TELKOM
NIKA
, Vol.11, No
.1, Janua
ry 2013, pp. 181
~18
6
ISSN: 2302-4
046
181
Re
cei
v
ed O
c
t
1, 2012; Re
vised
No
vem
ber 28, 201
2; Acce
pted De
cem
ber 4, 20
12
A Survey on Weighted Network Measurement and
Modeling
Xin Xia, Shu-
xin Zhu
Coll
eg
e of Information Sci
enc
e and T
e
chno
l
o
g
y
, Nan
jin
g A
g
ricult
ural U
n
iv
ersit
y
W
e
iga
ng 1, Na
njin
g,chi
an, 21
009
5
*corres
pon
di
ng
author, e-mai
l
: zzxia
xin@
nj
au
.edu.cn
A
b
st
r
a
ct
Real n
e
tw
orks have co
mpl
e
x
topolo
g
ica
l
structure
char
acteristics such a
s
pares of nod
es hav
e
different stren
g
t
h and ca
pacit
y connecti
on.
T
here are
ma
ny instanc
es i
n
our rea
l
life.
Every person
has
strong or w
eak
relatio
n
shi
p
w
i
th others in soc
i
al n
e
tw
ork and
different reacti
on paths
have
non-
unifor
m
flo
w
in
metab
o
lic
n
e
tw
ork. In the
food
ch
ain
pr
ed
ator a
n
d
pr
ey
have
div
e
rse
r
e
lati
on
an
d
in t
he
ne
ural
n
e
tw
ork
electric
al si
gn
a
l
trans
miss
ion
paths
have
diff
erent
c
apac
ity. All th
ese syst
ems
cou
l
d
be
descri
bed
w
e
ll
by
the co
ncept
of
w
e
ighte
d
n
e
tw
ork and
w
e
ig
ht of e
a
ch
ed
g
e
in
w
e
ig
hted
netw
o
rk stand
for the c
onn
ectio
n
strength a
m
o
n
g
ind
i
vid
u
a
l
s. This pa
per a
i
ms
to provid
e an o
v
erview
of rece
nt advanc
es in
such study ar
e
a
.
W
e
first intro
d
u
ce
a s
e
ries
of
w
e
ighte
d
netw
o
rk statis
tica
l c
haracter
i
stic p
a
ra
meter
an
d
conce
p
ts .T
he
n i
n
particu
lar, w
e
also
discuss
some pr
actical
netw
o
rk ap
plic
ation
exa
m
ples
. Emp
i
rica
l res
u
lts show
that
p
u
r
e
topol
ogic
a
l
mo
del is
not suffi
cient to ex
pla
i
n the a
bun
da
n
t
and co
mplex
characteristics
observ
ed i
n
r
e
a
l
system
s and it is necess
a
ry to im
pr
ov
e suc
h
m
o
del. So we finally focus
on some latest optimi
z
e
d weight
e
d
netw
o
rk mo
del
s and giv
e
co
mparis
ons.
Key
w
ords
:
Weig
hted net
work; Statistical characte
ri
stic; Mod
e
l;
Copyrig
h
t
©
2013
Univer
sitas Ahmad
Dahlan. All rights res
e
rv
ed.
1. Introduc
tion
Many thing
s
in re
al
worl
d
su
ch
as vari
ety of entities a
nd
com
p
l
e
x relatio
n
s
betwe
en
entities
coul
d
be exp
r
e
s
se
d by the fo
rm of wei
ghted network
. From Internet [1] to WWW [1],
from la
rge
po
wer net
work [
2
] to glob
al transportatio
n
netwo
rk [3], from b
r
ai
n in t
he o
r
ga
nism
to
variou
s meta
bolic
network [4], and fro
m
Scientif
ic
colla
boration
netwo
rk [5] to all kin
d
s of
eco
nomi
c
, po
litical and
so
cial relation
n
e
twork [5,6], we
can
say
we h
a
ve alre
ady have live
d
in
the worl
d filled with vario
u
s we
ight
e
d
net
wo
rk
s.
I
n
su
ch sit
u
at
io
n, weighte
d
netwo
rk top
o
l
ogy
stru
cture stud
y has beg
un to be the hotspot.
