Maximizing  the  spectral  gap  of  networks  produced  by  

node  removal

Naoki  Masuda  (University  of  Tokyo,  Japan)

Refs:  1.  Watanabe  &  Masuda,  Physical  Review  E,  82,  046102  (2010)2.  Masuda,  Fujie  &  Murota,  In:  Complex  Networks  IV,  Studies  in  ComputaUonal  Intelligence,  476,  155-­‐163  (2013)

Collaborators:Takamitsu  Watanabe  (University  of  Tokyo,  Japan)Tetsuya  Fujie  (University  of  Hyogo,  Japan)Kazuo  Murota  (University  of  Tokyo,  Japan)

Laplacian  of  a  network

x(t) = �Lx(t)

x1 =� 2x1 + x2 + x4

=(x2 � x1) + (x4 � x1)

1 2

3 4

L =

0

BB@

2 �1 0 �1�1 2 0 �10 0 1 �1�1 �1 �1 3

1

CCA

�1 = 0 < �2 �3 · · · �NEigenvalues:

Spectral  gap• If  λ2  is  large,  diffusive  dynamical  processes  on  networks  

occur  faster.  Ex:  synchronizaUon,  collecUve  opinion  formaUon,  random  walk.

• Note:  unnormalized  Laplacian  here

• Problem:  Maximize  λ2  by  removing  Ndel  out  of  N  nodes  by  two  methods.

• SequenUal  node  removal  +  perturbaUve  method  (Watanabe  &  Masuda,  2010)

• Semidefinite  programming  (Masuda,  Fujie  &  Murota,  2013)

• Note:  Removal  of  links  always  decreases  λ2  (Milanese,  Sun  &  Nishikawa  2010;  Nishikawa  &  Mober  2010).

PerturbaUve  method• Extends  the  same  method  for  adjacency  matrices  

(Restrepo,  Ob  &  Hunt,  2008)

• Much  faster  than  the  brute  force  method.

Lu =�2u

(L+�L)(u+�u) =(�2 +��2)(u+�u)

�u =�u� uiei

where ei ⌘ (0, . . . , 0, 1|{z}i

, 0, . . . , 0)

=) ��2 ⇡P

j2Niuj(ui � uj)

1� u2i

Select  i  that  maximizes  Δλ2  

Results:  model  networks(N  =  250,  <k>  =  10)

Goh

WS

HKBA

ER

f0 0.1 0.2 0.3 0.4 0.5

f0 0.1 0.2 0.3 0.4 0.5

f0 0.1 0.2 0.3 0.4 0.5

1

3

5

1

1.4

1.8

perturbative

betweenness-based

degree-based

optimal sequential

1

1.2

0.9

0.8

1.1

0.9

1

1.1

1.2

1

0.6

1.4

f0 0.1 0.2 0.3 0.4 0.5

f

Ȝ 2norm

ali

zed

0 0.1 0.2 0.3 0.4 0.5

Goh

Results:  real  networksperturbative

betweenness-baseddegree-basedoptimal sequential

e-mail

C. elegans

2

3

4

5

0.5

0

1

1.5

2

Ȝ 2Ȝ 2

E. coli

0

0.2

0.4

0.6

0.8

macaque

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.50

f f

0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5f f

N  =  279<k>  =  16.4

N  =  1133<k>  =  9.62

N  =  71<k>  =  12.3

N  =  2268<k>  =  4.96

Conclusions

• Careful  node  removal  can  increase  the  spectral  gap.

• For  a  variety  of  networks,  the  perturbaUve  strategy  works  well  with  a  reduced  computaUonal  cost.

• Ref:  Watanabe  &  Masuda,  Physical  Review  E,  82,  046102  (2010)

However,

• SequenUal  opUmal  may  not  be  opUmal  for  Ndel  ≥  2.

• An  obvious  combinatorial  problem  if  we  pursue  the  opUmal  soluUon.

min  t  subject  to

tI � F (x1, . . . , xn) ⌫ 0 (eigenvalues: t� �n · · · t� �1)

Semidefinite  programming

Eigenvalue  minimizaUon  using  SDP

nX

i=1

ciximin subject  to F0 +nX

i=1

xiFi ⌫ 0

F0, . . . , Fn :  symmetric  matrices

F (x1, . . . , xn) = F0 +nX

i=1

xiFi (eigenvalues: �1 · · · �n)

F0, . . . , Fn :  symmetric  matrices

DifficulUes  in  our  case• Discreteness:  xi  ∈  {0,  1}

• Ndel  (irrelevant)  0  eigenvalues  appear.

