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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory

Xu Zhang
• Corresponding author
• School of Traffic and ransportation Engineering, Dalian Jiaotong University, Dalian 116026, China
• Email
• Other articles by this author:
/ Bingzhi Chen
Published Online: 2017-03-01 | DOI: https://doi.org/10.1515/phys-2017-0001

## Abstract

Complex Network Theory can analyze the reliability of high-speed passenger traffic networks and also evaluate node importance. This paper conducts a systematic and in-depth research of importance of various nodes in the high-speed passenger traffic network so as to improve the high-speed passenger traffic network level. To study importance of network nodes can contribute to an in-depth understanding of the network structure. Therefore, the complex network is introduced and the node importance is evaluated. The characteristics of the complex network are briefly analyzed. In order to study the high-speed passenger traffic nodes, the network restraint coefficient, the network scale, the efficiency, the grade level, the partial clustering coefficient of degree and structural hole. Besides, the algorithm to calculate node importance is designed. Through analysis of the high-speed passenger network, the accuracy and practicability of the Complex Network Theory in evaluating node importance are pointed out. It is also proved that Complex Network Theory can help optimize high-speed passenger traffic networks and improve traffic efficiency.

PACS: 89.75

## 1 Introduction

Development of high-speed passenger traffic is the key to a country’s prosperity and a symbol of a country’s national comprehensive strength. In recent years, high-speed passenger traffic has entered a leapfrog developmental period, thus contributing to increasing improvement of the high-speed network. Therefore, it has been a research focus to study the complex high-speed network so as to guarantee reliability invulnerability of the complex network. Research suggests that networks of different topological structures show different degrees of invulnerability towards different methods. Compared with the stochastic network, the scale-free network has a higher degree of robustness towards stochastic, but is vulnerable towards a calculated attack. “Combination of robustness and vulnerability”—this is one of the many basic and important characteristics of the complex network. Thus, it is necessary to find key nodes. In this way, not only can the reliability of the whole network be improved through protection of key nodes, but also the high-speed passenger network system can be scientifically evaluated, thus providing suggestions for maintenance and optimization of the network system, and bases for macroscopic decision-making of the whole high-speed network system.

Foreign scholars have studied the complex network for a long time [1], having put forward the WS model and the BA model [2], and revealing the small-world effect and the scale-free attribute of the complex network [3, 4]. All these have provided a brand-new perspective for network development [5, 6]. Later, the complex network analysis was introduced to multiple fields, including mathematics and sciences [7, 8], life science and engineering science. Now, it has become a major analysis method of network research. The traffic network has significant time and space complexity, and bears most characteristics of the complex network [9]. Study on the traffic network based on the Complex Network Theory and Mechanism has drawn attention of an increasing number of scholars [10]. Based on analysis of topological attributes of the real traffic network [11], researchers have verified that the traffic network, the railway network, the urban traffic network and the civil aviation network are all typical complex networks. Based on the node deletion method, the node shrinkage method and the importance evaluation matrix, researchers have also ranked importance of traffic complex network nodes. However, most of them analyzed node importance from the perspective of a singular characteristic index of the traffic network, but ignored interplay among multiple attributes and characteristics. In view of this research gap, this paper applies the Structural Hole Theory to importance analysis of China’s high-speed passenger traffic network nodes, and ranks node importance based on multiple attributes and characteristics.

## 2.1 Small-world network model

In 1998, in order to realize the transition from the completely regular network to the completely stochastic network, two American scholars, Watts and Strogatz, designed a small-world network with a small average path length and a large clustering coefficient, and called it WS small-world network model.

Below is the construction algorithm of the SW small-world network model: (1) Start from the regular network: Assume that there is a nearest-neighbor coupling network with N nodes, and that the nodes form a ring. Every node is connected with K/2 nodes near to it on the left and right, and K is an even number; (2) Stochastic reconnection: Stochastically connect to every side of the network with the probability p. In other words, one endpoint of the side is maintained unchanged, and the other end is a node stochastically chosen in the network. It is regulated that there should be one side at most between every two different nodes. Besides, every node cannot be connected with any side.

