TELKOM
NIKA
, Vol. 11, No. 10, Octobe
r 2013, pp. 6
224 ~ 6
231
ISSN: 2302-4
046
6224
Re
cei
v
ed Ap
ril 19, 2013; Revi
sed
Jul
y
1
6
, 2013; Acce
pted Jul
y
25,
2013
An Optimal Routing Strategy Based on Specifying
Shortest Path
Yonghua Xu
*1
, Fei Shao
1,2
1
School of Infor
m
ation T
e
chno
log
y
, Jin
lin
g
Institute of
T
e
chnol
og
y,Na
nji
n
g
,
China
2
Jiangsu Infor
m
ation An
al
ysi
s
Engin
eeri
ng
Lab
orator
y, Jin
ling Institut
e
of T
e
chnolog
y, N
anji
ng, Ch
in
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: xyh@
jit.ed
u
.cn
A
b
st
r
a
ct
Unlik
e the sh
o
r
test path is rando
mly chose
n
in t
he trad
iti
ona
l shortest
path routi
ng st
rategy, a
nove
l
routi
ng s
t
rategy to i
m
pr
ove
the
netw
o
r
k
transportati
o
n
cap
a
city is
p
r
opos
ed i
n
this
pap
er. Accord
in
g
to the different
character
i
stics of t
he nod
es al
ong act
ual p
a
t
h
s, w
e
specify
the shortest pa
ths of all pa
irs o
f
nod
es
ai
mi
ng
at re
duc
ing
t
he
betw
een
ne
ss of th
ose hi
gh-b
e
tw
eenn
e
ss
no
des.
S
i
mu
lati
ons on
both
computer-
gen
e
r
ated a
nd r
e
a
l
-
w
orld
netw
o
rks
show
that the
new
routi
ng str
a
tegy c
an
enh
ance t
he
netw
o
rk
transportati
on
capac
ity greatl
y
. And it w
o
rks
better in
thos
e netw
o
rks w
i
th the fu
zz
y c
o
mmunity structure.
Ke
y
w
ords
:
o
p
timal routi
n
g
strategy, netw
o
rk transportati
on capacity,
comm
unity
structure, complex
netw
o
rks
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introducti
on
Due to
the
se
minal
work o
n
the
small
-
world
phen
ome
non by
Watts and Strogat
z [1] and
the scale
-
fre
e
pro
p
e
r
ty by Barab
á
si
an
d Albert [2], many peo
ple
take tremen
dou
s interest
s in
the structu
r
e
and dyn
a
mics of compl
e
x netwo
rks.
It h
a
s b
een
wid
e
l
y proved th
at the topolo
g
ical
stru
cture h
a
ve profoun
d ef
fects
on th
e
pro
c
e
s
ses ta
king
pla
c
e
on
the n
e
two
r
ks. Spe
c
ificall
y
in
recent years,
the study of info
rmatio
n traffic on co
mp
lex networks
is be
comin
g
more a
nd mo
re
importa
nt, as a
re
sult of th
e con
s
tantly i
n
crea
sing
im
portan
c
e
of l
a
rge
commu
nicatio
n
n
e
tworks
su
ch a
s
the
Internet
and t
he Worl
d Wi
d
e
We
b. A pa
rticular focus
of these
studi
es i
s
to ma
ke
the
netwo
rk tran
sportation
cap
a
city maximal
so as
to cont
rol the incre
a
s
ing traffi
c co
nge
stion.
