Partitioning a random graph that has a good partition

Model Panther U into Sot 151 41 7

Include s and t.tl edges with pot pit with prob 9 9 a p

key perturbationtheory for eigenvectors

Weyl If A has eiguals 2 2222 I da

B has eigrals B Z z Py

R A B then ki pile KRH

Recall HRH off.kz lRxH maxabs of eisual of R

khat about the eigenvectors

Adi didi 134 Piti O ang di Yi

Davis Kahan sin e

21112117Hi g I

will prove sin 0 e

2hr4minHi d j I

j i

proofi Assume wolog LEO Say by A it B d I

Let 8 hffin Hi att note i is fixed here

Compule HAYill two ways

HAY ll KETRIYHe11134ll 11124llFri TillaHRH e 211124

Setting Cj OGIYi Yi EjCjOj

KAUilf F.co df EieidIejEiei2E8jZiCf d2Cl ciY d sin 0

so Isin O e 2111211 sin e21112

Why dependence on difference between eignals

Consider 9 ando Ie with eigneafo and g

model A III iii If meant o Iii tart

s T independerty

Ula a p Acacal

So add pI to adjacency matrix Does not cleanseeigenvectors

M A t R where RCacD chosen from l p p sore sup

tf 7 diff gap

By slight extension of lastlecture

HRH 30pct with high probability if pecknd

some constant c

Will recover Sit from 42 if P f bgenoush

de de z 23 an _o be eigrals of A

Mc z z µ equals of µ

do E Pta A 1 Elect 1 Oh I

E III Is It Ada Elp Holz

2 gives thegroups if a dose to P

Az Lz Elp 94 e de LL 2qZ

let 42 be second eignec of M

Relate ang dared to misclassified

where classify a by sign Udal

really dial Eu aes so 11041 1fu aet

So when misclassify a decal 421911 e't

if misclassify K 110244 2 TEN

As Koh Yue E Sin Q get

Sino e TE

By Daud Kahan Sino e 2111211P fl

4FEN e ncf a Im

ncp ft

Te e 52 iz IIFp f

k e 288 PIM912

If p and q are constants and a grows

probably only misclassify a constant

If p t f z

ke 288 t ITSo misclassify at most a constant fraction