The re
st of pape
r is o
r
g
anized a
s
follows. Section
2 introdu
ce
s some
statist
i
cs a
nd
con
c
e
p
ts
of weig
hted n
e
twork. Se
ction
3 de
scri
be
s
some
weight
ed net
wo
rk m
odelin
g meth
ods,
inclu
d
ing YJBT [7], ZTZ
H [8] and AK [9]. Section 4 points o
u
t the defaul
ts of above three
modelin
g met
hod
s a
nd giv
e
s th
e imp
r
o
v
ements. In
se
ction 5
we
co
ncl
ude an
d
di
scuss
fut
u
re
wor
k
.
2. Weighted
Net
w
o
r
k M
e
asureme
nt
A weighted
netwo
rk
could be ex
pre
s
sed a
s
W
,,
GN
l
W
whe
r
e no
de
12
,,
,
N
Nn
n
n
and edg
e
12
,
,
,
K
ll
l
l
and
weight i
n
different
edge
s
12
,,
,
K
Ww
w
w
. The
weig
hted n
e
two
r
k
coul
d b
e
sh
own
a
s
Fig.
1 where thi
c
kne
s
s of
edge
s expresse
s the wei
g
ht. When in t
he form of m
a
trix,
W
G
is usua
lly expressed
as wei
ghted
matrix
W
with the si
ze of
NN
and the ele
m
e
n
t value
ij
w
sta
nds fo
r the
correspon
ding
edge weight.
ij
w
=0 wh
en no
d
e
i and j have no con
n
e
c
tion. In this paper we discu
ss o
n
ly two
situation
s
. On
e is
0
ij
j
i
ww
and th
e
other i
s
0
ii
w
i
. In certain
circu
m
stan
ce the
weight
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TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 181 – 1
8
6
182
of edge may be negative
such as the
strang
e deg
ree amo
ng p
eople in soci
al netwo
rk.
One
prima
r
y cha
r
a
c
teri
stic of we
ighted
network is the weig
h
t
distribution
()
Qw
, which
stand
s for the
prob
ability that the edge weight is
w
.
Figure 1. W
e
i
ghted net
work
2.1 Node Str
e
ngth a
nd Intensi
t
y Dis
t
ri
bution and c
o
rrelation
In weighte
d
netwo
rk, the
degree
i
k
of node
i
is call
ed nod
e stre
ngth whi
c
h
o
fte
n
expre
s
sed a
s
i
s
.The definitio
n of
i
s
is [10-12]
:
ii
j
j
sw
(1)
We
can
dra
w
a
co
ncl
u
si
on from
form
ula (1) th
at node
strengt
h incl
ude
s b
o
th nod
e
degree a
nd
e
dge
weig
ht conne
cting th
e
node.
Wh
en
t
he wei
ght an
d the top
o
logi
cal
stru
cture
of
the network i
s
not
releva
n
t
and if the
node
deg
ree
is
k
, the node
stre
ngth i
s
()
sk
w
k
whe
r
e
w
is the averag
e wei
g
ht. If we con
s
i
der the
correl
ation,
()
sk
A
k
.
Given a nod
e
i
with degre
e
i
k
and stren
g
th
i
s
,
ij
w
has the same o
r
de
r
of
/
ii
sk
.The
discre
pan
cy in different ed
ge wei
ght of node
i
could b
e
descri
bed a
s
[13-1
4
]:
i
ij
i
jN
i
w
Y
s
(2)
whe
r
e
i
N
indicates the fi
rst-orde
r n
e
ighb
or no
de
s set of node
i
.
i
Y
is well correl
ated
with no
de d
e
g
ree. If all
ed
ges
have
simi
lar
weight,
()
Yk
is prop
ortio
nal
t
o
1/
k
.If one edge coul
d
influen
ce formula (2) th
en
()
1
Yk
. In anothe
r
word,
()
Yk
is irrel
e
vant to
k
in s
u
ch sit
uat
io
n
[15]. Intensity distrib
u
tion
()
Rs
stand
s fo
r th
e pro
bability
of sele
cting
a
node
with in
tensity
s
.
()
Rs
and
()
Pk
,which
i
s
often
called
deg
ree
di
stri
buti
on ,d
escri
be the
struct
ural
prope
rtie
s of
weig
hted net
work well.
