TELKOM
NIKA
, Vol.11, No
.11, Novemb
er 201
3, pp. 6787
~6
793
e-ISSN: 2087
-278X
6787
Re
cei
v
ed Ap
ril 26, 2013; Revi
sed
Jul
y
1
6
, 2013; Acce
pted Jul
y
31,
2013
Scaling Behavior and Phase Change in Complex
Network
Wei Ch
eng*
1
, Tanzhen
Hua
2
, Guiran Chang
3
1,2
Soft
w
a
re Co
l
l
eg
e of Northe
astern Un
iversi
t
y
, Shen
ya
n
g
, CHINA
3
Computi
ng Ce
nter of Northe
a
s
te
rn Univ
ersit
y
, She
n
y
an
g, CHINA
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: cheng
w
@
s
w
c.neu.ed
u.cn, t
anzh
@
s
w
c.ne
u.edu.cn
A
b
st
r
a
ct
Scali
ng b
ehav
i
o
r is a extre
m
ely typica
l ph
e
n
o
m
e
non
in co
mp
lex syste
m
researc
h
, as w
e
ll as it
can act
that ma
ny
M
a
cro i
ndic
a
tors
i
n
sy
stem or di
strib
u
tion
functi
on
of so
me
vari
a
b
les
meet
exa
c
tl
y
pow
er-law
b
e
h
a
vior, w
h
ich
p
o
ssesses
diffe
rent kin
d
s of
Expon
ents. In
this articl
e, ac
cordi
ng to
Ph
as
e
Cha
nge c
onc
e
p
t in Phys
ics, it is rese
arch
ed that th
e n
a
t
ure in cr
itical
state of co
mplex
netw
o
rk w
i
th
Seep
ag
e
mo
de
l, an
d it
is total
l
y
stated th
at th
e b
a
sic
r
easo
n
of Se
lf-simil
ar
beh
av
ior, F
r
act
a
l
beh
avi
o
r, an
d
so on, an
d als
o
Phase C
h
a
n
ge in co
mpl
e
x
netw
o
rk in
critical state of compl
e
x netw
o
rk in accord w
i
t
h
pow
er-law
distr
i
buti
on.
Ke
y
w
ords
:
co
mp
lex n
e
tw
ork, phase ch
an
ge
, no scalin
g net
w
o
rk, seepag
e mo
de
l, pow
er-l
aw
distributio
n
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Scaling
beh
a
v
ior is
a imp
o
rtant a
r
ea
i
n
co
mplex n
e
twork
no
w, su
ch a
s
no
scaling
netwo
rk i
s
ju
st a powe
r-l
a
w
Di
stributio
n
network, al
so, for example, the incom
e
in soci
ety is in
acc
o
rd with t
he famous
P
a
reto theorem, which is
i
n
com
e
de
nsit
y function (f(x)~x-1.7
5
), b
e
long
to powe
r-l
aw Distrib
u
tion
[1]. Another, among En
g
lish word
s, the freque
ncy
of occurren
ce of
word ‘
r
’ in th
e sequ
en
ce
of frequ
en
cy
of occu
rren
ce from largest to s
m
alles
t
is
f(r)~
r
-1,that is
Zipf theorem.
In addition, the relation
of two va
riab
le is power-l
aw, giving a
n
example that
orga
nism Metaboli
s
m and i
t
s Body Size meet 3/4
po
wer-la
w (F
~M3/4), call
ed
Kleiber theo
rem.
The last example illustrat
ed with
i
s
no scaling network
as well
-k
nown. Many
complex network in
the re
al
wo
rld totally mee
t
power-la
w
distrib
u
ti
on
(p(x)~x-3
)
.Tha
t is A la
rg
e
cla
s
s of
scali
ng
behavio
r [2, 3].
Then, a
nothe
r cl
ass of
scal
ing be
havior i
s
that fractal i
s
familia
r to u
s
. Cal
c
ul
ating
fractal
dimen
s
ion of
Fractal imag
es, in fact, there i
s
a kin
d
of power-la
w
relatio
n
be
tween Me
asu
r
e
Value ’Y’ of fractal
and A
c
cura
cy valu
e ’x’ of sca
le
, as y~x-D, a
nd po
wer-la
w D is its fra
c
tal
dimen
s
ion
as talked a
bove
.