In order to guarantee the sparsity of the network, it is required N ≻≻ K. The network model thus built has a high clustering coefficient. The stochasticized reconnection process greatly reduces the network’s average path length, and the network model has a small-world characteristic. When the p value is small, the reconnection process has a slight influence on the network’s clustering coefficient. When p = 0, the model is degenerated into a regular network. When p = 1, the model is degenerated into a stochastic network. Through adjustment of the p value, the transition from the completely regular network to the completely stochastic network can be controlled. The clustering coefficient and the average path length of the WS small-world network model can be regarded as the function of reconnection probability, p, which are written as C(p) and L(p), respectively. Within the value scope of certain p, the WS network model can guarantee its short average path length (small-world characteristic) and a high degree of aggregation (high aggregation characteristic).

During the stochastic reconnection process of the WS small-world network mode, the network connectivity might be destroyed. In order to avoid isolated subnet caused by reconnection, American scholars, Newman and Watts, put forward the small-world network featuring the “stochastic bordering” to replace the previous one featuring the “stochastic reconnection,” and the new one was called the NW small-world network. Below is the construction algorithm of the NW small-world network model:

1. Start from the regular network: Assume that there is a nearest-neighbor coupling network with N nodes, and that the nodes form a ring. Every node is connected with K/2 nodes near to it on the left and right, and K is an even number;

2. Stochastic bordering: Add a side to the middle of a pair of nodes stochastically chosen at the probability of p It is regulated that there should be one side at most between every two different nodes. Besides, every node cannot be connected with any side.

When p = 0, the model is degenerated to a regular network; when p = 1, the model is degenerated into a stochastic network. Through adjustment of the p value, the model can be controlled to transit from the completely regular network to the completely stochastic network.

1. Degree of aggregation

Degree of aggregation of the WS small-world network [12]: $C(p)=3(k−2)4(k−1)(1−p)3$(1)

Degree of aggregation of the small-world network: $C(p)=3(K−2)4(K−1)+4Kp(p+2)$(2)

2. Average path length

Up to now, none have obtained an accurate analytical expression for the average path length of the WS small-world network model. Newman, Moore and Watts obtained the following approximate formula through the renormalization method and the sequence expansion method, respectively: $(p)=2NKf(NKP/2)$(3)

Where, f(u) stands for a universal scale function, and meets the following condition [13]: $Lf(u)=constu≺≺1(ln⁡u)/uu≻≻1$(4)

Up to now, there has not yet been an accurate analytical expression for f(u). Based on the mean field method, Newman et al. provided the following approximate expression: $f(x)≈12x2+2xarctan⁡hxx+2$(5)

3. Degree distribution

In terms of the WS small-world network, when kK/2, then $p(k)=∑n=0min(k−K2,K2)CnK/2(1−p)npK2−n⋅(pK/2)k−K2−n(k−K/2−n)!e−pK2$(6)

When kK/2, P(k) = 0. In terms of the NW small-world network, the degree of every node should be at least k [14]. Therefore, when kK, the probability of a stochastically chosen node, whose degree is k, is: $P(k)=Ck−KNKpNk−K1−KpNN−k+K$(7)

When kK, P(k) = 0.

To some up, the degree distribution of the ER stochastic network, the WS small-world network and the NW small-world network can be approximately expressed by the Poisson distribution. The distribution has a peak value at the mean value < k > of the degree, and then undergoes rapid exponential decline. The type of networks is called the homogenous network or the exponential network [15].

Table 1

Comparison of attributes of the small-world network, the scale-free network and the real network

## 2.2 Scale-free network model

In recent years, a large number of empirical researches have suggested that the degree distribution function of many large-scale real networks (such as WWW, Internet and metabolism network) all feature the power-law distribution, namely P(k) ∝ kγ. In such a network, most nodes have a small degree, but there are some nodes with a large degree and without a characteristic scale. Networks of this type whose node connection degree shows no significant characteristic scale are called scale-free networks. In order to explain the generation mechanism of the power-law distribution in the real network, Barabási and Albert put forward a scale-free network model in 1999, which is called the BA scale-free model. The construction of the model is mainly based on two internal mechanisms of the real network: (1) Growth mechanism: Most real networks are an open system. Along with the passage of time, the network scale keeps expanding. In other words, the number of network nodes and sides keeps on increasing; (2) Preferential connection: The newly-increased nodes prefer to connection with nodes with a higher connection degree [16].

