Applied Mathematics, 2013, 4, 1558-1562
Published Online November 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.411210
Open Access AM
Logistic Mapping-Based Complex Network Modeling
Xiaoling Yu1, Zhen Jia1,2, Xiangguo Jian3
1College of Science, Guilin University of Technology, Guilin, China
2Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin, China
3Department of Mathematics, Shanghai University, Shanghai, China
Email: moxin19890401@163.com
Received July 27, 2013; revised August 27, 2013; accepted September 5, 2013
Copyright © 2013 Xiaoling Yu et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, a new topological approach for studying an integer sequence constructed from Logistic mapping is pro-
posed. By evenly segmenting
0,1 into N non-overlapping subintervals which is marked as , representing
the nodes iden tities, a network is constructed for an alysis. Wherein the undirected edges symbolize their relation of ad-
jacency in an integer sequence obtained from the Logistic mapping and the top integral function. By observation, we
find that nodes’ degree changes with different values of
1, 2,,N
instead of the initial value—0
X
, and the degree distribu-
tion presents the characteristics of scale free network, presenting power law distribution. The results presented in this
paper provide some insight into degree distribution of the network constructed from integer sequence that may help
better understanding of the nature of Logistic mapping.
Keywords: Complex Network; Logistic Mapping; Power Law Distribution
1. Introduction
Graphical representations are widely used for displaying
relations among informational units because they help
readers to visualize those relations and hence to under-
stand them better. By representing individuals or organi-
zations as nodes and interaction or relation between them
with edges, complex networks can be constructed and
analyzed [1-3]. Examples include the Internet, World
Wide Web, social networks of acquaintance between
individuals, metabolic networks, food webs, etc. [4-8].
Recently, the general theory of complex dynamical
networks, which is an extension to the classical graph
theory, has been reconsidered for a better understanding
of the intrinsic relations, common properties and shared
features of possibly all kinds of networks in the real
world.
In this paper, a new topological approach is introduced
to analyze an integer sequence with sufficiently long
length generated by Logistic mapping and the top inte-
gral function. The integer sequence is used to construct a
network, from which its properties are studied under a
general complex network framework. We make research
on the degree distribution through computer simulation,
and come to the conclusion that the degree distribution
presents the characteristics of scale free network, pre-
senting po wer law distributi o n.
2. Network Modeling Based on Logistic
Mapping
2.1. Generating Nodes
Given a interval of
0,1
, 3,,
, we segment it evenly into
non-overlapping subintervals, each of which represents a
unique node identity in the network to be built, wherein,
the width of each subinterval is equal. In accordance with
the order from left to right, the subintervals are recorded
respectively as , so there are a total of
nodes in the network to be built, th e width of each subin-
terval is
N
N
1, 2N
1N, and the subinterval
-tkh
1,kNkN represents node k. For example, given
10N
, there are 10 nodes in the network. In detail,
evenly segment the interval of
0,1 into 10 non-over-
lapping subintervals, thus the width of each subinterval is
0.1. The subinterval—
0,0.1 re p res ents no de 1,
0.1,0.2 represents node 2,
0.2,0.3 represents node
3,, and
0.9,1 represents node 10. It should be noted
that, where the interval is left-open and right-closed. In
accordance with this principle, we get 10 nodes in the
network we studied, as is shown in Figure 1.
X. L. YU ET AL. 1559
Figure 1. Segment of [0, 1] and nodes generation.
2.2. Logistic Mapping
One-dimensional Logistic mapping from a mathematical
point of view is a very simple form of cha otic maps, a nd it
is given by (1). Here
11
nn
n
X
XX
 (1)
in which
0,4
is called Logistic parameter, and
wherein
0,1
n
X. Given different values of 0
X
and
, after several iterations, we will get different se-
quences. Studies have shown that the smaller the value of
4
is, the more evenly the corresponding sequence
distributed throughout the entire interval—
0,1 . More-
over, only when
0,1
n
X, the corresponding Logistic
mapping sequence generated by 0
X
is of non-periodic,
non-converge nce. If not, th e corresp onding seq uence will
converge at a certain specific value. In this paper, we
analyze the integer sequence with sufficiently long length
generated by Logistic mapping. Therefore, we only study
the particular situation, in which
0,1
n
X and
3.5
699456,4 .
2.3. Generating the Integer Sequence
First, computer generates a random number, which be-
longs to the interval—
0,1 , marked as 0
X
. Then, for
the given
, and the specific number of iterations,
which is denoted as , we use Logistic mapping itera-
tions to generate 1
Nm
12
,,,
n
X
,X
1,1 , which can be denoted
as k