In previou
s
works, there are two way
s
to impr
ove th
e netwo
rks transportatio
n
cap
a
city: making
approp
riate a
d
justme
nts to
the net
wo
rk topology
structure [3
-4]
or
d
e
si
gnin
g
efficient routi
ng
strategi
es [
5
-9]. Since it is
too expen
siv
e
and to
o
difficult to chan
g
e
the
st
ru
cture of so
me la
rge-
scale net
wo
rks, mo
st of p
r
evio
u
s
works have b
een
committed to
the develop
ment of effective
routing
strate
gies. S
o
me
o
f
them a
r
e
ba
sed
on
th
e
gl
obal i
n
form
ation: the
sh
ort
e
st p
a
th
routi
ng
strategy th
at pass th
roug
h
the minimu
m
numbe
r of
n
ode
s [5], the
efficient path
routing
strate
gy
who
s
e d
egre
e
sum of nod
es is a mini
m
u
m [6],
and the global dyna
mic ro
uting st
rategy ba
sed
on
the qu
eue
le
ngth of
no
de
s [7]. Som
e
o
t
hers fo
cu
s o
n
lo
cal
topolo
g
ical
info
rmat
ion
sin
c
e
glo
bal
informatio
n i
s
u
s
u
a
lly un
available i
n
large
-
scale n
e
tworks: nei
ghbo
r info
rm
ation [8], ne
xt-
nearest
-
nei
gh
bor info
rmatio
n [9].
The sh
orte
st path routin
g strat
egy,
as its n
a
me
impli
e
s, h
a
s the
shorte
st ave
r
a
ge p
a
th
length which
means a p
a
cket may reach its de
stination qui
cker tha
n
taki
ng other p
a
ths.
Ho
wever,
it i
s
p
r
ove
d
that
sh
orte
st p
a
th rout
ing
stra
tegy is
not
o
p
timal for scale-free
networks
becau
se con
gestio
n
will o
c
cur in
some
node
s with
highe
r deg
re
e [10]. As there may be
more
than one
sho
r
test path b
e
twee
n so
me p
a
irs
of
node
s
and on
e of them will be
sel
e
cted
ran
dom
ly
in the traditio
nal routin
g st
rategy, we can sp
ecify the sho
r
test p
a
th to enhan
ce the netwo
rk
transpo
rtation
capa
city.
With the
d
e
epeni
ng
of rese
arch
of
comple
x
networks, th
e p
r
operty
of co
mmunity
stru
cture a
p
p
ears to
be
co
mmon in
lots
of real
net
wo
rks [11
-
1
2
], which
mea
n
s the ten
den
cy for
node
s to divide into sub
s
e
t
s within whi
c
h node
-no
de
con
n
e
c
tion
s are de
nse bu
t between wh
ich
con
n
e
c
t
i
on
s
are
spa
r
ser.
S
e
v
e
ral r
eal
net
wo
rk
s h
a
v
e
mor
e
o
r
le
ss
co
mmunit
y
st
ru
ct
ure
w
h
ich
will impact the network transporta
tion
capacity [13]. The internal
structure of networks
will have
influen
ce on
our ne
w routi
ng strat
egy.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 622
4 –
6231
6225
2. Models
The traffic mo
del ca
n be de
scribe
d as foll
ows:
1) All
nod
es ca
n
create
packet
s
with
add
re
sse
s
of de
stinatio
n, re
ceive
p
a
ckets from
other
node
s, and fo
rwa
r
d the p
a
ckets to thei
r d
e
stinatio
ns.
2) At ea
ch ti
me ste
p
, the
r
e a
r
e
R
p
a
ckets gen
erated in th
e net
work,
with ra
ndomly
cho
s
en
sou
r
ces a
nd
destin
a
tion
s.
Once a
pa
cket is create
d
,
it is
placed
at
the
end
of th
e qu
eue
if the
node al
rea
d
y has
several p
a
ckets
waitin
g to be forwa
r
ded to their d
e
stinatio
ns.
3) At each time s
t
ep, the
firs
t
C
i
pa
cket
s at th
e he
ad
of the
queu
e
of ea
ch
nod
e are fo
rwa
r
d
e
d
one
step to
ward th
eir
de
stination
s
an
d t
hen pl
ac
ed
at the en
d of th
e que
ue
s of t
he
sele
cted
node
s. We
a
s
sume every node ha
s
the
sam
e
p
a
cket
delivery cap
ability
C
i
an
d s
e
t
C
i
=1 for
simpli
cit
y
.
4) A packet, upon rea
c
hin
g
its destin
a
tion, is rem
o
ve
d from the sy
stem.