Of course, i
t
is nor
m
a
l th
at scaling b
e
havior i
s
o
c
curred in
rand
om
fractal, like Browni
an Motio
n
and L
e
vy Flight [4-6]. Mo
re an
d mo
re
appe
ara
n
ces
of phen
omen
on
about scali
n
g
,
that makes rese
arche
r
s p
u
rsue to the n
a
ture.
What ki
nd of rese
arch is
origin
ated in
scaling be
ha
vior and po
wer-l
aw ph
eno
menon,
then? Of cou
r
se,
stri
ctly spea
king, in t
he er
a of cla
ssi
cal m
e
cha
n
ics, peopl
e
had di
scovered
power-la
w
ph
enome
non,
such a
s
the fa
mous
Grav
itation formula,
F~M1
M2/r2, i
s
a po
we
r-l
a
w
.
Ho
wever, th
e
r
e a
r
e t
w
o
so
urces ab
out t
hat the
wo
rd
‘
s
caling’ i
s
rea
lly mentione
d
and
a
relatively
large
-
scale rese
arch hav
e been d
e
vel
oped in p
h
ysi
cs. On
e is t
he Turbule
n
ce in liquid, when
peopl
e disco
v
ered that M
u
lti-scale
phe
nomen
a in
T
u
rbul
en
ce, which
wa
s surveyed by different
scale
s
, comp
letely sho
w
e
d
u
s
the
ana
logy re
gul
a
r
ity. And the ot
her
one
is P
hase
chan
ge
in
Statistical Ph
ysics, esp
e
ci
ally Phas
e ch
ange in
critical status [7
-8]
.
With the
discovery of
researchin
g o
n
co
mpl
e
x sy
stem in Ph
ase
ch
ange, li
ke
Phase
cha
nge b
eha
vior of a mag
net in high te
mperatur
e, m
any Accu
mul
a
tion of macro indicators can
give a seri
es of scalin
g b
ehaviors. Th
at is to
say, relation
s amo
ng so m
any indicators ca
n
be
carve
d
by po
wer-la
w, anot
her, the
syste
m
can al
so
show a l
o
t of similar be
havi
o
r to itself, while
clo
s
e to the
pha
se tra
n
siti
on point. So, critical
state
and scalin
g
behavio
r is o
ne of the mo
st
importa
nt bra
n
ch
es in
cla
s
sificatio
n
of Statistical Physics.
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Vol. 11, No
. 11, Novemb
er 201
3: 678
7 – 6793
6788
2. Percolation
Model
Creating
a gri
d
chart
consi
s
ts
of L*L l
a
ttices,
they will
be colored
by Probability p, whi
c
h
is the
colo
r of
every lattice
deci
ded
by o
ne Pro
bability
p. Whe
n
Pro
bability p ap
p
ears, the latti
ce
will be colored by black. T
he
cont
rary i
s
white. P i
s
0.4 and L is
10, arbitraril
y. As shown
in
Figure 1.
Figure 1. The
Grid Chart when p
=
0.4
Next, the bl
a
c
k lattice
s
wi
ll be
col
o
re
d
twice, an
d
some of
them
co
mmuni
cati
ng
with
each
othe
rs must
b
e
the same col
o
r u
nder
con
s
tr
ai
nt, but the op
posite i
s
diffe
rent. Fo
r the
nice
effect, comm
unicating latti
ce
s a
r
e all
co
lored
by bla
c
k, yet indep
e
ndent lattices is no
col
o
r.
The
two lattice
s communi
cate
with ea
ch oth
e
r mea
n
s th
ei
r edg
es a
r
e
conne
cted, not
points b
e
twe
en
lat
t
i
ces.
The
sam
e
colo
r
lat
t
i
ce
s con
nect
e
d wit
h
the othe
rs is call
ed
clu
s
t
e
rs.
