Durer-pentagon-based complex network
Rui Hou, Yuejiana Chang, Yuzhou Chang
College of Computer Science, South-Central University for Nationalities, Wuhan, People’s Republic of China
E-mail: [email protected]
Published in The Journal of Engineering; Received on 18th August 2015; Accepted on 22nd March 2016
Abstract: A novel Durer-pentagon-based complex network was constructed by adding a centre node. The properties of the complex network
including the average degree, clustering coefficient, average path length, and fractal dimension were determined. The proposed complex
network is small-world and fractal.
1
Introduction
Complex networks can describe many complex systems [1–3] and
have been widely applied in a variety of fields such as economics
and zoology. As such, they can directly affect societies and the
world [4–6]. Small-world networks, described by Warrs and
Strogatz [7], and scale-free networks described by Barabasi and
Albert [8], have had pivotal effects on the research of complex
networks.
Complex networks can be divided into two categories based on
the changeability of their topologies: random networks and definite
networks. The former have random topologies and the latter have
relatively stationary topologies [9–11]. Definite complex networks
have attracted significant attention for their predictability on an
expanding scale. Apollonian networks were the first definite
complex networks to be constructed by fractals [12]. Zhang et al.
[13] studied high-dimension definite Apollonian networks and proposed an incompatibility Sierpinski network with scale-free and
small-world properties [14]. Liu and Kong [15] researched Koch
curve networks. Le et al. [16] studied Sierpinski carpet-based
complex networks. However, these studies did not prove the
fractal nature of the networks studied, and the dimensions of
basic structures of the above models were small. Durer pentagons
integrate features of triangles and rectangles, achieving a high
fractal dimension; thus, they represent real complex networks
better than other models. We used the Durer pentagon as a basic
fractal structure to construct a novel definite complex network.
We analysed the constructed network’s average degree, clustering
coefficient, and average path length, and used different box dimension (BD) methods to calculate theoretical dimension and actual
dimension of the proposed complex network to prove its fractality.
2
Topologic properties of the Durer fractal pentagon
The Durer fractal pentagon can be constructed as follows [17]:
(A) Construct a basic regular pentagon. Then, build a new regular
pentagon along each edge of the basic regular pentagon outwards.
Thus, five new regular pentagons with the same size can be
constructed, creating a larger pentagon profile.
(B) Build a new larger pentagon profile along each edge of the previous larger pentagon profile outwards like step A. This results in
the second iteration structure.
(C) Iterate B using the latest set of pentagons to create new internal
pentagons to infinity. This results in a fractal structure with infinite
self-similarity.
As shown in Fig. 1, the vertices of the pentagons can be regarded
as nodes of a network, and the edges can be regarded as links
between nodes of the network. Thus, Durer pentagons can be
J Eng 2016
doi: 10.1049/joe.2015.0139
regarded as complex networks. The nth generation fractal Durer
pentagon is Dn.
In constructing Dn, the numbers of regular pentagons (pentagons
oriented in the same direction as D0) and inverse pentagons (pentagons and pentagon profiles oriented in the opposite direction as D0)
are both multiplied by five every iteration. In other words, NF(t) =
5NF(t − 1) and NUF(t) = 5NUF(t − 1), where NF(t) and NUF(t) are the
number of regular pentagons and inverse pentagons at the tth iteration, respectively. NF(0) = 1, NUF(0) = 0, NUF(1) = 1; therefore,
NF(t) = 5t, NUF(t) = (5t − 1)/4. Since each pentagon has five
nodes, and each node of each inverse pentagon is repeated once,
the total node number of Dn(n ≥ 0), Vn, is
3
5
Vn = 5 NF (n) − NUF (n) = × 5n+1 +
4
4
(1)
Similarly, since a pentagon has five edges, the total edge number of
Dn(n ≥ 0), En, is
En = 5NF (n) = 5n+1
(2)
Therefore, the main properties of the nth generation fractal Durer
pentagon Dn(n ≥ 0) are listed as follows:
(i) The clustering coefficient of Dn is 0.
(ii) The average degree of Dn is K = 8/3.
(iii) The diameter length of Dn is twice the diameter sum of the last
two iterations.