As
R
i
s
increased from
zero, the nu
mbers of
cre
a
ted and fo
rwarded
pa
ckets are bal
a
n
ce
d,
resulting i
n
a
steady free fl
ow of t
r
affic.
Since
pa
cket
delivery
cap
a
c
ity of nod
e i
s
limited, traffic
con
g
e
s
tion wi
ll
occu
r wh
en
R
is large
en
ough.
The
ph
ase
tra
n
sitio
n
from
the fo
rmer to
the
lat
t
er
is define
d
as the critical v
a
lue
R
c
which is the crite
r
ion of the network tra
n
spo
r
tation ca
pa
ci
ty.
We a
r
e i
n
tere
sted in
re
solv
ing critical
value
R
c
i
n
orde
r to ad
dre
s
s
whi
c
h
kind
of routing
strate
gy
is
more s
u
sceptible to traffic
c
o
nges
t
ion.
We i
n
tro
d
u
c
e
the al
go
rith
mic
between
ness
b
i
to
estimate the
po
ssi
ble t
r
affic
passin
g
throug
h a no
de
i
unde
r a g
i
ven routing
strategy whi
c
h
is defined a
s
below:
t
s
i
t
s
t
i
s
b
,
)
,
(
)
,
,
(
(1)
whe
r
e
)
,
,
(
t
i
s
is the numbe
r of p
a
ths un
de
r the gi
ven ro
uting strategy b
e
twee
n nod
e
s
s
a
nd
t
that pass th
rough
nod
e
i
and
)
,
(
t
s
is the
total numb
e
r
of path
s
u
nder the giv
en routing
strategy bet
ween
s
a
nd
t
and the sum i
s
over all pairs
s
,
t
of all distinct nod
es.
The
pro
babili
ty a pa
cket
will p
a
ss th
rough
the
no
de
i
is
n
j
j
i
b
b
1
/
, and
therefore
th
e
averag
e num
ber of pa
ckets that the node
i
will recei
v
e at each time step is
))
1
(
/(
n
n
Rb
i
where
n
is the node
s numb
e
r of the whol
e net
work. Wh
en
the numb
e
r of
incomin
g
pa
ckets is eq
ua
l to
or larger tha
n
the outgoin
g
packet
s
at the node
i
,
1
))
1
(
/(
i
i
C
n
n
Rb
, traffic congestion will
occur. So the
critical p
a
cke
t
-gene
ratin
g
rate
R
c
is
)
)
1
(
min(
i
c
b
n
n
R
(2)
The m
o
st
wid
e
ly used
algo
rithm b
e
twe
e
nne
ss is the
sho
r
test
path
betwe
enn
ess[14],
sb
i
,
base o
n
the
tradition
al
sho
r
test
path
rou
t
ing st
rategy.
Ne
wman
sug
geste
d a
noth
e
r b
e
twe
enn
e
s
s
measure by the na
me ran
dom
walk bet
wee
nne
ss[15
],
rb
i
, whi
c
h i
s
equal to
the
numbe
r of ti
mes
that a ran
d
o
m
wal
k
sta
r
ti
ng at
s
and e
nding at
t
passe
s thro
ugh
i
along th
e wa
y, averaged
o
v
er
all
s
and
t
. Th
e rand
om wal
k
between
ne
ss
rb
i
of node
i
is calculated as follows:
1) Con
s
truct the
matrix
D
−
A
, where
D
is
the diag
on
al
matrix with
eleme
n
ts
D
ii
=
k
i
and
A
is the
adja
c
en
cy matrix (if there i
s
an ed
ge bet
wee
n
i
and
j
,
A
ij
=1; oth
e
rwi
s
e,
A
ij
=0).
2) Re
move a
n
y single row
and the corre
s
po
ndin
g
col
u
mn.
3) Invert the resultin
g matri
x
and then ad
d back in a new ro
w and
column con
s
ist
i
ng of all zero
s
in the
po
sitio
n
where the
row
and
colu
mn
were
prev
iously
rem
o
ved. Call the resulting mat
r
i
x
T
, with elements
T
ij
.