2.2 Lowes
t-weight pa
th
Networks in
n-dim
e
n
s
iona
l Euclid
Spa
c
e are often t
w
o
or th
ree
d
i
mensi
onal
a
nd
so the
distan
ce of n
ode
i
and
j
cou
l
d be rep
r
e
s
e
n
ted by Euclidean di
stan
ce yardsti
ck. Edge length in
weig
hted n
e
twork
ca
n be
defined
a
s
a functio
n
of
weig
ht, for i
n
stan
ce
1/
ij
ij
lw
.In weighted
netwo
rk, an
optimal path is usu
a
lly not the lo
west
-h
op path for consi
deri
ng th
e edge weig
ht.
Cho
o
si
ng of the lowest-wei
ght path is co
mple
tely dep
ende
nt on the
definition of edge
weight.
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TELKOM
NIKA
ISSN:
2302-4
046
A Survey
on W
e
ighted Networ
k Meas
u
re
me
nt and M
odeling
(
Xin Xia
)
183
2.3 Weigh
t
cluster c
o
effic
i
ent
In the literatu
r
e [16], Barra
t
et al. define
d
clu
s
te
ring
coefficient o
n
node
i
in
wei
ghted
netwo
rk a
s
:
W
,
1
(1
)
2
ij
i
m
ii
j
j
m
m
i
jm
ii
ww
ca
a
a
sk
(3)
Weig
ht clu
s
te
ring
coeffici
e
n
t con
s
id
ers
not only the triangle
numb
e
r
s
adja
c
ent to
node
i
but al
so th
e relative weigh
t
of releva
nt
node
s i
n
ten
s
i
t
y or
stren
g
th
.
(1
)
ii
sk
is a
no
rmali
z
ation
fac
t
or to ensure
W
01
i
C
.
W
C
is th
e av
erag
e valu
e
of all n
ode
s’
weig
ht cl
uste
ring
co
efficie
n
t
and
W
()
Ck
is the a
v
erage val
ue
of node
s’ wei
ght clu
s
teri
ng
coeffici
ent wi
th the deg
ree
being
k
. In the ca
se
of larg
e-scal
e ran
dom
ne
twork(not co
nsid
erin
g de
gree
rel
e
vant
)
W
CC
and
W
()
()
Ck
C
k
whe
r
e
C
is th
e avera
ge val
ue of cl
uste
ri
ng coefficie
n
t of un-weig
hted net
work
with the
sam
e
topolo
g
y a
nd
()
Ck
is the a
v
erage val
u
e
of su
ch n
e
twork n
ode
s’
clu
s
terin
g
coeffici
ent wit
h
degree bei
n
g
k
.
In re
al
weight
ed n
e
two
r
k
we may m
eet t
w
o
re
cip
r
o
c
al
situatio
ns.
O
ne i
s
W
CC
and
then ed
ge
s formin
g trian
g
l
e
usually hav
e larg
er
weig
ht. The othe
r
is
W
CC
and the
s
e
edge
s
have small
e
r
weig
ht.
3. Weighted
net
w
o
r
k mo
deling
3.1 Yook–Je
ong–
Bara
bá
si–Tu (YJ
B
T)
The sim
p
le
st way to co
nstruct a weighte
d
netwo
rk is to build a rand
om network
with the
degree di
stri
bution is
()
Pk
a
nd then try to make it submit to mutual and in
depe
ndent
distrib
u
tion ,
s
uppo
sin
g
the
edge
s’
weig
ht are
inde
pen
d
ent of ea
ch
ot
her. O
ne int
r
e
s
ting
situation
is that the
n
ode
deg
ree
and
weig
ht a
r
e often
co
u
p
led. S.H. Y
ook et al. p
r
opo
sed th
e
Yook–
Jeo
ng–Ba
r
a
b
á
si–T
u (YJBT
)
free scale weighted
net
work mo
del. The topology o
f
the model and
the edg
e we
ight distri
buti
on in the m
odel a
r
e g
e
n
e
rated
ba
sed
on prefere
n
t
ial attachme
nt
mech
ani
sm
d
u
ring
the
peri
od of
network gro
w
th [7
]. T
he YJBT
Mo
d
e
l con
s
tru
c
tio
n
process i
s
as
follows
:
Starting fro
m
o
m
isol
ated n
o
d
e
s, eve
r
y tim
e
when i
n
tro
d
uce
d
a
ne
w n
ode
j
we
ma
k
e
it con
n
e
c
tabl
e to
m
existin
g
node
s wh
ere
0
mm
.The p
r
ob
abili
ty that nod
e
j
is
c
o
nn
ec
te
d to
the existin
g
node
i
obeys BA Net
w
ork prefe
r
e
n
tial
con
n
e
c
tion
chara
c
te
risti
c
.
