝐹′⊂𝐹 −1|𝐹\F′| 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”
lim𝑛→∞
1
|𝑉 𝐺𝑛 |log hom 𝐺𝑛, 𝐻 = 𝐹𝑢 𝐹,𝐻 lim𝑛→∞
𝑁 𝐹, 𝐺𝑛|𝑉 𝐺𝑛 |
Inverting this relation proves that right convergence implies left convergence
Summary so Far
(local) (local & global)
L-Convergence R-Convergence
Convergence of Quotients
(global)
x
+ + + +
8) Large Deviation Convergence
Convergence of Quotients: convergence of the sets
𝑆𝑞 𝐺𝑛 = {𝐺𝑛 𝜙 ∣ 𝜙: 𝑉 𝐺 → 1,… , 𝑞 } ⊂ 0, 𝐷𝑞2+𝑞
Large Deviation Convergence [BCG ‘12]: choose
𝜙: 𝑉 𝐺 → 1,… , 𝑞 uniformly at random, and study the
random variable 𝐹𝑞 𝐺 = 𝐺 𝜙 ∈ 0, 𝐷𝑞2+𝑞
Def: 𝐺𝑛 is large deviation (LD) convergent ⇔ for all 𝑞, 𝐹𝑞 𝐺𝑛 obeys a LD-Principle with suitable rate function 𝐼𝑞
Informally:
Pr 𝐹𝑞 𝐺𝑛 = 𝐹 ≈ 𝑒−𝐼𝑞(𝐹)|𝑉 𝐺𝑛 |
8) Large Deviation Convergence (cont.)
Def: 𝐹𝑞 𝐺𝑛 obeys a LD-Principle ⇔ ∃ rate function 𝐼𝑞 s.th.
Pr 𝐹𝑞 𝐺𝑛 ∈ 𝐴 ≈ sup𝐹∈𝐴 𝑒−𝐼𝑞(𝐹)|𝑉 𝐺𝑛 |
8) Large Deviation Convergence (cont.)
Def: 𝐹𝑞 𝐺𝑛 obeys a LD-Principle ⇔ ∃ rate function 𝐼𝑞 s.th.
− inf𝐹∈𝐴 𝐼𝑞(𝐹) = lim
log Pr 𝐹𝑞 𝐺𝑛 ∈ 𝐴
|𝑉 𝐺𝑛 |
8) Large Deviation Convergence (cont.)
Def: 𝐹𝑞 𝐺𝑛 obeys a LD-Principle ⇔ ∃ rate function 𝐼𝑞 s.th.
− inf𝐹∈𝐴0𝐼𝑞 𝐹 ≤ lim
log Pr 𝐹𝑞 𝐺𝑛 ∈ 𝐴
𝑉 𝐺𝑛
≤ lim log Pr 𝐹𝑞 𝐺𝑛 ∈ 𝐴
𝑉 𝐺𝑛≤ − inf𝐹∈𝐴𝐼𝑞 𝐹
Lemma: 1,2, … , 𝑛 𝑑 and ℤ 𝑛ℤ 𝑑 are LD-convergent
8) Large Deviation Convergence (cont.)
Thm: If 𝐺𝑛 is LD-convergent, then 𝐺𝑛 is right convergent
In fact, if 𝐻 is a soft-core graph with 𝑉 𝐻 = 𝑞, then
ℱ 𝐻 = sup𝐹 {log𝑊𝐻 𝐹 + log 𝑞 − 𝐼𝑞 𝐹 }
where
𝑊𝐻 𝐹 = 𝛼𝑖 𝐻𝛼𝑖 𝐹
𝑖
𝛽𝑖𝑗 𝐻𝛽𝑖𝑗(𝐹)
𝑖𝑗
So in the limiting free energy ℱ 𝐻 , the sequence 𝐺𝑛 only appears via 𝐼𝑞, and the “target graph” 𝐻 only appears via
𝑊𝐻
Summary
x Left Conv.
Conv. of Quotients
Right Conv.
LD Conv.