Analysis on topological features of deterministic hierarchical complex network

Physica A 524 (2019) 169–176

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Physica A journal homepage: www.elsevier.com/locate/physa

Analysis on topological features of deterministic hierarchical complex network ∗

Kai Li a , , Wei Wu b , Yongfeng He a , Fusheng Liu a a b

Army Armored Forces Academy, Beijing, 100072, China Beijing Special Vehicle Institute, Beijing, 100072, China

highlights • • • •

We introduce a complex hierarchical network model through network module replication. The complex hierarchical network model satisfies the small-world effect and scale-free characteristics. The clustering-degree correlations of the complex hierarchical network model satisfies the power-law. The complex hierarchical network model can reflect the characteristics of many complex networks.

article

info

Article history: Received 6 March 2018 Received in revised form 25 September 2018 Available online 29 March 2019 Keywords: Deterministic hierarchical network Growth Preferential attachment Small-world effect Scale-free feature Hierarchical modularity

a b s t r a c t Real complex networks usually have small-world effect, scale-free features and hierarchical modularity, a construction method for deterministic hierarchical network is proposed in this paper. The network model with growth and global preferential attachment characteristic and connected by the copy network modules to establish hierarchical network model. By theoretical calculation and numerical simulation about the deterministic hierarchical complex network model, the results illustrate that the complex network model satisfy the small-world effect, scale-free feature and hierarchical modularity, the calculation results show that the size of the hierarchical network model for exponential growth with the network size increased, and the average degree of nodes is shown as linear growth; at the same time, the model of scale-free feature and the hierarchical modularity with network parameters do not have correlation which is an inherent attribute of network model itself; the clustering-degree correlations in the network model satisfy power-law characteristics and the nodes contact closely together in the modules which are connected by the Hub nodes in the complex network model. . © 2019 Elsevier B.V. All rights reserved.

1. Introduction Recently, more and more attention has been paid to the study of complex systems in the real world by using complex network theory [1,2], such as the transportation network [3] and mail network [4]. Complex network modeling is the key to study complex system problems. The main network models at present include ER random network model [5], WS small-world network model [6] and BA scale-free network model [7]. Further studies show that the small-world effect and scale-free characteristics are the common global topological properties of many real complex networks, but the partial ∗ Corresponding author. E-mail address: [email protected] (K. Li). https://doi.org/10.1016/j.physa.2019.03.111 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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characteristics of the networks with the same global characteristics are usually different, it indicates that it is important to research the partial topological properties of the networks. A module is a combination of nodes composed of a small number of network nodes which are made up of a certain topology and replicated in a large number of networks [8]. In essence, a module is a sub-graph with the specific topology that is repeated in a network and there are modules in both the food chain network [9] and the traffic network [10]. The network sub-graphs describe the connected way within the network from the partial network level, and the whole network satisfy hierarchical modularity from the global network level. Ravasz and Barabási analyzed hierarchical modularity in complex networks, showing that scale-free feature and hierarchical modularity are the basic features of complex networks [11]. Furthermore, according to the small-world network model and scale-free network model, scholars analyzed the network topology characteristics with random reconnection and local preferential attachment in different hierarchical network models [12,13]. Different hierarchical network models can be built based on the types of nodes and edges, such as P2P network [14] and medical knowledge network [15]. There are many ways to build the deterministic hierarchical network (DHN), the module size in the DHN network model proposed by Chen et al. [16] is fixed constant 2, but the DHN network is not a general network model. In this paper, we propose a DHN network model based on arbitrary integer m(m > 1) as module dimension. The small-world effect, scale-free characteristics and hierarchical modularity of the DHN model are theoretically analyzed and numerically simulated to research the influence of network parameters on DHN topological characteristics. The structure of the paper is shown as follows. In Section 2 we describe the DHN model established method in detail. In Sections 3–5, we discuss the small-world effect, scale-free features and hierarchical modularity of the DHN model by the simulation analysis. Finally, a brief conclusion is presented in Section 6. 2. Deterministic hierarchical network model The hierarchical network model is composed by connecting replications of the small network modules, and the features of global network is obtained by the topology structure and connections of partial network module. In a sense, the simplification of global network information to partial network information can be used to study complex networks performance and the dynamic evolution characteristics in a more simple way [17,18]. The steps of the DHN (n, m) model are provided as following principle (1) Initial time: Starting with a single node which is called the root-node of the network, and the root-node layer is n = 0. (2) Growth: We add nodes layer by layer and the amount of nodes is mn in layer n, where m(m > 1) is the number of sub-nodes that a father-node contains, the nodes in the last layer is called leaf-node. (3) Preferential attachment: The degree of nodes is n(n ≥ 1) in layer n which will connect the n father-nodes from layer 0 to n − 1. The parameter m can be seen as the increase rate of the network, Fig. 1 shows the topology structure according to the hierarchical modularity of the DHN(n, m) model, and the parameters in the DHN(n, m) model are n = 4 and m = 3. The colors of nodes in the DHN model is 5 which means that the layers is n + 1 = 5, and n = 0 is the first layer in the network model. The root-node is the center point of the whole network, and whole network exhibits hierarchical modularity obviously. The parameter m = 3 means that the nodes in the network are divided into 3 modules according to the edges between nodes. We assume that the layers of DHN (n, m) model is n + 1, because n = 0 is the first layer and the amount of nodes in layer n is mn which compose a geometric sequence, so the total number of the nodes N in the network model can be obtained as N = m0 + m1 + m2 + · · · + mn = 1 +