3 Properties of a Durer-pentagon-based complex network
3.1 Construction of a Durer-pentagon-based complex network
A Durer-pentagon-based complex network can be constructed by
adding a centre node at the nth generation fractal of a Durer pentagon and connecting the centre node to the vertices of all the smallest
inverse pentagons. As a result, the resulting new network graph is
called EDn.
Fig. 2 shows the construction process of ED1 and ED2. ED1 was
obtained by adding a centre node in D1, and connecting it to the five
vertices of the inverse pentagon. Similarly, ED2 was obtained by
adding a centre node to D2, and connecting it to the 25 vertices
of the five smallest inverse pentagons.
3.2 Properties of Durer-pentagon-based complex networks
3.2.1 Average degree: The number of nodes EVn and the number
of edges EEn in EDn are
EVn =
3
9
× 5n+1 + ,
4
4
(3)
This is an open access article published by the IET under the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0/)
1
Fig. 1 Iterative constructions process of a Durer pentagon
(iii) All other nodes have adjacent nodes with no possible edges, so
the clustering coefficient of those nodes is zero.
Therefore, the clustering coefficient of EDn is
(2/(5n − 1)) + (1/5) × 5n
⇒ lim C
n1
(3/4) × 5n+1 + (9/4)
n−1
n
2+5
· 5 −1
4
4
≃ 0.053
= × n+1
=
3
5
+ 3 · (5n − 1) 75
, C .=
Fig. 2 Complex network based on a Durer pentagon
From (6), the clustering coefficient of the proposed complex
network EDn is not zero.
EEn = 5n+1 + 5n .
(4)
The average network degree can be calculated as
,K . =
(6)
2 × 5n+1 + 5n
2EEn
⇒ lim , K . = lim
≃ 3.2.
n1
n1 (3/4) × 5n+1 + (9/4)
EVn
(5)
Therefore, when n is large enough, the network is a sparse network.
From (3) and (4), EEn = 1.6EVn − 3.6.
3.2.2 Clustering coefficient: Clustering coefficients describe the
nodes integration scenario in a network graph: they are key parameters to measure the collectivised degree of networks. If there
are ki nodes directly connected to node i, the maximum number
of possible edges in ki is ki (ki − 1)/2. The clustering coefficient Ci
for node i is defined as Ei/(ki (ki − 1)/2), where Ei is the real
number of edges in ki [7]. The clustering coefficient of a network
<C> is the mean value of all node clustering coefficients in the
network.
There are three types of nodes in EDn:
(i) The added centre node has degrees 5n and 5n connected edges,
so the clustering coefficient of this node is 2/(5n − 1).
(ii) The 5n nodes connected to the centre node each have degree
5. The number of possible edges in those nodes’ adjacent
nodes is 2; hence, the clustering coefficient of those nodes is 1/5.
3.2.3 Average path length: In a network, the length between two
nodes refers to the number of passing edges from one node to the
other one. The maximum distance between all nodes is called the
diameter of the network D = Max(Dij ), where Dij is the shortest distance between nodes i and j. The average path length L in a network
is defined asthe average shortest
distance between any two nodes,
written L = 1/N (N − 1)
i=j Dij , where N is number of nodes in
the network. From the topology of EDn, the longest distance
between two nodes in each generation is the distance between
any two diagonal nodes of the pentagon profile. Since the centre
node connects to all the smallest inverse pentagons, the longest
distance between two nodes is 6. Therefore, the diameter D of
EDn is 6, indicating EDn has little information transmission delay.
Any two nodes in EDn can have distances of 1, 2, 3, 4, 5, or
6. Table 1 shows the number of paths of each distance from the
nodes in detail.
From Table 1, we can calculate the total distance between nodes
of degree 2 to be (295 × 52n − 922 × 5n + 587)/8, the total distance
between nodes of degree 4 to be (74 × 52n − 1227 × 5n + 4285)/16,
the total distance between nodes of degree 5 to be 11 × 52n −
12 × 5n, and the total distance of nodes of degree 5n to be (29 ×
5n + 15)/4.