Figu
re
2 is
obtaine
d by colorin
g
figure
above.
Figure 2. The
Commu
nicating Lattice
s
The figure ab
ove is achiev
ed in L=10
，
p=0.4. Fo
r a better situatio
n, it can be like figure
2.3, 2.4 and 2
.
5, if while L expand
to 100,
and p is 0.4, 0.6 and 0.7.
Figure 3. The
Grid Chart when p
=
0.4, p
=
0.6
Figure 4. The
Grid Chart when p
=
0.7
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TELKOM
NIKA
e-ISSN:
2087
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Scaling Be
ha
vior a
nd Pha
s
e Cha
nge in
Com
p
lex
Net
w
ork (Wei
Ch
eng)
6789
In these fig
u
res, the
bi
gge
st
cluste
r i
s
marked
by
black. As we
h
a
ve seen,
the
sm
aller p
is (p
<0.6
), the less these
cluste
rs i
s
. More
ov
er, they sho
w
a tende
ncy to not conn
ect with,
therefo
r
e, diff
erent
graysca
l
e
values ma
rked
the
clu
s
t
e
rs.
When
th
e big
ger p
is,
they tre
nd to
a
large
one. E
s
peci
a
lly, whil
e is 0.6, the
r
e
is the la
rg
e
r
clu
s
ter of th
e
m
. At the sa
me time, all kinds
of and so ma
ny types of clusters a
r
e be
ginnin
g
to take sha
pe, and
also a la
rge q
uantity.
3.
Seepage a
n
d Phase Ch
a
nge
3.1. Seepage
The so
-called
Seepage is
one large clu
s
ter in
sy
ste
m
can get through an
d permeate left
and ri
ght, or
up and
do
wn
bound
ary of
these latti
ce
s.
At that situation of p with
three differe
n
t
values
as
ab
ove, the clu
s
t
e
r is sm
aller
and n
o
com
m
unication
with each othe
r, whe
n
p i
s
0.4.
That i
s
to
say, there
is
no
see
pag
e
formatio
n
in
system. But,
wh
en
p i
s
0.6 or 0.7, t
h
e
phen
omen
on
of that cluste
r is mu
ch la
rger an
d
intercon
ne
cted is
obviou
s
. In
particul
a
r, whe
n
p
is 0.7, the se
epag
e occu
rs.
Therefore,
while p is f
r
om
0.4 to 0.7, t
he cl
uste
r turns fro
m
sm
al
l and
di
sco
nn
ected to
large
and int
e
rconn
ecte
d. That is the
seepa
ge
growi
ng out of not
hing. The
ph
enome
non i
s
so-
calle
d pha
se
cha
nge, which mean
s pha
se chan
ge ha
ppen
s while p is from 0.4
to 0.7 in system.
The natu
r
e of
system ha
s
cha
nge
d
duri
ng the proce
ss of ph
ase chang
e. It is illustrate
d
that the
see
pag
e of system mu
st be formed
wh
e
n
there i
s
a
critical p
r
ob
abi
lity Pc is equ
al or le
sser th
an
p.
3.2. Phase Change
T Fo
r g
e
tting
the value
of p
c
, when
calcu
l
ating
p
from
0.1 to 0.9,
th
e big
g
e
s
t si
ze of the
clu
s
ter is Sm
ax. P is as abscissa, and
the size
i
s
as ordinate. Be
cau
s
e
syste
m
is rand
om,
th
e
result is not
same to th
e di
fferent value
of
p. To
avoi
d ra
ndo
m di
sturban
ce,
en
semble
averag
e
can b
e
to execute. The
n
, in different si
ze
,
it shows the
Smax-p cu
rv
e, like Figu
re
5.
Figure 5. Cha
r
t the Biggest
Size of Clust
e
r as p
Cha
n
g
ing
Whateve
r
the
p is, all curve is monoton
ically
increa
si
ng. And also,
the more L is, the
steep
er the
curve i
s
. Esp
e
cially, whe
n
L is bi
gge
r (L = 15
0), it is happe
ned
that a sudd
en
cha
nge i
s
at 0.6. Some macro
state of
system sud
denly turn
s a
s
one vari
abl
e has
cha
ngi
ng,
whi
c
h is
calle
d cha
nge p
h
rase.
