Journal of Software Engineering and Applications, 20 12, 5, 42-45
doi:10.4236/jsea.2012.512b009 Published Online December 2012 (http://www.SciRP.org/journal/jsea)
Dimension al Measurement of Com pl et e-connective
Network under the Condition of Particle’s Fission and
Growth at a Constant Rate*
Jinsong Wang, Beibei Hu
Department of management sci ence and engin eer ing, Bei jing University of Technology, Beijing100124, P.R. China.
Received 2012
ABSTRACT
We construct a complete-connective regular network based on Self-replication Space and the structural principles of
cantor set and Koch curve. A new definition of dimension is proposed in the paper, and we also investigate a simplified
method to calculate the dimension of two regular networks. By the study results, we can get a extension: the formation
of Euclidean space may be built by the process of the Big Bang’s continuously growing at a constant rate of three times.
Keywords: Particle’S Fis sio n; Regular Frac ta ls; the Complete-Connective Network; N etwor k Dimension
1. Introduction
The recent research shows that the fractal structure exists
in massive networks, and there are different methods to
calculate the fractal dimension of the network. Song et al.
[1-3] adopted the Box counting method and Burning al-
gorithm to measure the fractal dimensions of the net-
wor ks; Ki m et al . [4] also applied the Box counting method
to the measurement of fractal dimensions of scale-free
networks. However, bui lding the ne twork is stil l an under
explored field which has not yet been adequatel y s t ud ied ,
especially by using a fractal set to identify the network
and calculate its dimension.
In this paper, we attempt to make the further analysis
on the construction of the network in the perspective of
using the nodes of so me regular fractals to build the net-
works. The study is based on the theory of
self-replication algebraic space, and assumes the par-
ticle’s fission grow at a constant rate. To build the net-
wor k, we take the nodes from some regular fractals, such
as the end point of cantor set and the apex of modified
Koc h curve . T he st ud y result s su gges t tha t the j oi nt fu nc-
tion of the correlation and connectivity among network
nodes might influence the measurement of network di-
mensi on and sp ace str ucture . Thus, thi s work c ontrib utes
to re-measuring the dimension of the network from the
perspective of the joint function of correlation and con-
necti vi t y amo ng ne t wor k no de s.
2. Problem Introduction
If a particle A generates m particles after one fission, and
get m2 particles after twice fissions,

, there will be mn
articles after n times of fissio ns, and these n m particles
could form a space. In this space, the n m particles are
taken as the nodes, which mean the salient points, and
the network will form through the interaction among the
m particles. We use n
m
m
C to describe the total correlation
degr ee of t he net work, so the next step is to calculate the
dimension of the constructed regular network space. To
calculate the dimension from a new perspective of corre-
lation and connectivity among network nodes, three hy-
potheses are introduced:
1) Self-replication Space: Particle A generates N par-
ticles after n times fission. The particles are independent
mutually in the proceeding of fission, at the same proba-
bility, and the correlation degrees among these particles
are also the same with each other. T hus, the space formed
by the N particles is regarded as a Self-replication Space.
2) Network Connectivity: Regular fractals, such as the
cantor set and the Koch curve, describe the fragmented
and singular graphics which are self-similar with unli-
mited multi-levels. Following the principle of the cantor
set, and by using its end point as the nodes, correlation is
established among these nodes and then a complete-
connective network is formed, as shown in Figure
1-((a)(b)). And in Figure 1-((c)(d)), the paper rebuilds
the Koch curve based on its basic principles and then
takes the apex of the modified Koch curve as the nodes.
Finally, a complete-connective network is also formed
*P
roject supported by the Major Program of the National Natural
Science Foundation of China (Grant No. 71261026) National Key
Techn ology R & D Program of China (Gran t No. 201 2BAJ11B03)