Below is the construction algorithm of the BA scale-free network model:

1. Growth: In the initial moment, it is assumed that there have been m0 nodes. In every follow-up time step, a node with the connection degree of m(mm0) is increased, and the newly-increased node is connected with m different nodes. There is no repeated connection [17, 18].

2. Preferential connection: During the selection of the connection point for a new node, the probability of a new node to be connected with an existing node, i, is Πi, which is in direct proportion to the degree, ki, of the node, i.

$∏i=ki∑jkj$(8)

After t steps, the algorithm can generate a network with N = t + m0 nodes and mt sides [19].

1. Average path length

The average path length of the BA scale-free network: $L∝log⁡Nloglog⁡N$(9)

This indicates that the BA scale-free network also has the small-world characteristic.

2. Coefficient of the aggregation degree

Below is the aggregation degree of the BA scale-free network: $C=m2(m+1)24(m−1)lnm+1m−1m+1ln⁡(t)2t$(10)

Similar to the ER stochastic network, when the network scale is large enough, the BA scale-free network will show no significant clustering characteristic.

3. Degree distribution

There are three approaches to working out the degree distribution of the BA scale-free network: a. Mean-Field Approach; b. Master-Equation Approach; c. Rate-Equation Approach. The approximate results obtained by the three approaches are the same. Among them, results obtained by the Master-Equation Approach and the Rate-Equation Approach are equal in value. Through analytical calculation, $P(k)=2m(m+1)k(k+1)(k+2)∝2m2k−3$(11)

This suggests that the degree distribution of the BA scale-free network can be approximately described by the power-law function whose power exponent is 3.

## 3 Selection of indexes to evaluate the network node importance of the high-speed passenger network

Based on some indexes of degree and structural hole, such as the network restraint coefficient, the network efficient scale, efficiency and grade, the contribution of various passenger stations towards node importance under constrictions of various indexes is analyzed.

## 3.1 Network degree

The degree of Node i is defined as the number of neighboring nodes of the node. Below is the specific expression: $ki=∑j∈Gaij$(12)

Degree can reflect the degree of direct influence of a node on other nodes. The higher the numerical value is, the more important it is in the network. For example, in a high-speed passenger traffic network, a high-speed passenger station has 50 stations connected with it, then the degree of the high-speed passenger station is 50. Under general conditions, the more the neighboring nodes a high-speed passenger station has, the greater the influence of the high-speed passenger station, the larger its scale is, and the more important the node is.

## 3.2 Network restraint coefficient

A network restraint coefficient is used to evaluate the degree of reliance of one node on other nodes. The higher the network restraint coefficient is, the stronger the restriction is, the smaller the structural hole is and the higher probability for it to become the central node. The degree of constraint is shown below: $Cij=(Pij+∑qPiqPqj)2$(13)

In the above equation, Node q is the shared neighboring node of Node i and Node j; Pij stands for the proportion intensity of Node j on all connecting nodes of Node I; stands for the indirect investment of Node i on Node j. For example, in the high-speed passenger network, Passenger Station q stands for the shared connection node of Passenger Station I; Pij stands for the proportion of Passenger Station j among all connecting stations of Passenger Station i. The total restraint coefficient of Node i is: $Ci=∑jCij$(14)

Grade is used to describe the concentration degree of node restriction. The higher the grade is, the more likely it is within the scope of the node, and the more likely the restraint concentrates on the node. Below is the calculation formula: $HIi=∑jCijC/NlnCijC/NNln⁡N$(15)

Where, N stands for the number of all nodes; C stands for the restraint coefficient of nodes.

## 3.4 Network scale

Network scale is used to describe the general influence of nodes, which can measure the importance of structural hole’s nodes to some extent. Below is the calculation formula: $ESi=∑j(1−∑qPipPqj)=n−1n∑j∑qPjq$(16)

Where, n stands for the degree of Node i; j stands for the neighboring nodes of Node i; q stands for the shared neighboring nodes of Node i and Node j; Pip and Pqj stands for the proportion of Node q among the neighboring nodes of Node i and Node j.