Acco
rdi
ng to
the Figure 5,
at the value
of p about 0
.
59, there is
a sud
den
ch
ange in
c
u
rve
when L is
at s
o
me point.
4. Scaling
Beh
a
v
i
or
Whe
n
p i
s
a
t
about critical point n
earby
pc, all
ki
nds
of scalin
g beh
aviors
will be
achi
eved (th
a
t
is power-la
w
behavio
r).
Fi
rstly, it is su
rveyed that
pro
bability
distrib
u
tion belo
ng t
o
the si
ze
s of e
v
ery clu
s
ters i
n
se
epa
ge
system. In
the clusters
in se
e
page system (every
differe
nt
colo
rs pa
rt), the si
ze
s a
r
e
extremely different. So
, to
get the dive
rsity, a si
ze of
one
clu
s
ter i
s
see
n
as a
ra
ndom p
a
ram
e
ter, and al
so it can get
p
r
oba
bility
distribution of th
e paramete
r
. The
figure below has
shown that different param
e
ters
can deci
de probability
distribution of clust
e
rs’
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e-ISSN: 2
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TELKOM
NIKA
Vol. 11, No
. 11, Novemb
er 201
3: 678
7 – 6793
6790
size. clu
s
te
rs’ si
ze i
s
a
s
a
b
scissa,
and
p is
as
ordin
a
te. Figu
re 6
sh
ows that
unde
r L
=
15
0,
probability distribution of cl
usters’
sizes
at different values of p.
Figure 6. The
Probability Different Clu
s
ters’ Si
ze
s
In figure
6,
e
v
ery data
poi
nt rep
r
e
s
e
n
ts, unde
r
a
fixed si
ze
x, there are
som
e
p
r
opo
rtion
of clu
s
ters in
Inter-cell x+dx. As we
sh
own, a
s
the
p is mo
re a
n
d
more to be
approa
chin
g
the
critical point 0
.
59, the distri
bution curve i
s
to
be a line
eventually. Double log
a
rith
mic co
ordinat
es
that is ab
scissa a
nd o
r
di
na
te are
both lo
garithmi
c
. Th
us, the lin
e m
ean
s that two
variable
s
are
to
meet the p
o
w
er-la
w
relati
onship. Unde
r p
=
0.58, th
e
distri
bution d
ensity fun
c
tio
n
of its
size
can
be marke
d
by p(x) =0.3
7*x
-1.72
.
In a word, the distrib
u
tion of system’s
si
ze is
po
we
r-l
aw at criti
c
al point as a co
nclu
sio
n
,
calle
d a scali
ng beh
avior.
Nea
r
by critical
state,
the large
r
clu
s
ter
in
se
epa
ge
system i
s
ju
st the
similar fracta
l
obje
c
t. For in
stan
ce, it can
be the bigge
st
one un
der
L=1
50, p=0.6
.
As Figure 7.
Figure 7. Grid
Figure u
nde
r L=15
0
，
p
=
0.
6
The bla
c
k clu
s
ter i
s
the big
gest on
e in
seepa
ge. To
shown cle
a
rly,
others is
col
o
red
by
light gray. Accordi
ng to t
he
cluste
r, it
is ex
tremely
simila
r to
ra
ndom t
r
aje
c
tory of
comm
on
Brownian m
o
tion. In fact, it is a rand
om
fracta
l g
eom
e
t
ry, which
ca
n cal
c
ul
ate th
e bla
ck
bigge
st
fractal dim
e
n
s
ion by box covering.
So-called bo
x covering i
s
so simpl
e
, that
is the dimensi
on is
covered by different
resolution
s b
o
xes. Th
en a
t
a fixed re
solution, it
will
be te
sted th
at the num
b
e
r of b
o
xes l(s)
need
ed i
s
as app
roximate
area
of thi
s
dime
nsi
on.