We
ca
n a
s
si
g
n
each edg
e weight of node
j
()
j
ii
j
ww
:
'
'
i
ji
i
i
k
w
k
(4)
YJBT co
uld g
enerate a fre
e
scale n
e
two
r
k
who
s
e d
e
g
r
ee di
strib
u
tio
n
is
()
~
Pk
k
and
3
.The network’s po
we
r law intensity dist
ribution i
s
()
~
s
R
ss
where th
e valu
e
s
of is
clo
s
ely relate
d to the choi
ce of
m
.
3.2 Zheng–
T
r
imper–Zhe
n
g
–Hui (Z
TZH)
Zheng
–Tri
mp
er–Z
hen
g–
Hu
i (ZTZH) is
the gene
rali
zed form of YJBT. It combine
s
rand
om
wei
g
ht allo
cation
mech
ani
sm
b
a
se
d o
n
the
node
de
gre
e
distrib
u
tion
[8
]. We
add
ne
w
edge
j
i
l
on the probability of
p
and on the p
r
oba
bility of
1
p
add ed
ge
j
i
l
weight:
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TELKOM
NIKA
Vol. 11, No
. 1, Janua
ry 2013 : 181 – 1
8
6
184
'
'
i
ji
i
i
w
(5)
whe
r
e
i
is th
e optimu
m
p
a
ram
e
ter
an
d is ra
ndoml
y
sele
cted
o
n
the
pro
bab
ility of
between 0 and 1
[17].When
1
p
,it is YJBT model.
Z
TZH
also lea
d
s to
po
we
r
law
intensity di
stribution’
s bei
n
g
()
~
s
R
ss
.
s
is
sensitiv
e to
p
and if
p
i
n
cr
ea
se
s f
r
o
m
0 t
h
e
n
s
falls down from 3 contin
u
ously.
3.3 Antal–Kr
apiv
sk
y
(AK)
Antal–Krapiv
sky (AK) p
r
op
ose
d
a stru
ct
ural g
r
owth
weighted net
work m
odel ba
sed o
n
edge
and
weig
ht cou
p
ling [9]. The model ge
ne
ration pro
c
e
s
s is a
s
follo
ws: Every st
ep a ne
w n
ode
con
n
e
c
ts to a
n
existing no
de
i
and the conne
ction p
r
o
bability is
i
ji
l
l
s
s
(6)
This p
r
eferen
tial conne
ctio
n rule is a strengt
h-pri
o
r ru
le. That is the new nod
e is alway
s
inclin
ed to conne
ct the n
ode with hig
her st
re
n
g
th. Many real
netwo
rks nat
urally use this
con
n
e
c
tion
m
e
ch
ani
sm.
Due to
the
co
nsid
eratio
n o
f
the b
and
wi
dth an
d flo
w
,
in Inte
rnet,
new
route
r
s
sho
u
l
d
be co
nne
ct
ed to central routers [1].
In the scientific rese
arch net
work, auth
o
rs a
r
e
more e
a
sily
to search
co
operating wit
h
ot
hers. Fo
r each edg
e
adding the
weig
ht whi
c
h
is
positive and
obedi
en
ce to distributio
n
()
w
,the final sha
pe of the network is the tree form.
Whe
n
the co
nne
cting process
last
s a long time and with
s
,the intensity distrib
u
tion
()
Rs
of the network app
roa
c
h
e
s a stable dist
ribution
3
()
~
R
ss
and is irrel
e
vant to
()
w
,the edge
weig
ht distrib
u
tion.
4. Model comparison an
d impro
v
ement
Above YJBT
，
ZTZH and
AK model
s
are
all ba
se
d
on g
r
o
w
th
mech
ani
sm.
Weig
ht is
assign
ed wh
en edg
e is fi
rst e
s
tabli
s
he
d and
kee
p
s unchang
ed.
These mo
del
s all ign
o
re the
dynamic evol
ution
cha
r
a
c
t
e
risti
c
s of th
e
edg
e
we
ig
ht
whe
n
n
e
w no
des an
d e
d
g
e
s
are a
dde
d
to
the net
work.