4) The
rand
o
m
walk b
e
twe
enne
ss of no
de
i
is define
d
as
2
/
)
1
(
)
(
n
n
st
I
rb
t
s
i
i
(3)
otherwise
t
s
i
T
T
T
T
A
st
I
j
jt
js
it
is
ij
i
1
,
)
(
2
1
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Optim
a
l Routing Strate
gy Based on
Specifyi
ng Sh
ortest Path (F
ei Shao)
6226
Our ne
w routi
ng strat
egy
is described as follow:
1) Calculate t
he de
gre
e
, the sh
orte
st pa
th
betwe
enn
e
ss,
a
nd
the random wal
k
betwe
enn
ess
of
every node a
s
k
i
,
sb
i
,
rb
i
a
c
cordingly.
2) Obtain all
sho
r
test path
s
between n
o
des
s
and
t
u
s
ing b
r
ea
dth first search al
gorithm.
3)
Cal
c
ulate t
he sum of
de
gree
s
k
i
of all
node
s
on e
a
c
h
sho
r
te
st p
a
th, and
sel
e
ct the o
ne
with
minimum
su
m a
s
o
u
r ro
u
t
e path.
We
call thi
s
SK
routing
strate
gy, and SS
routing
strate
gy for
the minimum
sum of
sb
i
, SR routin
g stra
tegy for the minimum sum of
rb
i
4) Obtain
rout
e paths fo
r all node pai
rs.
To observe t
he influen
ce
of network int
e
rnal
co
mm
u
n
ity structu
r
e
on our
strate
gies, we emp
l
o
y
the modul
arit
y measu
r
e,
Q
, which is d
e
fined a
s
follo
ws[11]: co
nsi
d
er a divi
sion
of a netwo
rk i
n
to
m
ods
comm
u
n
ities, define
an
m
ods
×
mod
s
symm
etric matrix
E
whose ele
m
ent
e
ij
is the fracti
on
of all edge
s that link no
de
s in co
mmuni
ty
i
to nodes i
n
comm
unity
j
. So we get
i
i
ii
a
e
Q
)
(
2
(5)
whe
r
e we
ha
v
e
j
ij
i
e
a
. Different division will
result in
different
Q
, th
e
maximum of
them,
Q
max
, can describ
e the internal com
m
unit
y
structu
r
e th
e netwo
rk h
a
s
.
3. Simulatio
n
s and An
aly
s
is
To obtain th
e critical pa
cket gene
rati
ng rate
R
c
i
n
simulatio
n
s, we use the ord
e
r
para
m
eter [3]
:
t
R
t
lim
(6)
whe
r
e
)
(
)
(
t
t
t
, with
indicating averag
e over ti
me wind
ows
of width
t
, and
)
(
t
is the total n
u
mbe
r
of pa
ckets i
n
the n
e
twork at tim
e
t
. At the early stage,
wh
en
R
is
ve
r
y
small, the ge
nerate
d
pa
ckets ca
n be de
livered,
is less than zero a
nd so i
s
η
. Where
η
is
greate
r
than
zero, we
can
obtain the critical pa
cket ge
neratin
g rate
R
c
.
Firstly, we
inv
e
stigate th
e e
fficiency of
o
u
r routing
strategie
s
in
ho
mogen
eou
s
n
e
tworks.
We g
ene
rate
a serie
s
of
pse
udo
-rand
om net
works with
n
=12
8
node
s whi
c
h
are divided into
m
ods
=4 com
m
unities with
n/m
ods
=3
2 n
ode
s in ea
ch
comm
unity. Each
node
ha
s on average
Z
in
edge
s
con
n
e
c
ting it to n
ode
s of the
same
co
m
m
unity and
Z
out
edge
s to nod
es
of other
comm
unitie
s
. Whil
e
Z
in
i
s
varied, the
value
of
Z
out
is ch
osen to
keep th
e total
averag
e d
e
g
r
ee
<k
>
=16. We increa
se
Z
in
from 8 to 15 edge
s to gene
rate the network, an
d use DA comm
unit
y
detectio
n
alg
o
rithm [16] to gain the maxi
mum modul
arity
Q
ma
x
as sh
own in Fig
u
re
1.