From the definition of L and (3), we find
L=
896N 2 − 12, 620N + 87, 321
16 × 15 × N (N − 1)
(7)
Table 1 Number of each possible distance between two nodes in EDn
Distance
1
2
3
4
5
6
Five initial vertices,
each with degree 2
Centre node
with degree 5n
5n Nodes connected to the centre
node, each with degree 5
Remaining (5n+1 − 5)/2
nodes, each with degree 2
Remaining (5n − 5)/4 nodes,
each with degree 4
2
2
5
5n + 2
2 × 5n − 6
(3 × 5n − 15)/4
5n
2 × 5n
(3 × 5n + 5)/4
0
0
0
5
5n + 3
2 × 5n − 4
(3 × 5n − 11)/4
0
0
2
5
5n + 1
2 × 5n − 4
(3 × 5n − 11)/4
0
4
4
9
5n + 4
2 × 5n − 12
(3 × 5n − 31)/4
This is an open access article published by the IET under the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0/)
2
J Eng 2016
doi: 10.1049/joe.2015.0139
Fig. 3 Relationship between average path length and log10 N
Fig. 5 Relationship between iteration n and dimension of the network dB
Table 2 Network dimension using box-counting method
3
5n
lB
NB
6
5n−1
15
5n−2
…
…
network diameter+1
1
Fig. 3 shows the relationship between the average path length in
EDn and log10N. L increases monotonically with log10 N, but
approaches a fixed value asymptotically after the value of log 10
N exceeds 3. As n → ∞
896 · N 2 − 12, 620 · N + 87, 321 56
=
≃ 3.73.
n1
16 × 15 × N · (N − 1)
15
(8)
lim L = lim
n1
If the value of dimB F is equal to dimB F, that value is the BD of F,
i.e. dimB F = lim log Nd ( F )/−log d.
d0
For EDn, the ratio between the edge size of pentagons of a given
iteration and the edge size of pentagons
of the previous iteration
√
obeys the golden cut, r = 1 − ( 5 − 1)/2 . Since each pentagon
in each iteration is divided into five small pentagons, by the kth
iteration, F has been covered by (5k + 1) sets with edge length
0.382k.
If 0.382k < δ ≤ 0.382k−1, then, Nδ(F) < < 5k + 1, so
log 5k + 1
log Nd ( F )
,, lim
≃ 1.67. (11)
d0 −log d
k1 −log 0.382k−1
dimB F = lim
If 0.382k+1 ≤ δ < 0.382k, then, Nδ(F ) >> 5k + 1, and
Therefore, since the clustering coefficient does not equal zero, and
the average path length is short, EDn is a small-world network.
3.2.4 Network dimension: First, we use BD method [18] to calculate the theoretical dimension value of the proposed complex
network.
To calculate the BD-based theoretical fractal dimension of EDn,
let F be a non-empty bounded subset in n-dimension Euclidian
space R n; Nδ(F ) denotes the least number of sets that can cover F
with a maximum length δ. The upper and lower BD of F can be
defined as
dimB F = lim
log Nd ( F )
−log d
(9)
dimB F = lim
log Nd ( F )
−log d
(10)
d0
d0
log 5k + 1
log Nd ( F )
.. lim
≃ 1.67.
k1 −log 0.382k+1
d0 −log d
dimB F = lim
(12)
Since dimB F equals dimB F, the theoretical dimension value of EDn
is 1.67.
Then, we use box-counting method [19] to calculate the actual
dimension of the proposed complex network. Using the boxes of
size lB to cover the network, to make the distance between any
two nodes in a box no greater than the given lB − 1, in which the
distance means the minimal number of edges from one node to
another node, then obtain the number of boxes as NB. For EDn,
we can obtain the following table (Table 2).
Fig. 4 gives the result of EDn with n = 2.
The fractal dimension dB of EDn can be calculated as (see
equation (13) at the bottom of the next page)
where
Fig. 4 When n = 2, using lB = 3 and lB = 6 to cover network
J Eng 2016
doi: 10.1049/joe.2015.0139
This is an open access article published by the IET under the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0/)
3
√ √n−1 √ √n−1 √
9+5 3 × 1+ 3
− 9−5 3 × 1− 3
/2 3
is the value of network diameter.
It can be seen from Fig. 5 that, dB approaches 1.60 with the increase of n, thus we can obtain the dimensions of EDn is 1.60
which is almost close to its theoretical value of 1.67. Therefore, it
can be concluded that EDn is fractal.