• Not  interested  in  the  zero  eigenvalue    λ1=0.

• So,  let’s  start  with  the  following  problem:

max  t  subject  to

λ1=0  →  λ1’=αNew  zero  eigenvalue  →  βBut,  a  nonlinear  constraint  

�tI +X

i<j;(i,j)2E

xixjLij + ↵J + �

NX

i=1

(1� xi)Ei ⌫ 0

NX

i=1

xi = N �Ndel, xi 2 {0, 1}

where Ei = diag(0, . . . , 0, 1|{z}i

, 0, . . . , 0)

L =X

1i<jN ;(i,j)2E

Lij

(Lovász,  1979;  Grötschel,  Lovasz  &  Schrijver,  1986;  Lovasz  &  Schrijver,  1991)

• Xij,  where  (i,j)  is  not  a  link,  is  a  “free”  variable.

• We  can  reduce  the  number  of  variables  using  Xii  =  xi.  But  sUll  O(N2)  terms  exist,  and  the  algorithm  runs  slowly.

• For  a  technical  reason,  we  set  α  =  β/N

• Challenges

• Discreteness  of  xi  →    “relax”  the  problem

• Nonlinear  constraint  →    introduce  new  vars

Xij ⌘ xixj

� tI +X

i<j;(i,j)2E

XijLij + ↵J + �

NX

i=1

(1� xi)Ei ⌫ 0

NX

i=1

xi = N �Ndel

Y ⌘1 x

>

x X

�⌫ 0

0 xi(= Xii) 1(1 i N)

SDP1

←  actually  not  needed

�tI +X

i<j;(i,j)2E

xixjLij + ↵J + �

NX

i=1

(1� xi)Ei ⌫ 0

max  t  subject  to

An  improved  method  SDP2:  “local  relaxaUon”

�tI +X

i<j;(i,j)2E

XijLij + ↵J + �

NX

i=1

(1� xi)Ei ⌫ 0

x1x2 �0

x1(1� x2) �0

(1� x1)x2 �0

(1� x1)(1� x2) �0

X12 �0

x1 �X12 �0

x2 �X12 �0

1� x1 � x2 +X12 �0

IntuiUve  comparison• Consider  N=1  (unrealisUc  though).

• SDP1

• Note:  In  fact,  X11  =  x1.

• SDP2

• Linear!

1 x

>

x X

�=

1 x1

x1 X11

�⌫ 0 () X11 � x

21

8>>><

>>>:

Xij � 0

xi �Xij � 0

xj �Xij � 0

1� xi � xj +Xij � 0

with i = j = 1 =)

8><

>:

X11 � 0

X11 x1

X11 � 2x1 � 1

• Number  of  vars  reduced.

• Size  of  the  SDP  part  reduced.

• Constraint  0  ≤  xi  ≤  1  unnecessary.

SDP2 max  t  subject  to�tI +

X

i<j;(i,j)2E

Xij˜

Lij+↵J + �

NX

i=1

(1� xi)Ei ⌫ 0,

NX

i=1

xi =N �Ndel,

For links (i, j)

8>>><

>>>:

Xij � 0

xi �Xij � 0

xj �Xij � 0

1� xi � xj +Xij � 0

Small  networks

Karate  club(N=34,  78  links,  β=2)Data:  Zachary  (1977)  

Macaque  corUcal  net(N=71,  438  links,  β=2)

Data:  Sporns  &  Zwi  (2004)    

0

1

2

0 10 20

h2

Ndel(a)

sequentialSDP1SDP2

0

1

2

3

0 10 20

h2

Ndel(b)

RelaUvely  large  networks

BA  model  (scale-­‐free  net)(N=150,  297  links,  β=2)

C.  elegans  neural  net(N=297,  2287  links,  β=2.5)Data:  Chen  et  al.  (2006)

0.5

0.6

0.7

0 10 20

h2

Ndel(c)

1

1.5

2

2.5

3

0 10 20 30

h2

Ndel(d)

SDP2

sequenUal

ObservaUon:  SDP1/SDP2  may  work  beber  for  sparse  networks.

Possible  direcUons

• Go  violate  convexity

• (1-­‐xi)  →  (1-­‐xi)p,  and  increase  p  gradually  from  p=1.  By  the  Newton  method

• Parameter  tuning?

�tI +X

i<j;(i,j)2E

XijLij + ↵J + �

NX

i=1

(1� xi)Ei ⌫ 0