,0,2,
X
kn, wherein k
X
means the
value of the Logistic mapping at the iteration.
Seen from above, if is given large enough, then there
are plenty of
-thk
m
k
X
in the corresponding sequence, namely
k

,,01,2,,1
X
kn is an integer sequence with
sufficiently long length.
It is noteworthy that, for given , each k
N
X
in the
sequence above corresponds to a unique node, which is
denoted as k. Based on the rule of node’s generation,
for each k
Y
X
, we can get its corresponding node in the
network, according to the subinterval it belongs to. For
example, given , m and 10N0.X70.798Xn
,
since m
X
and n
X
respectively belong to the two sub-
intervals—
0.6,0.7 and
0.7,0.8 , so Xm and Xn re-
spectively correspond to node 7 and node 8, namely,
and Y. 7
m
Y8
n
If we regard k
X
as the independent variable, k as
the dependent variable, then we can use the top integral
function to describe the correspondence relationship be-
tween them. The top integral function is the function that
its value is the smallest integer greater than the inde-
pendent variable or equal to it, is given by (2). Here
Y

min
kk k
YNX mZNXm  (2)
in which k
X
represents the value of the Logistic map-
ping at the iteration, k
Y represents the correspond-
ing node of k
-tkh
X
, means the total number of nodes in
the network and Z represents the set of integers. There-
fore, for any sequence of

N
,0,1,2,,
k1
X
kn, we
can use Equation (2) to obtain the corresponding integer
sequence—
Yk,0,1,2,,1n
k
, through which we
can get the rule of edge connection in the network we
constructed.
2.4. Generating Networks
In the network we constructed, whether two nodes are
connected or not, depends on their locations in
,0,1,2,,
k
Yk n1
. In detail, it’s determined by their
adjacency in the integer sequence above. For node k
X
,
it is just connected to its adjacent nodes—node 1k
X
and node +1k
X
. Note that the edge we studied is undi-
rected and unweighted.
For example, given 10N
, 3.8
, and X0
= 0.7, we got the integer sequence, as is shown in Table
1.
10n
Then we get the rule of edges’ generation as follow.
The first edge is (7, 8), the second one is (8, 7), the third
one is (7, 10) and th e fourth one is (10, 4),, and so on.
Namely, the first edge is connected between node 7 node
8, the second is also connected between node 8 and node
7, the third is connected between node 7 and node 10,
and so on. Wherein, reconnection is allowed. But in case
of 1kk
YY
, we skip over it, since it is not allowed that
a node can be connected to itself. Connect other nodes in
accordance with the rule above, and then we get the net-
work topology, as is shown in Figure 2.
3. Parameters’ Influence on Nodes’ Degree
It mainly refers to two parameters in the process of the
network’s generating, namely the initial value 0
X
and
Table 1. Integer sequence .
0 0.7 7
1 0.798 8
2 0.6125448 7
3 0.901867938 10
4 0.336308208 4
5 0.84817899 9
6 0.489331286 5
7 0.949567478 10
8 0.181978513 2
9 0.565676868 6
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X. L. YU ET AL.
1560
Figure 2. Network topology generated from sequence: 7, 8,
7, 10, 4, 9, 5, 10, 2, 6.
the Logistic parameter
. Seen by the nature of the cha-
otic sequence, slight difference between the initial values
will come to completely different chaotic sequences, and
it’s the same with the Logistic parameter. So, the com-
plex networks obtained by different initial values and
parameters are completely different. What about the n o de s ’
degree? By simulation, we find that the nodes’ degree in
various networks showed particular properties as follow.
3.1. Initial Value’s Influence on Nodes’ Degree
Given 3.9
, N = 200 and n = 1000, we study nodes’
degree influenced by 100 different initial values. We car r y
out a hundred of experiments through computer simula-
tion, since there are a hundred of different values of 0
X
.
In each experiment, the computer first generated a ran-
dom number—0
X
, then carried out the iterations of a
thousand times. Based on the result of the iterations and
the top integral function, we get the final inte ger seque nce
of nodes. Then we carry out the connection of adjacent
nodes according to the final sequence, and get the data of
nodes’ degree. The data is shown graphically in Figure
3. As is shown in the figu re ab ove, th ere a re 200 no des in
the network, wherein the horizontal axis represents the
200 nodes, while the vertical axis represents the 100 ex-
periments. In the right side of this figure, we use a color
bar to describe different values of the degree by different
colors. The color bar shows that when the color changes
gradually from blue to red, the corresponding value of
the degree also synchronously increases. Wherein the co lo r
of dark blue represents the value of the degree is zero,
namely the dark blue nodes in the figure are all isolated
nodes.
Figure 3. Nodes’ degree influenced by 100 different initial
values.
As is shown in Figure 3, in each of the 100 experi-
ments, the value of each node’s degree almost has no
change although 0
X
takes different values. In particular,
the number of the isolated nodes in the figure is almost
the same, which is the initial value has little effect on the
number of isolated nodes. It reveals that in the condition
of
is given; the initial value has little effect on the
nodes’ degree.
3.2. Logistic Parameter’s Influence on Nodes’
Degree
Given 200N
and 1000n
, we study the influence
on nodes’ degree caused by a total number of 100 dif-
ferent Logistic parameters. Since there are a hundred of
different values of
, so we carry out a hundred of ex-
periments through computer simulation, w 11iii


