in the binomial random graph Tel Aviv University Wojciech Samotij - - PDF document

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in the binomial random graph
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in the binomial random graph Tel Aviv University Wojciech Samotij - - PDF document

Feb 28, 2024 349 likes •507 views

Deviations of triangle counts in the binomial random graph Tel Aviv University Wojciech Samotij joint work with Matan Harel Frank Mousset SC MS online seminar Our setting Gmp the binomial random graph with lo n's and edge prob p Vx set X EEN

slide-1
SLIDE 1

Deviations of triangle counts

in the binomial random graph

Wojciech Samotij

Tel Aviv University jointwork with MatanHarel Frank Mousset

SCMS online seminar

slide-2
SLIDE 2

Our setting

Gmp thebinomial randomgraphwith

Vx set

lo n's and edgeprob p

X

triangles in Gmp

EEN

131ps

Problem Forevery8 0 determinetheasymptotic of

logPIX Htd EIN

logarithmicuppertailprobability

slide-3
SLIDE 3

Theorem Chatterjee DeMarco Kahn 20121

If p

loginIn then for every d O

logPIXHtoHEINI Odn'p tog

Holt Question What is the implicitconstant

Definition For d O

yo

minded EIXI G Gnp aHOHEN

EoEXT

slide-4
SLIDE 4

Proposition If 4181

00

then

PIX24 81EEN

z p't

t t401

Proof Pol I Pf IGE

Gmp

Pick a small E O and a

Ge kn with YldtE edges

st EoENzHotE ELI

Notethat

lil PlutoIs PIGEGmp PLUTO

pH IR

UTd.CiilEoIXIePolUTo4

lltdHEMtlPdUTd lb

Finally Iiit

Potato

EEEN15

Ep potto

O

slide-5
SLIDE 5

Theorem l

If n'Epal

then

PIX HOMELY ept

  • Wo

Chatterjee Dembo 12014

9 42

Eldan120161

a _if

Cook Dembo 12018

2 3

Augen12018

a

L

slide-6
SLIDE 6

Theorem Hard Mousset S 2019

tf

n login

pal

then for everyd 0

PIX HSIEH

pl

401

Theorem LubetzkyZhao 20141 For every8 0

41811n'p

842

if n park

min 8442,83

if

n'keep 1

To

p

g

3np

clique

Mfdp

ggBTnp.is

hub

slide-7
SLIDE 7

Proof Fix a small E O and a large C

Definition A graph GE kn

is

a seed if

he

G

E Crip log

Hp

41

d

01n'p4

a EoIN

Hd E EEN

Lemma t Pl UT

E HoH IP Gmp contains a seed

Unionbound over all seeds Too naive

Go

a seed with m NotE edges

e G

2M

Gou G is a seed

Eli seeds in Gmp

Ibm psms nI

Rfp

2man

slide-8
SLIDE 8

Definition A graph

is

a

core if

41 e E

s Criplog

14 p

a EoEX

Hd 2E IE

EX

Have

IEo EXT

IE until off gap

Fact Everyseed contains a

core Proof Iterativelyremove from a seed all edges failing 3 Lemma2 For every M cores with m edges

e p

Em

slide-9
SLIDE 9

Assuming both lemmas

Ll

P UT

EP Gmp contains a seed

F

IP Gmp contains a

core

EL cores in Gnp

Em EI cores with m edges in Gmp

L2

8 EmsWoee p

Em

p

m

e pl e Hd2E

4182E E edgesin a core

slide-10
SLIDE 10

Proofof Lemma1 based on Janson OleszkiewiczRucinski

Let 2

D Gmp does notcontain a seed

As ZELOlb

P1Xs HotEIN

E PIZ O

PIX2 Hor EEN

Enough to show

4 pm

EP Xs HOHEN

Markov's inequality

He I

1EUR14

HOY EEN

Claim 31 Criplog

Hp

ELIX2 YeHo E EEN

slide-11
SLIDE 11

Claim 31

Crip'logHp

1EUR19ECHOEYELID

Proof

EUR14 PIT

u

uTe

a Guys a 2

1

E

2

PIT u ute

a Gmp

Ti

Te

Tu

uTe notaseed

eltu.vee3eeCrip4ogl4pl

Et uaTX

Hd E EN

E 2 PIT u ute.ieGmp

PITeeGn.plTu uTeieGn.p

T

Te

Tu

uTe nota

seed

e

Htc E EEN

slide-12
SLIDE 12

Foofof Lemma 2

Suppose G

core graphwith medges

aura

aw

Earth Eo w X

Ours EEEX

Clip'loglyp I

840

log

4p

Owe nlp f

duidvllp p4

du.ir t p

U

V

UN Yu

u

Knt

either duidrzg.nl ogHplordu.vZjnp logHpl

slide-13
SLIDE 13

Cor Have

dutdrzjnllogllplordu.vs.jp logHpl

Define A

well

dw28h1

1210g'll

pH

B

we VIE

dw 2 jnpllogHp

Obs 2M 2461 Eno dw Z Max At271g

1131 Engin's

IAte amaxi

4M98

IBI e b

ma

2m10

8 1

y n p

every

edge

has2endptsin B

A B

  • r Zl endpt inA

cores w medges E lanna

Imax

Ibm imam le

e p

D

slide-14
SLIDE 14

Problem Calculate

loglplx.zdtdHEL

xt.tl

where Xp

copies of H in Gmp

State of the art

H Kr

Haret Morisset S

H connected

regular

HMS

Basak Basu

Hi irregular

Cook Dembo