Next,
so
me smalle
r boxe
s
s’ are
u
s
e
d
by
coveri
ng the
s
e lattice
s an
d
a ne
w a
ppro
x
imate are
a
i
s
a
c
hieve
d
. Rep
eating th
e process, it can
get a
curve
depi
cted th
e rel
a
tion b
e
t
ween l
(
s)
a
nd s a
s
cha
nge of
different s. Fo
r t
w
o-
dimen
s
ion
a
l geomet
ry, like a circle, the
curve g
o
t by
this actio
n
is
also a p
o
wer-law on
e. That
is
l(s)~s-D,
and
D
= 2.
However, for fra
c
tal
geomet
ry, althoug
h a
po
wer-l
aw
can be
achieved, D is
less than 2 a
s
usual. So it is a fra
c
tal.
As belo
w
, the fractal dimen
s
ion of the re
d clu
s
ter is g
o
t by Box Covering, re
sult sho
w
n
by the Figure
8.
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TELKOM
NIKA
e-ISSN:
2087
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Scaling Be
ha
vior a
nd Pha
s
e Cha
nge in
Com
p
lex
Net
w
ork (Wei
Ch
eng)
6791
Figure 8. The
Fractal
Dime
nsio
n
It is the li
ne
unde
r p
o
wer-law th
at the
whol
e figu
re i
s
covered
by ch
angi
ng th
e si
ze
of
boxes, an
d al
so the sl
ope
of line, that is fractal di
me
n
s
ion, is 1.9
5
, less than 2. It is asse
rted th
a
t
it can get
a
smaller cl
ump
of fractal dim
ensi
on
wh
en
L is tested as a bi
gger val
ue. It will
refl
ect
that Complexi
ty of Clusters can d
e
viate far from
conve
n
tional ge
om
etry.
Thus, it can
be con
c
lu
ded
that clusters is a
similar fractal
stru
ctur
e unde
r criti
c
al state,
and that is a
scaling b
ehav
ior.
5. Renorm
a
liza
t
ion
Equatio
n
To any pe
rcolation mo
de
l, the param
eter t
hat de
cide
s its n
a
ture i
s
P, an
d every
reno
rmali
z
ati
on ope
ratio
n
make P
cha
nge a time
t
o
a pe
rcolation mod
e
l. It
assume
s that
in
original scal
e
S probability of black lattices i
s
P(
s), and after renormalizat
ion operation, the
scale
S becom
es S
’
, which is la
rger. Also
, the
P(s) chan
ge
into P(s’). Th
e probl
em is that what rel
a
tion
is between P(s) an
d P(s’).
s
P
f
s
P
(
1
)
That f, the function
nature
is lay in Coa
r
se G
r
aini
ng
rule, as
sh
own
Figure 3, tha
t
is wh
a
t
is provisio
n
of ignori
ng in
formati
on. Attention i
s
that
the left of
these rul
e
s i
s
the situatio
n
of
many bla
c
k l
a
ttices
occu
p
i
ed pa
rtly un
der
origi
nal
scale S
n
in fact. The right is the s
i
tuation of
new scale
S
n+1
occu
pied.
Then, the rel
a
tion of between
1
n
s
P
and
n
s
P
ca
n be cal
c
ul
ated
by the equati
on as follo
w:
2
2
3
4
1
1
2
1
4
n
n
n
n
n
n
s
P
s
P
s
P
s
P
s
P
s
P
(
2
)
4 po
we
r item
s of
n
s
P
is
th
e la
s
t
r
u
le in
co
r
r
e
s
p
o
n
d
s
w
i
th
on
es
(
t
ha
t me
an
s b
l
ac
k
lattices appear consecutively four ti
mes and the
probab
ility is obviously
4
n
s
P
i
n
original
scale), 3 po
wer item is the
situation tha
t
there
are th
ree bla
c
k lattice
s on left of rule, whi
c
h i
s
totally four rules
.