As
F
i
gu
r
e
1
s
h
ow
n
,
w
h
en
Z
in
is 8,
Z
out
=8
=
Z
in
, the
netwo
rk is
random
net
work with
Q
max
=0.2162. While
Z
in
i
s
increa
sing, there
are m
o
re ed
ge
s co
nne
cting no
d
e
s of the
sa
me
comm
unity which result in the stron
g
e
r
community structure and th
e highe
r
Q
max
. When
Z
in
is 15,
Q
max
reaches
0.6771.
The relatio
n
ship betwe
en
critical pa
cket generat
ing rates of three
routing
strate
gies an
d
Z
in
is sh
own
in Figu
re 2.
From Fi
gure
2 we
ca
n
se
e that all the
three
routin
g strate
gie
s
can
enha
nce the
netwo
rk t
r
an
sportation
ca
p
a
city and th
e
routing
strate
gy by spe
c
ifying the
sho
r
te
s
t
path with
the
minimum
su
m of the n
o
d
e
sh
orte
st
pat
h between
ne
ss is th
e mo
st rema
rkable
one.
Figure 1 a
n
d
Figure 2
also sh
ow th
at
the str
ong
er
comm
unity st
ructu
r
e th
e n
e
twork
ha
s the
more in
effecti
v
e our strateg
i
es are.
As we all
kn
own, m
any real net
wo
rks inclu
de the
Internet di
spl
a
y a hete
r
og
eneo
us
stru
cture with
a scale
-
fre
e
deg
re
e
di
stri
bution.
We
u
s
e BA
mod
e
l
to valid
ate o
u
r
strate
gie
s
in
hetero
gen
eo
us netwo
rks. We gene
rate
a
serie
s
of
scale-free n
e
tworks
usi
ng B
A
model
with
BA
para
m
eter
m
sets to 2 an
d also u
s
e
DA algor
ith
m
to get the highe
st modula
r
ity
Q
ma
x
as sho
w
n
in
Figure 3. And the relation
ship
R
c
and
Z
in
is sho
w
n in F
i
gure 4.
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Figure 1. Maximum modul
a
r
ity
Q
ma
x
ver
s
us
Z
in
Figure 2.
R
c
ver
s
us
Z
in
for d
i
fferent routin
g strate
gies
Figure 3. Maximum modul
a
r
ity
Q
ma
x
versus no
de
s nu
mber
n
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ei Shao)
6228
Figure 4.
R
c
versus n
ode
s
numbe
r
n
for
different ro
uting strat
egie
s
From Figu
re 3
and
Figu
re
4,
we ca
n se
e
that
the SS
strate
gy al
so
gives th
e b
e
s
t re
sult
s
in hete
r
og
en
eou
s n
e
two
r
ks. And
thre
e
strategi
es
work
better
in networks
with fuz
z
y
c
o
mmunity
stru
cture whi
c
h is
con
s
i
s
te
nt with analysis above.
Then we ch
e
ck the imp
a
ct
of BA param
eter
m
on our routing
strat
egie
s
. We ge
nerate
four BA networks
with
n
=100 no
de
s a
nd BA para
m
eter
m
is 2, 3, 4, and 5. The maximum
modula
r
ity
Q
ma
x
is 0.4
4
9
5
, 0.351
5, 0
.
2858, a
nd
0.2592
corre
s
po
ndin
g
ly. The
relatio
n
ship
betwe
en
R
c
a
nd
m
is shown in Figure 5.
Fig. 5.
R
c
versu
s
BA para
m
eter
m
for different routin
g strate
gies
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The results i
n
Figu
re 5
al
so a
g
re
e wit
h
our
co
ncl
u
sion. Th
e efficien
cy of ou
r routing
strategi
es de
cre
a
se with
the in
cre
a
si
ng
of BA param
eter
m
. When BA parameter
m
is 2,
R
c
is
increa
se by 5
5
.5% with SS routing st
rate
gy while ne
arl
y
69.7% whe
n
m
is 5.