## 3.5 Efficiency

Efficiency is used to stand for the degree of influence of nodes on other relevant nodes in the network. Under general conditions, nodes in the structural hole have a higher efficiency. Below is the calculation formula: $EFi=ESin$(17)

n stands for the number of nodes. When the network is fully connected, the efficiency is 1; otherwise, the efficiency is 0.

## 3.6 Local clustering coefficient

Local clustering coefficient can reflect the tendency of station points to form a cluster with the neighboring nodes. Generally speaking, only nodes with a small clustering coefficient might become nodes of the structural hole. Below is the calculation formula: $C(i)=2E(i)k(i)[k(i)−1]$(18)

Where, E(i) stands for the number of actually existing sides of neighboring sides of Node i; k(i) stands for the degree of Node i.

## 4 Building of the evaluation model

From the perspective of spatial autocorrelation, the nearer the two objects are to each other, the stronger the reliance is between them. Based on the space autocorrelation theory, it is thought that nodes neighboring current nodes contribute more to the importance of the nodes. Existing research has suggested that some characteristics of many complex systems are positively correlated with the degree of the node. Thus, during the node importance evaluation process, the index importance contribution matrix of neighboring nodes is positively correlated with the value of degree.

When, the influence of only one characteristic (such as value of degree) on node importance is considered, the node importance evaluation function of Eq. (19) is introduced in terms of any Node i: $Ii=aδi+b1∑j∈πi1δj+b2∑j∈πi2δj+⋯+bm∑j∈πimδj$(19)

Where, Ii stands for the importance index of Node i; δ stands for the node attribute value, which can be the degree of node or the network constraint coefficient; a and b are two adjustable parameters, which are used to adjust the degree of reliance of node importance on the node attributes and the neighbouring nodes from the first order to the nth order, respectively. From the perspective of autocorrelation, in order to fully consider the contribution of importance of the node and the neighboring nodes, here the value of a and b meets the conditions of $\overline{k}\cdot b>a>b$ and 1 > b > 0, where $\overline{k}$ stands for the node’s mean degree. From Eq.(19), it can be seen that the evaluation function comprehensively considers the contribution of the node to the importance of the node itself and the neighboring nodes of the m order. Besides, the farther the node is away from Node i, the less contribution it is to the importance of Node i. Considering the influence of neighboring nodes of the m order on node importance, the location information of the node is utilized. For the convenience of understanding, here, neighboring nodes of the m order to be evaluated are regarded as the depth of neighboring nodes to be observed for the node importance evaluation, and their value should meet the condition of Dm ≥ 0.

Generally speaking, in real life, when importance of target objects is evaluated, the influence of multiple factors will be taken into consideration. To network nodes, their importance is not fully decided by their degree and structural hole index. The remaining factors should also be accommodated to so as to achieve an accurate evaluation of node importance.Thus, it is assumed that every node selects n evaluation indexes, and that δi, j is used to stand for the j index value of Node i. Therefore, the importance evaluation model of Node i can be defined as below: $Ii=AEi$(20)

Where, Ii stands for the degree of importance of Node i; A stands for the evaluation coefficient matrix or the importance degree contribution matrix; Ei stands for the contribution of the node itself and its neighboring nodes of various orders to the importance of Node i. Here, it is assumed that neighboring nodes of the same order contribute the same amount to Node i. In other words, their evaluation coefficient is the same. W stands for the evaluation index matrix of Node i, which includes the index value of Node I and various neighboring nodes; W stands for the index weight matrix, which is used to stand for the degree of reliance of the importance of Node i on various indexes. Here, the proportion of the weight of n evaluation indexes is written as, w1, w2, …, wn. Under the multi-factor situation, the single-factor evaluation function is maintained. The method to calculate the contribution of neighboring nodes of the same order to the importance of Node i is shown below: $δi,jm=∑j∈πimδj,n,$(21)