So the
co
efficient is fou
r
. T
he p
r
ob
ability of con
s
e
c
uti
v
ely appe
arin
g three time
s
and th
e
l
a
s
t
is
wh
i
t
e
la
tti
c
e
. T
h
a
t
is
n
n
s
P
s
P
1
3
. 2 p
o
we
r ite
m
s
co
rre
sp
on
ds
with that t
w
o
kind
s
of
two bla
ck vert
ical rul
e
s
con
necte
d, and the co
efficient
is 2.
By compa
r
in
g the three
situation:
wh
en P=
0.5,
0.6 and
0.7,
there i
s
m
u
ch la
rg
er
influen
ce i
n
ren
o
rm
aliza
t
ion e
s
pe
cial
ly P
=0.5
and
0.7. F
o
r in
stan
ce,
whil
e P=
0.5
,
reno
rmali
z
ati
on make pro
bability turn smalle
r and t
he colo
re
d lattices can be
more and m
o
re
spa
r
se.
Whe
n
P
= 0.7, th
e situ
ation i
s
oppo
site.
Oth
e
rwi
s
e,
these
two
situatio
n
s
can
affect
the
prop
ortio
n
of the biggest
clu
s
ter. Wh
e
n
P = 0.5,
th
e prop
ortion
of black lattices turn
small
e
r
quickly, and
whe
n
0.7 rapi
dly incre
a
se nearly to 1.
Howeve
r, in critical statu
s
, even if P= 0.
6 is simila
r to Pc, there is hardly influence in
the bigge
st cl
uster to reno
rmalizatio
n op
eration
to the
den
sity of black lattices. As Figure 8.
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Figure 8. The
Curve of the
Den
s
ity of Co
lored L
a
ttice
s from Ren
o
rm
alizatio
n Steps
In Figure 8,
the density of colored
lattices (o
ri
ginal bla
ck
one
s) chan
g
e
from
renormali
z
ati
on steps. In
these three
statuses, when only in
P =
0.6, the curv
e still stay same,
and in othe
r status, the den
sity is either l
a
rge
r
or
small
e
r.
Therefore, in
different stat
us, the re
sult
of
renormali
zation of pe
rcolatio
n mod
e
l is got,
then, and ho
w to cal
c
ulate
proba
bility Pc in
the critica
l
status by re
norm
a
lization
?
6.
Renorm
a
liza
t
ion and Fix
e
d Point
While the
sca
l
e of model t
u
rn
s from Sn
to
Sn+1 to e
v
ery operatio
n of Ren
o
rm
alizatio
n,
and the prob
ability is from
n
s
P
to
1
n
s
P
, which i
s
acco
rdin
g refer to (2). T
hen, iterative
equatio
n can
be a
c
hieve
d
,
which sho
w
s every
ren
o
r
mali
zation
o
peratio
n ma
ke bla
ck l
a
ttice
s
cha
nge. Diffe
rent initial probability ca
n deci
de the ev
olutiona
ry tra
c
ks of this ite
r
ative equatio
n.
Also Figu
re
7 shows track of
reno
rmal
i
z
ation equati
on in different initial
point
7
.
0
,
6
.
0
,
5
.
0
0
s
P
. It is seen that the curve
grad
ually de
clines
whe
n
5
.
0
0
s
P
. If original
lattice is infin
i
ty and reno
rmalizatio
n ke
eps
cont
in
ue
d, then, the curve ap
pro
a
ches to 0.
Wh
en
7
.
0
0
s
P
, the curve is near 1. Othe
rwise, when
6
.
0
0
s
P
, although it decline so sl
owly
, it
may be
ne
ar
0. In fa
ct, the
behavio
urs of
these
cu
rves are
compl
e
te
ly deci
ded
by
fixed poi
nts
of
reno
rmali
z
ati
on gro
u
p
s
. So-called the fi
xed point is a
spe
c
ial
*
s
P
.
2
*
2
*
*
3
*
4
*
1
2
1
4
*
s
P
s
P
s
P
s
P
s
P
s
P
n
(
3
)
Whe
n
N a
p
p
r
oa
che
s
infini
ty, we can g
e
t the equati
on on the to
p, and solving tchis
equatio
n, we
can g
e
t four a
n
swers of
*
s
P
.