m(1 − mn ) 1−m

=

1 − mn+1 1−m

(1)

The number of edges E in the DHN (n, m) model is E = m1 + 2m2 + · · · + nmn

(2)

mE = m2 + 2m3 · · · + nmn+1

(3)

E(1 − m) = m1 + m2 + m3 · · · + mn − nmn+1 =

E=

nmn+2 − (n + 1)mn+1 + m (1 − m)2

nmn+2 − nmn+1 − mn+1 + m 1−m

(4)

(5)

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171

Fig. 1. The topology structure of complex deterministic hierarchical network.

So the average degree ⟨k⟩ of the DHN (n, m) model can be expressed as

⟨k⟩ = 2

E N

=2

mn+1 (nm − n − 1) + m (m − 1)(mn+1 − 1)

≈2

nm − n − 1 m−1

= 2(n −

1 m−1

)

(6)

The node vi−j means the jth(1 ≤ j ≤ mi ) node in layer i(0 ≤ i ≤ n), and the degree ki−j of the node vi−j is ki−j = kout + kin

(7)

where kout is the amount of father-nodes that connected to the node vi−j when the node vi−j is the sub-node, and kin is the amount of sub-nodes that connected to the node vi−j , when the node vi−j is the father-node. Then we can obtain the degree ki−j as follows

⎧ k =i ⎪ ⎨ out n−i ∑ ⎪ k = ma in ⎩

(8)

a=1

m(1 − mn−i )

ki−j = i +

1−m

(9)

The maximum and minimum value of the degree ki−j is

⎧ ⎨k

max

⎩

=

m(1 − mn ) 1−m

=N −1

(10)

kmin = n

where kmax is the root-node when i = 0 and kmin is the leaf-node when i = n.

3. Small-world effect Watts and Strogatz found that with the transition from the completely regular network to completely random network, adding partial randomness in regular network can produce a network model with small-world effect, which is called WS network model [6]. The WS network is created from the regular network, the nodes in the network are rewired by the rewiring probability p and the repeated connections and self-loop are excluded. In other words, small-world networks are part random and part regular, when the rewiring probability p = 0, the network correspond to the complete regular network and the network turn into the complete random network as p = 1 [19]. The WS network mainly contains two features, the smaller network average path length and the larger clustering coefficient. In this section, the small-world effects of the DHN (n, m) model are verified with the two features.

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Fig. 2. The relation between the average path length and network size in three network models.

3.1. Average path length In the DHN (n, m) model, we can find that the sum path length Li−j from the node vi−j to other N − 1 nodes in the network model can be expressed as Li−j = [i +

m(1 − mn−i ) 1−m

] + 2[

m(1 − mn ) 1−m

−

m(1 − mn−i ) 1−m

− i]

(11)

where the first item means that the path length is 1 that the node vi−j connectes its neighbour nodes directly, and the second item means that the path length is 2 with the remaining nodes, because all the remaining nodes could connect the node vi−j across the root-node. Eq. (11) can be rewritten as Li−j =

2mn+1 − mn−i+1 − m m−1

−i

(12)

Therefore, the average path length L in the DHN (n, m) model can be obtained as L=

=

=

1

1

2

( )

n ∑

N 2

i=0

1

1

[

2

( )

1

1

2

( )

=2

Li−j mi

−(m − mn+1 − nmn+1 + nmn+2 ) (m − 1)2

N 2

[

−

mn+2 − m − 2mn+1 (mn+1 − 1) (m − 1)2

2m2n+2 + (2n + 4) mn+1 − 2 (n + 1) mn+2

−

mn+1 (n + 1)

]

m−1

]

(m − 1)2

N 2

m2n+1 + (n + 2) mn − (n + 1) mn+1

(

mn+1 − 1 (mn − 1)

)