Dimensional Measurement of Complete-connective Network under the Condition of Particle’s
Fission and Growth at a Constant Rate
43
thro ugh t he co rre lation among these kinds o f node s .
3) Net work tr ansfor ms based on the c antor set a nd the
Koch curve: It has been revealed that some of the regular
fractals have a feature of the particle’s fission at a con-
stant rate. For example, the end points of the cantor set
always grow at a constant fission rate of two times, and
the apex of the modified Koch curve with the constant
rate of four times. Therefore, on the premise of retaining
the topological feature of fractals, the networks built in
this paper are still based on the fundamentals of the can-
tor set and the Koch curve, and gain through the 2 or 3
times fission of particle A which still exists during the
fission. More specifically, we regard that all particles
after fi ss io n own t he homo ge n eo us mut ua l c or r el ati o n, so
the particles
A
,1
A
,2
A
, or
A
,1
A
,
2
A
,
3
A
will be gained
after the first fission of particle
A
,…, continuously, the
new particles will be generated after one more iteration,
and finally there are N particles after several fissions.
These N particles will form a complete-connective regu-
lar network. However, there are two additional assumed
cond itions: a) in o rde r to get the net work with sel f- simi-
larity from N particles, we first generate the network in
accordance with an identically independently-distributed
process with the same probability. (b)Additio nally, when
the N particles are taken as the network nodes to build
regular network, there is random distance between each
nodes.
3. A New Definition of Dimension and Mea-
surement of Network Dimension
Definition 1. We assume there are N nodes, and they
compose a network based on the interaction of the mul-
tiple nodes. The relevance
K
is used to describe the
corre-
lation degree of this network space, and its dimension
d
is represented as:
ln
lim ln
n
K
dN
→∞
=
. (1)
Theorem1. If particle
A
generates
m
particles 1
A,
2
A
, ···, m A (
2m≥
) during the time 1 t where the
correlation degree among these
m
particles is totally
1
=C
m
m
K
; and then after time
2
t
, each of the particle i A
(
,1, 2,,ij m=
) generates m particles
ij
mA
( )
,1, 2,ij m=
with
2
2m
m
KC=
,···, continuing to re-
peat to timen t, there are
=
n
n
Nm
particles from
A
with
n
m
nm
KC=
. Accordingly, the dimension d of the
regular network here is formed by the n N particles
generated from particle
A
duri ng time
n
t
:
ln
ln
lim lim
ln ln
n
m
nm
nn
n
C
K
dm
N nm
→∞ →∞
=== (2)
Proof: Mathematical induction is applied to solve this
problem.
Firstly, we have an explanation,
i
d
represents the
dimensions of network space composed by
n
i
particles.
(1) When
m
is equal to 2.
Through the first fission, there are 2 particles. Thus,
the total correlation degree 1 K is
2
2
C
.
Through the second fission, there are
2
2
particles.
Thus, the total correlation degree 2 K is
2
2
2
C
.

Through the
n
times fission, there will be
2
n
par-
ticles. Thus, t he total corr e la tion degree n K is
2
2
n
C
.
( )( )
2 -1
2
22-1
===22-1
2
n
nn
nn
n
KC
Figure 1. Fractal and comple te -connective net work graph.

Current Distortion Evaluation in Traction 4Q Constant Switching Frequency Con verters
44
According to the Equation (2), we can attain:
( )
( )
( )
-1
2
ln22 -1
ln
= limlim
ln ln 2
-1ln2+ ln2-1
= lim=2=
ln 2
nn
nn
nn
n
n
n
K
dN
nm
n
→∞ →∞
→∞


=
(2)
(2) When m is equal to 3.
Through the first fission, there are 3 particles. Thus,
the total correlation degree
1
K
is
3
3
C
.
Through the second fission, there are 2 3 particles.
Thus, the total correlation degree 2
K
is
2
3
3
C
.

Through the
n
times fission, there will be 3n par-
ticles. Thus, t he total corr e la tion degree n
K
is
3
3
n
C
.
( )( )()
3 -1
3
33 -1
===33-1 3-2
3
n
nn
nn n
n
KC
According to the Equation (2), we can attain:
( )()
( )
( )()
-1
2
ln33-1 3-2
ln
= limlim
ln ln3
-1 ln3+ln3 -1 +ln3 -2
= =3=
ln3
nn n
nn
nn
n
nn
K
dN
nm
n
→∞ →∞