5
1
2
1
,
5
1
2
1
,
1
,
0
(
4
)
Among th
ese
an
swers, th
e third
is omi
tted be
cau
s
e
of neg
ativity. The
r
efore, there
a
r
e
three
an
swers: {0, 1, 0.6
1
8
}, wh
i
c
h
are
thought
as t
he fixed poi
nt
s of reno
rmal
ization
equ
ation.
For the
case
of 0 an
d 1,
they are
call
ed a
s
tr
ivial f
i
xed point
s. They are correspon
ded to
the
den
sity 0(no
l
a
ttice)
and
1
(all a
r
e lattice
s)
of last latti
ce
s after
ope
ration
ren
o
rm
alizatio
n. Both
are
stable
attracto
r, which
is
0
s
P
initial. After re
no
rmali
z
ation
unlimit
ed op
eratio
n, they will
conve
r
ge
to t
he two attract
o
rs.
And
on
ce they
c
onve
r
ge, the
r
e i
s
n
o
e
s
cape.
Fo
r the
last
an
swer
0.618, it is non-trivial fixe
d point. The situati
on i
s
exactly fractal. That is to sa
y, when
0
s
P
is
about 0.61
8, reno
rmali
z
ati
on ope
ration
doe
s not affluent the see
p
a
ge gra
phi
cs.
So, the attrac
tor
*
s
P
is the
critical
poin
t
Pc that
we ne
ed to
find. Unde
r t
h
i
s
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prob
ability of critical poin
t, seepag
e
system i
s
wi
th scale inv
a
rian
ce.
Ho
wever it zoo
m
s
(re
normali
zati
on), it can be
got a simila
r system.
7. Conclu
sion
Percolation
model i
s
a
ki
nd of si
mple
rule m
odel, b
u
t its beh
avior is so com
p
lex and
inclu
d
e
s
man
y
critical
pha
se transitio
n
phen
om
en
a
and all
kind
s of scaling
b
ehaviors. On
e of
the Meth
od
s
is u
s
e
d
in
n
a
t
ures a
nd
de
tails of
so m
any compl
e
x network, the
r
eby, it
can
be
overall g
r
a
s
p
ed the funda
mental natu
r
e
of complex n
e
twork.
Acco
rdi
ng to
calculating
exactly, now,
it is
widely reco
gni
zed th
at Pc=0.5
93,
but it is
0.618 by me
ans of
reno
rmalizatio
n eq
uation. Altho
ugh it is
simil
a
r, it is yet not the sam
e
. The
rea
s
on i
s
that
cal
c
ulatin
g renormali
zatio
n
equ
ati
on i
s
an a
c
t of app
roximate o
p
e
r
ating, an
d n
on-
trivial fixed point is simil
a
r
to Pc. The
sources of
error mainly h
app
en in
Coarse
Graini
ng
rul
e
. In
that rule, it is see
n
ap
proximat
ely as
a
black lattice
whe
n
the n
u
m
ber
of bla
c
k lattice
s i
s
l
a
rge
r
than and
equ
al to three. A
nd wh
en it is
two, we o
n
ly
con
s
id
er that
it is a bla
ck la
ttice throu
gh
Up
and do
wn. Ever, the fact is that the approximat
e
operating ma
y destroy st
atus of origi
nal
clu
s
ters. If two
clu
s
ters
are n
e
a
r
by
on same lev
e
l, Coa
r
se
Graini
ng
rule
may igno
re
it.
Therefore, th
e error ap
pe
ars. If the operatin
g
mu
ch roug
he
r, only in acco
rdan
ce with t
h
e
majority pri
n
ciple, wh
en th
e numb
e
r
of origin
al bla
c
k
lattices i
s
la
rg
er than
3, it can be m
app
e
d
a
black lattice,
otherwise, white. At that situation,
we can get
a probability dev
iate from P
c
far.
In
oppo
site, if Coarse G
r
aini
n
g
is mu
ch finer, we
can ge
t better result.