4
Conclusion
A Durer-pentagon-based complex network was constructed with fractal
and small-world characteristics. Through analysis and calculation, we
found that the proposed network had an average degree of 3.2, clustering coefficient of 0.053, average path length of 3.73, and a fractal
network dimension of 1.60. Compared with previously studied
pentagon-based complex networks, our proposed complex network
had a greater clustering coefficient and smaller average path length.
5
Acknowledgments
This study was supported by the National Natural Science
Foundation of China under grant no. 60841001; the Scientific and
Technological Projects of Wuhan, China, under grant no.
2015010101010008, and the Special Fund for Basic Scientific
Research of Central Colleges, South-Central University for
Nationalities, under grant no. CZW15034. The authors thank all
the reviewers for their useful comments.
6
References
[1] Steven H.S.: ‘Exploring complex networks’, Nature, 2001, 410,
(6825), pp. 268–276
[2] Newman M.E.J.: ‘The structure and function of complex network’,
SIAM Rev., 2003, 45, pp. 167–224
[3] Zhang Z., Zhou Y., Fang J.: ‘Recent progress of deterministic models for
complex networks’, Complex Syst. Complex. Sci., 2008, 5, (4), pp. 29–46
dB = log 5n /log
[4] Zhu D., Wang D., Hassan S., ET AL.: ‘Small-world phenomenon of
keywords network based on complex network’, Scientometrics,
2013, 97, pp. 435–442
[5] Jin X., Li J.: ‘Research on statistical feature of online social networks
based on complex network theory’. 2014 Seventh Int. Joint Conf. on
Computational Sciences and Optimization, 2014, pp. 35–39
[6] Shenela B.L.: ‘It’s a small world: fusion of cultures in genetic counseling’, J. Genet. Couns., 2012, 21, (2), pp. 20–208
[7] Watts D.J., Strogatz S.H.: ‘Collective dynamics of ‘small-world’ networks’, Nature, 1998, 393, (6684), pp. 440–442
[8] Barabási A.L., Albert R.: ‘Emergence of scaling in random networks’, Science, 1999, 286, (5439), pp. 509–512
[9] Barabási A.L., Ravasz E., Vicsek T.: ‘Deterministic scale-free networks’, Physica A, 2001, 299, pp. 559–564
[10] Comellas F., Ozón J., Peters J.G.: ‘Deterministic small-world communication networks’, Inf. Process. Lett., 2000, 76, (1–2), pp. 83–90
[11] Knora M., Škrekovski R.: ‘Deterministic self-similar models of
complex networks based on very symmetric graphs’, Physica A,
2013, 392, (19), pp. 4629–4637
[12] Andrade J.S., Herrmann H.J., Andrade R.F.S., ET AL.: ‘Apollonian networks: simultaneously scale-free, small world, Euclidean, space
filling, and with matching graphs’, Phys. Rev. Lett., 2005, 94, (1),
p. 018702
[13] Zhang Z., Comellas F., Fertin G., ET AL.: ‘High dimensional
Apollonian networks’, Physics A, 2006, 39, (8), pp. 1811–1818
[14] Zhang Z., Zhou S., Zou T., ET AL.: ‘Incompatibility networks as
models of scale-free small-world graphs’, Eur. Phys. J. B, 2007,
60, (2), pp. 259–264
[15] Liu J., Kong X.: ‘Establishment and structure properties of the scalefree Koch network’, Acta Phys. Sin., 2010, 59, (4), pp. 2244–2249
[16] Le A., Gao F., Xi L., ET AL.: ‘Complex networks modeled on the
Sierpinski gasket’, Physica A, 2015, 436, pp. 646–657
[17] http://www.mathworld.wolfram.com/Pentaflake.html, accessed August
2015
[18] Falconer K.: ‘Fractal geometry: mathematical foundations and applications’ (John Wiley & Sons, Inc. Press, 2007, 2nd edn.)
[19] Song C., Havlin S., Makse H.: ‘Self-similarity of complex networks’,
Nature, 2005, 433, pp. 392–395
√ √n−1 √ √n−1
√
9+5 3 × 1+ 3
− 9−5 3 × 1− 3
2 3 +1
This is an open access article published by the IET under the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/3.0/)
4
/
(13)
J Eng 2016
doi: 10.1049/joe.2015.0139