[ ≈2

m2n+1 m2n+1

+

n+2 mn+1

−

n+1

] (13)

mn

When the network model layer n → +∞, we can find that the average path length in the DHN (n, m) model is L ≈ 2. Fig. 2 shows the average path length L in the DHN (n, m) model, WS model and BA model, and the average path length L increase with the network size N in WS model and BA model. Whatever, the average path length L → 2 in the DHN (n, m) model means that the efficiency and connectivity are better than WS model and BA model, the simulation results show that the small-world effect of the DHN model is better. 3.2. Clustering coefficient The clustering coefficient Ci−j of node vi−j with ki−j edges in the DHN (n, m) model can be written as follows [20] Ei−j ) Ci−j = ( ki−j 2 where Ei−j is the amount of edges among the ki−j neighbors of node vi−j , and edges among the ki−j neighbors of node vi−j .

(14)

(

ki−j 2

) is the largest probable number of

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173

Fig. 3. The relation between clustering coefficient and network size in three network models.

There are three probable edges among the ki−j neighbors (1) different father-nodes of node vi−j connect directly E1 =

(

kout 2

)

= 21 i (i − 1)

(2) the father-node and sub-node of node vi−j connect directly in the DHN (n, m) model E2 = kin × kout =

im(1 − mn−i ) 1−m

(3) different sub-nodes of node vi−j connect directly E3 = m ×

(n − i − 1)mn−i+1 − (n − i)mn−i + m (1 −

m)2

=

m2 + [(m − 1)(n − i) − m]mn−i+1 (m − 1)2

(15)

So the clustering coefficient Ci−j of node vi−j in the DHN (n, m) model is Ci−j =

=

2(E1 + E2 + E3 ) ki−j (ki−j − 1) i(i − 1)(m − 1)2 + 2mi(1 − m)(1 − mn−i ) + 2[(m)(n − i + 1) − m] + 2m2 ki−j (ki−j − 1)(m − 1)2

(16)

The clustering coefficient of nodes in the same layer is equal, so the average clustering coefficient C of the DHN (n, m) model is C =

n 1 ∑

N

Ci−j mi

(17)

i=0

Fig. 3 shows the relation between clustering coefficient C and network size N in the DHN (n, m) model, WS model and BA model, the results indicate that the clustering coefficient C in the DHN (n, m) model is larger than the WS model and BA model, and the clustering coefficient C in BA model reduce with the network size N, there is no correlation between the clustering coefficient C and the network size N in the DHN (n, m) model and WS model. Although the network size N → 104 , the clustering coefficient C ≈ 1 in the DHN (n, m) model, it means that the DHN (n, m) model has large numbers of triangular topological structures and nodes are closely connected to each other to compose lots of clusters, so the whole DHN model has a strict organizational structure. In this part, we prove that the DHN model has the important characteristics of small-world effect from the two aspects of average path length and clustering coefficient. The average path length satisfies L → 2, and the clustering coefficient approximately C ≈ 1, indicating that the small-world effect of DHN (n, m) is better than WS network and BA network. 4. Scale-free feature Barabási and Albert pointed out that the real complex network usually has two important features: growth and preferential attachment, like the North American power grid. Growth means that the size of the network is expanding, and new nodes are constantly added to the network, preferential attachment mainly refers to the newly added nodes are connected mainly to the Hub nodes with higher degree, this phenomenon is called the Matthew effect [21]. The connection mode of the scale-free network follows preferential attachment, while the edges of the small-world network are random. The scale-free networks present the power-law degree distribution which means that the probability that the degree is k of a randomly selected node follows p(k) ∼ k−γ , and γ is the degree exponent. In nonmathematical terms, a scale-free

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Fig. 4. The degree distribution in the DHN model (a) the degree distribution in the DHN model and BA network (b) the degree distribution in the DHN model with different network parameters.

network is one model consisting of small number of high-degreed nodes and large number of low-degreed nodes, which results in a skewed degree sequence distribution. The degree of node vi−j is ki−j which means that the node vi−j in the layer i, and the probability p(i) that randomly select one node just in layer i is p(i) =

mi

N The degree distribution p(k) of the DHN (n, m) model satisfy two qualifications: (1) normalization:

∑n

i=0

(18)

p(k) = 1

(2) equivalence: p(k) = p(i), when k = i +

m(1−mn−i ) 1−m

Eq. (18) can be rewritten as p(i) =

mi (1 − m)

(19)

1 − mn+1

Then we can establish the p(k) to satisfy p(k) = p(i) as follows k−i=

m(1 − mn−i )

(20)

1−m

(k − i)−1 =

1−m

(21)

m(1 − mn−i )