herein the value of
in the experiment is noted
as i
thi
, 1i100
. For each i
, we set, namely the
value of i
increases with the increasing of . And the
value of each i
i
belongs to the interval that we ob-
tained hereinabove—[3.5699456, 4].
In each experiment, the computer first generated a
random number—0
X
, then carry out a thousand of itera-
tions according to each value of i
. Based on the result
of the iterations and the top integral function, we get the
final integer sequence of nodes. Then connect the adja-
cent nodes according to the final sequence, and get the
data of nodes’ degree. The data is shown graphically in
Figure 4. Here in which the horizontal axis represents
the 200 nodes, while the vertical axis represents the 100
experiments. As is shown in the figure, in most of the
100 experiments, the value of nodes’ degree increase
with the increase of
. What’s more, in the whole net-
work, the total number of the isolated nodes decreases
with the increase of the
. It reveals that the larger
is, the better the connectivity of the network is.
From what has been discussed above, we come to the
conclusion that
have more influence on nodes’ de-
gree and the connectivity of the network than 0
X
, al-
though it has important influence on the corresponding
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X. L. YU ET AL. 1561
Figure 4. Nodes’ degree influenced by 100 different Logistic
parameters.
integer sequence. In detail, the larger
is, the less the
isolated nodes’ number is, and the better the connectivity
of the network is, while the initial value has little effect
on the nodes’ d egr ee.
4. Degree Distribution
On the basis of the conclusion above, we conduct further
research on the degree distribution by constructing six
typical networks. In the network we constructed, we need
to choose applicable parameters to decrease the number
of isolated nodes, since the less the number of isolated
nodes is, the better the connectivity is, and it is of little
significance to the study of degree distri bution.
Given N = 100,00, and the value of is respectively
as follow, 10,000, 50,000, 100,000, 500,000, 1,000,000,
and 5,000,000. Therefore, there are ten thousands of nodes
in the network, the total number of iterations is large
enough, and the result of iteration is the corresponding
integer sequence with sufficiently long length. Then we
get the figure of degree distribution in each of the net-
works above through computer simulation, and the result
is shown in Figure 5.
n
As is shown in Figure 5, in dual-logarithm coordinates
system, the degree distributions of all the networks above
approximate a straight line, presenting the characteristics
of power law distribution. It is similar to the degree dis-
tribution of scale free network. Therefo re, in terms o f the
degree distribution, the network we constructed from in-
teger sequence presents the characteristics of scale free
network.
The results presented above provide some insight into
distributions of the integer sequence that may help better
understanding of the nature of the Logistic mapping.
5. Conclusion
In this paper, we have studied the degree distribution of a
network which is constructed from Logistic mapping. It
is found that
has more influence on nodes’ degree
and the connectivity of the network than 0
X
. Further
Figure 5. Probability versus degree.
more, the degree distribution of the network shows pow er
law distribution, presenting the characteristics of scale
free network. The results presented in this paper also
provide some insight into how networks are constructed
from integer sequences, as well as how integer sequence
is generated from the Logistic mapping.
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