Where, ${\delta }_{j,n}^{m}$ stands for the total contribution amount of the set of neighbouring nodes of the m order belonging to Node i to the importance of Node I under the restriction of the n index; ${\delta }_{j,n}\left(j\in {\pi }_{i}^{m}\right)$ stands for the value of the n index of the central node j among neighboring nodes of the m order belonging to Node I. It is assumed that the value of the n indexes of Node i is {δi,1, δi,2, …δi,1}. Therefore, the evaluation index matrix in Eq. (19) can be expressed as below: $Ei=δi,10δi,20⋯δi,n0δi,11δi,21⋯δi,n1⋮⋮⋯⋮δi,1mδi,2m⋯δi,nm=δi,1δi,2⋯δi,n∑j∈πi1δj,1∑j∈πi1δj,2⋯∑j∈πi1δj,n⋮⋮⋯⋮∑j∈πimδj,1∑j∈πimδj,2⋯∑j∈πimδj,n$(22)

The value scope of different indexes might vary greatly. For example, the degree of the same node might be several hundred, but its network restraint coefficient might be smaller than 1. The physical meaning and the measurement unit of various indexes might not necessarily be the same, thus might lead to a different data dimension and magnitude. Therefore, it is necessary to conduct normalization of the evaluation index matrix. From Eq. (21), it can be seen every column of elements corresponds to one index. There are n indexes in total. The following normalization equation is adopted to process every index value: $δi,j′k=δi,jk−minδi,jkmaxδi,jk−minδi,jk,j=1,2,…n$(23)

In order to work out the normalization evaluation index matrix based on the index value after normalization, it is assumed that the importance of any Node i in the network can be expressed a ${I}_{i}=A{E}_{i}^{\prime }W,$ namely: $Ii=a,b1,b2,...,bn×δi,10δi,20⋯δi,n0δi,11δi,21⋯δi,n1⋮⋮⋯⋮δi,1mδi,2m⋯δi,nm×w1w2⋮w∂$(24)

## 5 Algorithm flow

Based on comprehensive consideration of contribution of the node itself and nodes of the m order to node importance under restrictions of various indexes, relatively accurate evaluation results can be obtained. To consider the degree value of the node itself is critical to evaluating the importance of the node. To some extent, the value of degree reflects node importance. To accommodate to the information of neighboring nodes of the mth order can help analyze the importance of a node in the whole network and reflects the importance of the node location. The network topology, G ={V, L}, the evaluation index set and the index weight matrix, W, have been already known. Below is the algorithm flow to evaluate node importance:

1. Extract the node set of any Node i according to the network topological institutions ${\pi }_{i}^{k};$

2. Calculate every index value of the set of neighboring nodes of various orders belonging to Node $i:{\delta }_{i,1}^{k}=\sum _{i\in {\pi }_{i}^{k}}^{{\delta }_{i,1}},k=1,2,...,m,i=1,2,..,n$ and the evaluation index matrix of Node i is Ei confirmed;

3. Conduct normalization of indexes of Ei to obtain the evaluation index matrix${E}_{i}^{\prime }$ after normalization;

4. According to the ideal value of various indexes, bj*(j = 1, 2,…, n) and work out the weighting coefficient of various indexes, wj*(j = 1, 2, …, n);

5. Output the target value of various indexes according to Eq. 24.

After the target value of nodes is obtained, the target value is ranked from large ones to small ones. The former nodes are much more important than latter nodes. According to the ranking results of the node importance, the node or the node set most important to the network can be confirmed.

Based on the above node importance evaluation method, it can be seen that evaluation indexes and neighboring nodes to be observed are the key to the whole evaluation effect. The evaluation process relies on characteristics of nodes, thus ignoring the location information of nodes in the network. The evaluation results are similar to those obtained through the traditional connection degree approach. As to selection of evaluation indexes, the degree and structural hole of nodes is a key to reflecting node importance. Thus, based on the evaluation indexes, the specific evaluation model is built.

## 6.1 Construction of China’s high-speed passenger traffic network

While building the high-speed rail and civil aviation compound network, this paper makes the following hypotheses:

1. a high-speed passenger transport network is a complex network built in space p. If there is an airline or a high-speed railway between two cities, then it is considered that two cities have an edge.

2. the high-speed passenger transport network is an undirected weighted network. If a city has a high-speed rail station and an airport simultaneously, then this city is considered as a node.