Seepag
e mo
del is a
kind
of simple
rule
s, but it
s beh
aviour i
s
so
complex, even
includ
es
pha
se tra
n
sit
i
on and
criti
c
al ph
enom
e
na, and al
so
all kind of
scaling b
eha
viours. In two-
dimen
s
ion
a
l
percolatio
n
m
odel, ma
ny a
c
ts
ca
n expa
nd in
com
p
le
x netwo
rk. Al
though P
h
ysi
c
ist
can cal
c
ulate analytical sol
u
tions
of
two
-
dimen
s
io
n
a
l percolatio
n
problem
s, peop
le reali
z
e ha
rdly
to lots
of sca
ling ph
eno
me
non from a
traditional
poin
t
of view of ti
me an
d
spa
c
e. In additio
n
,
reno
rmali
z
ati
on is
a
simila
r a
c
t, but its e
n
try point
i
s
very de
ep. It can g
r
a
s
p the
nature
of scal
ing
behavio
ur, scale invaria
n
ce.
That is al
so
fractal
cha
r
acteri
stics
an
d sc
al
e inva
riance of
syst
em. Wh
atever initial
dynamics
rul
e
s of system
is, wh
atever se
epa
ge
mo
del o
r
I
s
ing
model
is, as long as
sy
stem
turns to th
e
critical
statu
s
,
whe
n
it
can
prod
uc
e all
ki
nds of
scalin
g be
haviou
r
s and
omit
so
me
rest
rictio
ns
of
dynami
c
s rul
e
s, it can co
mplete
ly portray scalin
g be
haviour n
e
wl
y from the point
of view
of the
re
normalization e
quatio
n.
So, ren
o
rm
ali
z
ation
meth
o
d
is likely to
b
e
a
ne
w
starti
ng
point, rathe
r
than a sim
p
le
techni
cal me
ans.
Referen
ces
[1]
Barab
a
si AL. B
ona
be
au E. Scal-free
net
w
o
rk
s. Scientific American. 2
003;
50-59.
[2]
Ne
w
m
a
n
MEJ.
T
he structure and fu
nctio
n
of net
w
o
rks.
C
o
mputer P
h
ysi
cs Co
mmu
n
ica
t
ions
. 20
02
;
147: 40-
45.
[3]
F
a
loutsos M
F
a
loutsos C.
On po
w
e
r-
la
w
relatio
n
shi
p
s
of the Internet
topolo
g
y
.
ACM SIGCOM
M
Co
mp
uter Co
mmu
n
ic
ation R
e
view
. 1999; 29(
4): 251-2
62.
[4]
Pastor-Satorra
s R Vespi
gna
ni A. Epid
emic
s d
y
namics
an
d en
d
y
nam
ic states in com
p
le
x n
e
t
w
or
k
.
Phys Rev. E.
2001; 63: 0
661
1
7
.
[5]
Pastor-Satorra
s R. Vespi
gna
ni A. Epi
d
mics
and
immun
i
zi
ation
in sca
l-free n
e
t
w
orks. In Ha
ndb
ook
o
f
Graphs and Net
w
orks, Bor
nholdt S. Schuste
r H.G. (etc.)
, WILE-YVCH publisher. 2003.
[6]
Pastor-Satorra
s R. Ves
p
ig
na
n
i
A. Ep
id
emics
d
y
nam
i
cs in finite siz
e
scale-fr
ee
net
w
o
rks.
P
h
y
s
.
Rev. E.
200
3; 68:03
51
03.
[7]
Coh
en R, H
a
vl
in 5. Prob
ab
ilis
tic Predictio
n
i
n
Scal
e- Free
Net
w
orks: Di
a
m
eter Cha
n
g
e
s
. Ph
y
s
. Rev.
Lett.
2003; 90:
058
70
1.
[8]
Andre
a
C, Al
essio C, Iren
e G
，
Giorgio
P, S Raffaele, F
abio
S, MassimnilianoV.
Scale-free
correlati
ons i
n
starlin
g flocks, PNAS. 2010; 1
07(2
6
): 118
65.
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