Post on 26-Mar-2020
transcript
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