Figure 6. p
r
ovides insi
gh
t into ho
w t
he
routing
strategy works by
com
p
aring the
betwe
enn
ess
distributio
ns with
different routing st
rate
gies in the
ca
se of a BA network
with 1
0
0
node
s and a
v
erage n
ode
degree <
k
>=4. Figure 6.(a) sh
ows the
between
n
e
s
s plotted aga
inst
the no
de i
n
d
e
x whil
e Fi
g
u
re
6.(b
)
sh
o
w
s hi
stog
ram
s
of
the
bet
wee
nne
ss di
stributio
n
. In
TS
routing
st
rate
gy, the majo
rity of the nod
es
hav
e ve
ry low
between
ness, b
u
t a
small num
be
r
of
them have ve
ry highe
r b
e
twee
nne
ss. In SS routi
ng strategy,
nod
e betwe
enn
ess
are confin
ed
to
a narro
w ban
d.
Fig. 6. Distrib
u
tion of node
betwe
enn
ess wi
th two different ro
uting st
ragtegi
es
(a) b
e
twe
enn
ess of every node (b)
di
stri
bution of nod
e
betwe
enn
e
s
s
0
10
20
30
40
50
60
70
80
90
10
0
0
50
0
10
00
15
00
20
00
25
00
30
00
35
00
40
00
no
de
i
n
d
e
x
bet
w
eennes
s
(a
)
SS
TS
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TELKOM
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An Optim
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ortest Path (F
ei Shao)
6230
Finally, we va
lidate ou
r rout
ing st
rategie
s
on
the
real
world
netwo
rk. We e
m
ploy th
e jazz
netwo
rk [17]
with 19
8 n
o
d
e
s
who
s
e
Q
ma
x
is 0.4452 a
nd the
email
netwo
rk [18]
with 11
33
no
des
who
s
e
Q
max
is 0.57
38. Th
e critical
pa
cket gene
rating
rate
s of thre
e ro
uting
stra
tegies
are sh
own
in Table 1.
Table 1.
R
c
of
real wo
rld n
e
t
works fo
r
different routing
strategi
es
SS
SR
SK
TS
Ja
zz
18.9
17.5
14.7
6.8
Email
37.8
36.1
31.2
26.2
As Tabl
e 1
sho
w
n, in
re
al networks,
t
he SS strategy ca
n al
so e
nha
nce
netwo
rk
transpo
rtation
ca
pa
city mo
re g
r
eatly tha
n
othe
rs
,
es
p
e
cially
in
net
wor
k
s
wit
h
f
u
zzy
com
m
uni
ty
st
ru
ct
ur
e.
4. Conclusi
on
Acco
rdi
ng to
the different
cha
r
a
c
teri
sti
cs
of
the no
des
along
actual sh
orte
st path, we
prop
ose thre
e ro
uting
strategie
s
to e
n
han
ce th
e
ne
twork tran
sp
ortation
ca
pa
city. The o
n
e
by
spe
c
ifying
th
e
sho
r
test pa
th
with
the m
i
nimum su
m of
the nod
e shorte
st
p
a
th betwe
enn
ess
is
proved to b
e
the most re
m
a
rkable o
ne.
All of our
strategie
s
are l
e
ss effective
in netwo
rks
with
st
ron
g
e
r
com
m
unit
y
st
ru
ct
ure.
Ackn
o
w
l
e
dg
ments
This work was pa
rtially suppo
rted by the Na
tional
Natural Scie
nce Fo
und
ation of China
(Grant No. 60874
091
), the Six Project
s
Spon
so
ring
Talent Sum
m
its of Jia
n
g
s
u Province,
Chin
a
(Grant No. SJ20
900
6), th
e Natu
ral S
c
i
ence Fou
nda
tion of Jia
n
g
s
u Provin
ce,
Chin
a (G
ra
nt No.
BK20120
82)
and spon
so
re
d by Qing La
n Proje
c
t.
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