Presentation slides for the following two papers (currently available in the pdf format only). (1) T. Watanabe, N. Masuda. Enhancing the spectral gap of networks by node removal. Physical Review E, 82, 046102 (2010). (2) N. Masuda, T. Fujie, K. Murota. Semidefinite programming for maximizing the spectral gap. In: Complex Networks IV, Studies in Computational Intelligence, 476, 155-163 (2013).

1. 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)

2. Laplacian
of
a
network
˙x(t) = Lx(t)
˙x1 = 2x1 + x2 + x4
=(x2 x1) + (x4 x1)
1 2
3 4
L =
0
B
B
@
2 1 0 1
1 2 0 1
0 0 1 1
1 1 1 3
1
C
C
A
1 = 0 < 2  3  · · ·  NEigenvalues:

3. 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).

4. 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 ui ˆei
where ˆei ⌘ (0, . . . , 0, 1|{z}
i
, 0, . . . , 0)
=) 2 ⇡
P
j2Ni
uj(ui uj)
1 u2
i
Select
i
that
maximizes
Δλ2

5. Results:
model
networks
(N
=
250,
<k>
=
10)
Goh
WS
HKBA
ER
f
0 0.1 0.2 0.3 0.4 0.5
f
0 0.1 0.2 0.3 0.4 0.5
f
0 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
f
0 0.1 0.2 0.3 0.4 0.5
f
2normalized
0 0.1 0.2 0.3 0.4 0.5
Goh

6. Results:
real
networks
perturbative
betweenness-based
degree-based
optimal sequential
e-mail
C. elegans
2
3
4
5
0.5
0
1
1.5
2
22
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 0.5
0
f f
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
f f
N
=
279
<k>
=
16.4
N
=
1133
<k>
=
9.62
N
=
71
<k>
=
12.3
N
=
2268
<k>
=
4.96

7. 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)

8. However,
• SequenUal
opUmal
may
not
be
opUmal
for
Ndel
≥
2.
• An
obvious
combinatorial
problem
if
we
pursue
the
opUmal
soluUon.

9. 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

10. 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’=α
New
zero
eigenvalue
→
β
But,
a
nonlinear
constraint
tI +
X
i<j;(i,j)2E
xixj
˜Lij + ↵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
1i<jN;(i,j)2E
˜Lij

11. (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
Xij
˜Lij + ↵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
xixj
˜Lij + ↵J +
NX
i=1
(1 xi)Ei ⌫ 0
max
t
subject
to

12. An
improved
method
SDP2:
“local
relaxaUon”
tI +
X
i<j;(i,j)2E
Xij
˜Lij + ↵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

13. 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 x2
1
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

14. • 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

15. 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
2
Ndel
(a)
sequential
SDP1
SDP2
0
1
2
3
0 10 20
2
Ndel
(b)

16. 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
2
Ndel
(c)
1
1.5
2
2.5
3
0 10 20 30
2
Ndel
(d)
SDP2
sequenUal
ObservaUon:
SDP1/SDP2
may
work
beber
for
sparse
networks.

17. 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
Xij
˜Lij + ↵J +
NX
i=1
(1 xi)Ei ⌫ 0