Not only th
at, the wei
ght
evolution a
n
d
edge
re
co
nn
ection i
s
a
common
natu
r
al
cha
r
a
c
teri
stics of
network. For
examp
l
e in
th
e avi
a
tion n
e
two
r
k, when
a
n
e
w
route
wa
s
establi
s
h
ed it would
affect
other route
s
traffi
c volum
e
. In such sit
uation an im
proved
wei
g
h
t
ed
netwo
rk m
o
del appe
are
d
. That is Barrat–B
a
rth
é
lemy–Ve
s
pi
gnani
（
BBV
）
model which
con
s
tru
c
ted
b
a
se
d on weig
ht dynamic e
v
olution
ca
used by the local netwo
rk
growth an
d ed
ge
reconnec
tion
[18-20]. It s
t
arts
from
0
m
nod
es
and th
e e
dge
weig
ht is
0
w
.Then every
step a
new node co
nne
cts
exi
s
tin
g
m
node
s o
n
the proba
bility as
sho
w
n in
formula
(7
), suppo
sin
g
the
origin
al wei
g
h
t
of
m
edge
s is
0
w
.
0
()
1
kK
Nk
K
kK
Pk
N
Kp
Kp
kK
kK
NN
(7)
Let
j
i
l
be a ne
w edge a
nd its
appe
ara
n
ce make
s
weig
h
t
of edges b
e
t
ween n
ode
i
and
its neigh
bor n
ode set cha
n
ge as follo
ws:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Survey
on W
e
ighted Networ
k Meas
u
re
me
nt and M
odeling
(
Xin Xia
)
185
i
l
il
il
i
ww
w
l
N
(8)
w
h
er
e
il
il
i
w
w
s
(9)
Whe
n
a new edge with the weig
ht of
0
w
is conne
ct
ed to node
i
it would ma
ke
comm
uni
cati
on traffic on
the othe
r ed
g
e
s of
i
incre
a
s
e by
.Duri
n
g
the process
each si
de’
s
increa
sed
a
m
ount i
s
p
r
opo
rtion
a
l t
o
the
edg
e
wei
ght a
n
d
p
r
odu
ce
s the
re
sult
o
f
0
ii
ss
w
in the end. Specifi
c
evoluti
on pro
c
e
s
s is sho
w
n in figu
re 2.
0
ii
ss
w
i
j
0
w
Figure 2. Edge weig
ht evolution pro
c
e
ss of BBV
Intensity and weight generated by BBV model
both obey exponential dist
ribution. In
orde
r n
o
t to
lose th
e g
e
n
e
ral
we
ca
n
make
0
1
w
and t
hen the
mod
e
l is
determi
ned by
para
m
eter
.After a few steps
we
coul
d get th
e weig
ht distribution a
s
()
~
w
Qw
w
,
21
/
w
and deg
re
e distrib
u
tion a
s
()
~
Pk
k
and inte
nsity distributio
n as
()
~
s
R
ss
,where
43
/
2
1
s
.This
res
u
lt s
h
ows that
BBV
c
o
uld generate
a free sc
ale
netwo
rk
who
s
e power la
w expone
nt
[2
,
3
]
.When
0
BBV is
t
he s
a
me as
B
A
.
5. Conclusio
n
and Futu
r
e
Works
Weig
hted n
e
twork
ha
s be
en an
active
and difficult re
sea
r
ch to
pic fo
r re
ce
n
t
years.
Some re
sea
r
che
r
s o
b
tain fairly good results. Bu
t there still exist so
me unresolve
d and sca
r
cel
y
addresse
d problem
s su
ch
as comm
uni
ty detecti
on ,con
se
nsus p
r
oblem an
d node impo
rtan
ce
evaluation
an
d so o
n
. At the same tim
e
,
some
ki
nd
s o
f
statistics
of
weig
hted n
e
tworks h
a
ve n
o
t
a unified defi
n
ition and the
i
r physi
cal
sig
n
ifican
ce
s ar
e not very cle
a
r. To solve su
ch qu
estio
n
s,
we
need
to l
ook for m
o
re
relia
ble m
e
thod to
de
scri
be the
wei
g
h
t
ed net
work.
More
over,
we
should try to find m
o
re effective methods to bu
ild weighted
network end-to
-end reli
ability in the
con
d
ition of n
ode failure.
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