=
(4)
(3) For the same reason, when m is equal to k. Through
the n times fission, the dimensions d of network space
composed by n k particles are equal to k.That’s to say,
=
k
dk
.
When
= +1mk
Then
( )
( )
+1
+1
=n
k
nk
KC
,
( )
= +1
n
n
Nk
First ly, we k now when
n→∞
,
( )
( ) +1 -1
..
+1
nk
as
n
Kk→
(5)
Therefore, we can attain:
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )( )
() ()
+1
+1
+1
+1
+1
+1
+1
ln
ln
= lim= lim
ln ln +1
lnln ln
= limlnln+1 ln
ln ln
lim ln+1 ln
ln+1 -1ln+1
= limln+1-1 ln
1
=1+=+1=
n
nn
n
n
n
k
k
n
kn
nn
n
k
k
kk
k
nk
k
k
k
nk
n
C
K
dNk
C kC
nk kC
kC
kkC
k nkk
kk nkk
k km
k
→∞ →∞
→∞
→∞
→∞
⋅⋅
⋅⋅
⋅
= ⋅⋅
⋅⋅ 

⋅⋅⋅



(6)
We have got the conclusion that, through n times fis-
sion, the dime nsion d of the network space composed by
the n m new particles is equal to m. The demonstration
is completel y shown in the Equa tion(2) .
4. Simple Calculation method of Dimension
of Regular Network Based on the New De-
finition of Dimension
As predicted above, two new particles can be generated
from only one particle, and each of the two new ones can
generate extra two particles, so the correlation could be
established among any two of these
2
2
particles to form
a complete-connective regular network(Figure 2-(a)).
According to the Equation (2), the dimension of the ne w
network here could b e calculated as
2
2
2
ln ln6 1.293
ln 22ln 2
C
d== ≈
.
Furthermore, if the newly generated
2
2
particles get
two more particles separately, the correlation could be
established among any two of the
3
2
particles, and the
dimension of this complete-connective regular network
could be calculated as
3
2
2
3
ln 2 ln71.602
3ln 2
ln 2
C
d+
= =≈
. To
get correlation among the fresh
4
2
particles split from
those
3
2
particles, then we can get another regular
network with complete-connectivity, its dimension can
be
calculated as4
2
2
4
ln 2 ln71.727
4ln 2
ln 2
C
d+ ==≈ ;···; To put it
briefly, the dimension of the complete-connective regular
network will increase gradually, as the number of newly
generated particles grows at a constant rate of 2 times.
When the number of particles becomes
2n
, and
n→∞
,
the complete-connective regular network has the dimen-
sion =2
d
.
Figure 2. (a) shows Complete-connective regular networks
built by particles growing at a constant rate of 2 times. (b)
shows complete-connective regular networks built by par-
ticles g rowing at a const ant rate of 3 times.

Dimensional Measurement of Complete-connective Network under the Condition of Particle’s
Fission and Growth at a Constant Rate
45
To consider the situation that one particle can produce
3 new particles, and each of the 3 new ones can have
other extra 3 particles, so totally there are 2 3 particles
and a complete-connective regular network based on the
correlation among these particles (Figure 2-(b)). Also
accord ing t o the d e fi niti o n o f di men sio n, we ca n me a sur e
the network with dimension
2
3
3
2
ln 2ln 2ln 212.016
2ln3
ln3
C
d+ = =≈
.
Another network will be formed from the fission of the
2
3, and the dimension of this network with totally
3
3
particles will be calculated as
3
3
3
3
ln 2(ln3 ln5)+ln132.421
3ln3
ln3
C
d+ = =≈
.
If we continue, there will be the network composed by
4
3
particles with dimension
4
3
3
4
ln 2.584
ln3
C
d= ≈
; ···; In
short, the dimension of the complete-connective regular
network will increase gradually, as the number of newly
generated particles grows at a constant rate of 3 times.
When the number of particles becomes 3n and
n→∞
,
the complete-connective regular network has the dimen-
sion
=3d
.
5. Conclusion
Based on the Self-replication Space and the principle of
the cantor set and the Koch curve, we establish a com-
plete-connective regular network, and propose a new
definition of the dimension due to the network correla-
tion:
ln
lim ln
n
K
dN
→∞
=
.
We identify that the number of network nodes estab-
lished by particles after fissions at a constant rate of m
times is
=
n
n
Nm
, and the original particle will not exist
after the fission. Furthermore, our examination suggests
that the total co rrelation degree among no des is
=
n
m
nm
KC
,
and as
n→∞
, the dimension
=dm
. Moreover, we also
give a simplified method for the measurement of the di-
mension of the two regular networks. T hen we can get a
extension: the formation of the three-dimension Eucli-
dean space may be built by the process of the Big Bang's
continuously growing at a constant rate of 3 times, to
some extent, this can provide more supportive explana-
tion and judgment on the causal relationship between
materials and space.
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