Under the above hypotheses, the operation data of China’s high-speed rail and civil aviation in 2015 are statistically analyzed. The data source of the civil aviation sub-network is the Summer Flight Schedule 2015. The database contains 10,093 flights of more than 20 airlines in China (excluding flights to Hong Kong, Macau and Taiwan) and 196 cities. (See the topological diagram of civil aviation network in Fig. 1). The HSR (high-speed rail) data are from operation plans of high-speed trains, bullet trains and intercity railways formulated by the Railways Bureau since 2015. The network contains about 4,000 high-speed train numbers and 425 high-speed rail stations all over China. (See the topological diagram of HSR network in Fig. 2) Based on relevant data of the civil aviation sub-network and the high-speed rail sub-network, the high-speed passenger traffic network is built. The network contains 579 nodes and 14,312 sides. Refer to Fig. 3 for the topological diagram of high-speed passenger traffic network.

Figure 1

Topological diagram of civil aviation network

Figure 2

Topological diagram of HSR network

Figure 3

Topological diagram of passenger traffic network

## 6.2 Importance evaluation of major nodes

Based on a comprehensive analysis of the degree, the average path length and the aggregation degree coefficient of China’s high-speed passenger traffic network nodes, 15 important nodes are chosen. See below: Beijing-v1, Shanghai-v2, Nanjing-v3, Guangzhou-v4, Shenzhen-v5, Xiamen-v6, Changsha-v7, Jinan-v8, Chengdu-v9, Hangzhou-v10, Wuhan-v11, Zhengzhou-v12, Nanchang-v13, Hefei-v14 and Kunshan-v15. According to Part 4 of this paper, the value of various evaluation indexes of these nodes is shown in Table 2.

Table 2

Value of node evaluation indexes

Table 2 shows the calculation results of every index of nodes: Build the evaluation matrix based on the value of every index: $Ei=31775.47170.2132316.24200.9976157.621031375.13150.2137312.24850.9976155.624022463.49120.1883223.27370.9968116.675023569.73500.2539234.35010.9972116.142022467.15920.3510222.22680.9921111.137017444.42470.2314173.31150.9960110.613518975.07510.2062188.34070.9965108.179016962.85360.2168168.36180.996293.670517756.75370.2636176.41080.996787.705523454.05410.1881233.28380.996986.156021867.29480.1781217.35770.997184.738516758.75440.2224166.18660.995183.183017159.34720.2236170.47620.996983.183016459.24170.2143163.18750.995082.593516859.88020.2142167.36550.996281.0930$

Conduct normalization of every index: $Ei′=110.20300.9935110.97390.98900.20590.967610.97390.46410.61410.46960.46190.92730.46500.45750.81520.05780.45500.87270.45800.39220.73230.05900.39000.85450.39260.3922010.383200.38570.35290.987200.35160.90910.35390.16340.59360.16250.16330.80000.16440.08500.39710.49450.08580.83090.08640.06540.31020.30830.06570.70910.06620.04580.73660.26320.04730.87270.04760.03270.46150.22380.03360.74550.03380.02610.48060.20880.02710.74550.02730.01960.47720.25620.02080.54550.019600.49780.209400.52730$

In terms of the matrix, the ideal point is b* = (1, 1, 1, 1, 1, 1, 1). Thus, the index weighting vector of the matrix worked out according to the formula is W* = (0.0859, 0.2242, 0.0879, 0.0858, 0.4300, 0.0860).

Calculate the target value of evaluation: $di=0.9294,0.9204,0.681,0.7033,0.6378,0.4536,0.5336,0.4185,0.512,0.6973,0.5757,0.3568,0.4523,0.1877,0.3691$

Table 3 uses the algorithm and the algorithm respectively proposed by Literature [20], Literature [21] and Literature [22] to obtain evaluation results of node importance of the high-speed passenger traffic network. Besides, a = 1 and b = 0.5. The node importance evaluation results obtained by the four algorithms are different, because they have different focuses. The basic idea of the algorithm proposed in this paper is to evaluate importance of various nodes based on various node indexes and the analytic hierarchy process. The algorithm proposed in Literature [20] is based on changes of networks generated after removal of nodes. The algorithm proposed in Literature [21] decides the key nodes based on the capability of nodes to provide the shortest available path of the network. The algorithm proposed in Literature 23 evaluates node importance based on contribution of various nodes to network information transmission in the network. It just accommodates to contribution of nodes to importance of neighboring nodes. Evaluation Results suggest that the algorithm proposed in this paper comes up with the most important node in the high-speed passenger traffic network, which is v1.

