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.