So we can obtain the degree distribution p(k) of the DHN (n, m) model as follows p(k) =

mi+1 − mn+1

(22)

(k − i)(1 − mn+1 )

And the Eq. (22) satisfy the normalization N −1 ∑

p(k) =

k=n

n ( i+1 ∑ m − mn+1 i=0

1 − mn+1

×

(1 − m) m(1 − mn−i )

) =

n ∑ mi i=0

N

=

n ∑

p(i) = 1

(23)

i=0

So the degree distribution p(k) of the DHN (n, m) model is p(k) =

mi+1 − mn+1 (k − i)(1 − mn+1 )

(24)

Eq. (24) shows that the degree distribution is p(k) ∝ k−1 which decays as the power-law, so the DHN (n, m) model is a scale-free network. The degree distribution p(k) in the DHN (n, m) model and BA network described in Fig. 4(a) predict that

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175

Fig. 5. The curve variation of clustering-degree correlations in DHN model.

the slope of p(k) line is negative in log–log coordinate, the slope of the p(k) being γ = 1 in the DHN (n, m) model, which is smaller than γ = 3 in the BA network. Numerical simulations in Fig. 4(b) can indicate that the degree distribution p(k) in the DHN (n, m) model keep consistency with different network parameters, the scaling exponent of degree distribution p(k) is independent of the network parameters n and m in the model, and when the parameters n → +∞, the degree distribution p(k) satisfy power-law characteristics. 5. Hierarchical modularity Ravasz and Barabási considered the hierarchical modularity of complex network is the clustering-degree correlations follow the scaling law C (k) ∼ k−α , C (k) is the average clustering coefficient of the nodes with k links and α is the exponent [22,23]. In other complex network models, the random network and WS network are homogeneous networks, the degree distribution function p(k) satisfies Poisson distribution and there is no correlation between C (k) and k. For the BA networks, the degree distribution function satisfies the power-law distribution, but its clustering coefficient is approximate C (k) → 0, the results show that the BA network is not a hierarchical network. In this section, we will prove that the DHN (n, m) model possess hierarchical modularity. The clustering coefficient of nodes in the same layer is equal, and using Eqs. (9) and (16), the clustering coefficient of the nodes with k edges in the DHN model is Ci (ki ) =

=

{

1

i(i − 1) + 2i(ki − i) +

ki (ki − 1)

[

1 ki (ki − 1)

2m2 (m − 1)2

2ki i − i − i2 + 2(n − 2)(ki − i) +

+

2m m−1

}

2 (m − 1)2

[m + (ki − i)(m − 1)] [(m − 1)(n − 1) − m]

] (n − ki ) =

2n + 2i −

2m m−1

−2

ki − 1

+

2mn m−1

− (2n − 3) i − i2 ki (ki − 1)

(25)

When the network layer n → +∞, the second item is approximately equal to 0, then Ci (ki ) ≈

2n + 2i −

2m m−1

ki − 1

−2

(26)

The scaling law C (k) ∝ k−1 in Eq. (26) means that the DHN (n, m) model is one hierarchical network. Fig. 5 is the hierarchical analysis results of DHN (n, m) model under different network parameters, for different network parameters n and m, all the clustering-degree correlations satisfy power-law characteristics, it shows that the DHN model has hierarchical modularity and there is no correlation between network parameters and C (k). The scaling law C (k) ∝ k−1 means that with the increase of the nodes degree, the clustering coefficient of the nodes decreases gradually. The nodes with smaller degrees have larger clustering coefficients and belong to different network modules, the clustering coefficient of the Hub is smaller, and it only plays the role of the connection among the network modules. The results indicate that in the DHN (n, m) model, the internal nodes of the network modules are closely connected, but the different modules are loosely connected. 6. Conclusion Considering the characteristics growth and preferential attachment of the real complex network, make a copy of the network modules and connect to each other according to the preferential attachment, the algorithm is proposed to study the topology characteristics of the hierarchical network model. The theoretical analysis of the DHN model indicate that the DHN (n, m) network model is satisfied the small-world effect, scale-free feature and hierarchical modularity. The simulation results show that the average degree in the DHN (n, m) network increases linearly with the network parameter n and is independent of parameter m. The connectivity of DHN (n, m) network is better than WS network and BA network, the degree distribution p(k) and clustering-degree correlations C (k) satisfy power-law distribution. In the hierarchical

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network modules, the nodes are connected more closely, and the Hub node plays the role of the connection among different network modules. The hierarchical network model proposed in this paper is proved to satisfy the small-world effect, scale-free characteristics and hierarchical modularity through theoretical calculation. We expect further studies of the practical application about the DHN model in the real network. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant No. 41374068. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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