Table 3

Node importance evaluation based on the algorithm in this paper

From Table 3, it can be seen that the most important node worked out by the algorithm in this paper is v1, which is different from that obtained by the algorithm proposed in Literature [22], but the same to that proposed in Literature [20] and Literature [21]. This suggests the algorithm proposed in this paper is accurate in some way. From the above table, it can be seen that the algorithm based on the network constraint coefficient, efficient scale, efficiency and local clustering coefficient, respectively, the algorithm proposed in Literature [20] (node removal approach) and the algorithm proposed in Literature [21] (the capability of providing the shortest available path) all obtain the most important node to be v1. Among them, the algorithm based on the network coefficient, efficient scale, efficiency and local clustering coefficient belongs to the single-index calculation, thus being limited, low in evaluation accuracy and large in evaluation error. In terms of the node removal approach, if it deletes too many nodes, it will result in network disconnection and failure of accurately evaluating the degree of node importance. The algorithm based on the capability of providing the shortest available path can increase the node evaluation accuracy, but its calculation process is complex and its accuracy is poor in finding the key node. To sum up, the algorithm proposed in this paper comprehensively considers the global importance of nodes, combines the node structural hole attribute and accounts for the structural hole indexes of various aspects of nodes to evaluate the node importance.

## 6.3 Analysis of algorithm efficiency

This paper uses four operation methods, i.e., the proposed algorithm, algorithms involved in References [20, 21] and [22] to evaluate the node importance of the proposed network. The operating time of each algorithm is as shown in Fig. 4. It can be seen from the figure that, with the increase of number of nodes, the slope of the proposed algorithm declines and the time decreases, suggesting that the proposed method is significantly better than the other three methods. So the proposed evaluation method of node importance has a high efficiency and is suitable for the calculation of large-scale complex networks.

Figure 4

Operation hours of different numbers of nodes in the network

Table 4

High speed passenger transport network node importance degree (descending) results

## 7 Conclusions

Complex Network Theory has vital research value and scope for further research. To apply the theory to traffic analysis can help scholars get an in-depth understanding of the operational rules governing traffic networking systems and help master the complexity of traffic networks both microscopically and macroscopically. This paper mainly introduces basic knowledge of the complex network, the calculation of various indexes of the complex network nodes and the evaluation of node importance. Below is a brief review of the research work in this paper:

1. Analyze the research status of the complex network both at home and abroad based on relevant literatures and research findings, and analyze problems of the complex network in traffic;

2. Briefly introduce the basic theories of the complex network and the structural hole theory, including some indexes to calculate the structural hole node, such as the network constraint coefficient, grade, network scale, efficiency and local clustering coefficient;

3. The complex network is important in that it is a reliable index to measure the traffic network’s stability. The influence of some commonly-used measuring indexes of the road network’s reliability on nodes is analyzed.

4. Introduce the structural hole indexes and use the multi-attribute decision-making to combine the structural hole and degree to analyze characteristics of different indexes. Analyze the high-speed passenger traffic network based on analysis of concepts of different indexes and approaches to calculate the node importance.

5. Analyze findings and put forward reasonable building plans.

The traffic system, either in terms of its topological structure or in terms of its evolution process, has significant complex network characteristics. The combination of the complex network system and the traffic network has good application prospects. Besides, research done in the traffic field can deepen, enrich and expand the Complex Network Theory.

The high-speed passenger traffic network is a relatively complex system. Compounded by characteristics of the traffic field, it becomes an even more complex system. Based on the high-speed passenger traffic network, this paper conducts a study on the complex network. The topological structures are analyzed, the extraction and ranking of key nodes are explored and suggestions to optimize key nodes are put forward. However, there are still some problems calling for further research. For example, research on nodes when the path weighting is considered, and the influence of different network structures and network scales on key nodes are the research direction in the future.

## Acknowledgement

This research was supported by the National Social Science Foundation major research project(No.13&ZD170), and LiaoNing Doctoral Scientific Research Foundation(No.201501182).

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Accepted: 2016-12-27

Published Online: 2017-03-01

Citation Information: Open Physics, Volume 15, Issue 1, Pages 1–11, ISSN (Online) 2391-5471,

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