The modeling of attraction characteristics regarding passenger flow in urban rail transit network based on field theory

Aimed at the complicated problems of attraction characteristics regarding passenger flow in urban rail transit network, the concept of the gravity field of passenger flow is proposed in this paper. We establish the computation methods of field strength and potential energy to reveal the potential attraction relationship among stations from the perspective of the collection and distribution of passenger flow and the topology of network. As for the computation methods of field strength, an optimum path concept is proposed to define betweenness centrality parameter. Regarding the computation of potential energy, Compound Simpson’s Rule Formula is applied to get a solution to the function. Taking No. 10 Beijing Subway as a practical example, an analysis of simulation and verification is conducted, and the results shows in the following ways. Firstly, the bigger field strength value between two stations is, the stronger passenger flow attraction is, and the greater probability of the formation of the largest passenger flow of section is. Secondly, there is the greatest passenger flow volume and circulation capacity between two zones of high potential energy.


Introduction
Urban rail transit(URT) has become people's first choice due to its convenience, efficiency and safety. A networked operation in URT has taken place in some international metropolis such as Beijing, Shanghai, and Tokyo, with the scale of passenger flow increasing and a congestion forming in a continual way, especially in rush hours. Taking Beijing as an example, the subway lines has reached 19 by the end of 2016, with the mileage of 574km. The increasing correlation among stations and lines has led to an increasing complexity in the characteristics of passenger flow change. Currently, 4 large stations, including Dawanglu Station, Xierqi Station, Dongzhimen Station, and Beijing Railway Station, are more liable to form congestion. The reasons are that these stations have served as transfer ones and meanwhile there are usually apartment buildings, office buildings and shopping malls around them.
Thus, how to conduct an effective research into the attraction relationship and the spacetime distribution regarding passenger flow among stations, has become an urgent issue, the a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 solution of which will render a support in project planning and network operation management.
The internal interaction and potential attraction of passenger flow among stations in the network is a relatively complicated phenomenon, with the main existing problems in this research as follows. 1. With respect to the representation of the space-time distribution characteristics regarding the passenger flow in the network, the indirect method has been used in the traditional research to carry out an analysis of the changing trend of the passenger flow [1][2][3]. Some scholars have utilized decaying models of the road performance to establish the distribution model [4]. Passenger flow distribution is a result of passenger route choice decision-making. Referring to passenger route choices, train delays has been considered as an influence factor [5][6][7]. Moreover, the analysis of passenger travel behaviors is considerably important [8]. Zhu (2011) [9] studied a scenario-based route choice model and calculation method against the background of the 2010 World Expo in Shanghai, China. Some research has been conducted regarding to passenger flow patterns analysis based on smart card data [10][11][12][13], passenger flow distribution [14][15][16], congestion evolution under interrupted operation [17,18], and passenger flow forecasting [19][20]. These researches have some limitations as it depends on the changes of the distance between the stations and the city center. However, there is no uniqueness of the city center any more, with the deepening of urbanization and increasing passenger flow.
2. In terms of the analysis of characteristics of URT networks [21][22][23][24][25][26], most research has been focused on the network degree, the complexity and robustness of metro networks [27][28][29][30][31], betweenness centrality [32][33][34][35]. While in the definition of betweenness centrality, it has been assumed that the weights of all the edges are equal (the value is 1). However, for most actual cases of networks with additional flows, such as those in logistics, road transport, and subway, we can't just assume the equality of all the edges.
3. With the application of the field theory to the logistics field [36,37], mainly including its concept, basic characteristic quantity, etc., relevant scholars have conducted a research into the internal laws of logistic operation system. The application of the field theory to the transportation field [38][39][40][41] mainly involves the modeling analysis of the effect of passage congestion and diffusion, the space distribution characteristics of the flow and the framework of passenger flow. What the field theory describes is about the distribution of physical quantity in an area of some given space realm, with all the elements having their own specific distribution laws and interactions among them. However, there is not much research into the systematic characterization of the internal potential attraction and interaction regarding passenger flow.
In view of what is mentioned above, this paper has made a quantization of the passenger flow attraction, with the introducing field theory to reveal the internal characteristics and operation mechanism in respect to passenger flow distribution. The contributions made in this research are as follows.
1. For the purpose of a comprehensive analysis of the potential attraction and space-time distribution characteristics of passenger flow, this paper has introduced a new concept of the gravity field of passenger flow and proposed the computation methods of field strength and potential energy from the perspective of topological attributes and flow attributes of the network.
2. This paper offers a computation method of betweenness centrality based on the optimum path as the key parameter of field strength, with the simultaneous verification of the optimum path through the employment of OD matrix. In order for the computation of potential energy, we have applied Compound Simpson's Rule Formula to get a solution to the function.
The structure of this paper is as follows. Part One focuses on the construction of the model of the network, the concept of gravity field, field strength and potential energy, as well as the relevant methods of computation involving the parameters of field strength, the definition of the betweenness centrality of optimum path, the parameter computation of potential energy function based on Compound Simpson's Rule Formula. Part Two deals with the simulation and verification of all theories in Part One, including the source and background of data, the determination and verification of related parameters in Part One and the force analysis of the spatial linkage of passenger flow.

The modelling of URT network
The URT network consists of some operational lines which are composed of some stations and zones among stations. Taking station i as node v i and zones among stations i,j as edge l ij = {v i ! v j }, considering the operational directions of the trains, we construct the pattern of topological structure of URT network as is shown in Fig 1. Definition 1 neighbors: As for any node v i ,v j , if there is an accessible path between them, v i and v j are defined as neighbors. r ij represents the distance, with the calculation of the minimum numbers of stations between v i and v j . Definition 2 optimum path: It refers to the path with the smallest weight of w e st between the two nodes of v i and v j in the network, with the mathematical formula as follows.
d v i !v j is the impedance function of the path.
In the URT network, d v i !v j is two-tuples of travel distance and transfer times. For the convenience of computation, the travel distance is replaced by the train operation time among the zones, with the omission of time for the passengers to get on and off the train. The mathematical formula is as follows.
Where G v i !v j stands for all the stations attached to the path of δ between the node-pairs OD.
T tr i stands for the time of the ith transfer and T v i !v j refers to the operation time of the train according to the timetable. κ is used to stand for the traveling impedance coefficient of the transfer times. Definition 3 The betweenness centrality BC i of the optimum path: d jk is set as the total number of the optimum path between the v j and v k and d jk (v i ) is the total number of the optimum path when the node of v j travels through the node of v i to the node of v k , i.e., 8j,k 2 G and j 6 ¼ k. If the optimum path travels through the node of v i , we have d jk The betweenness centrality of the optimum path regarding the station reflects the degree of topological significance of node v i , a large value represents its high capacity to attract passenger flow in the network.
Definition 4 The source and gathering point of passenger flow: As for the node of v i in the network, the classification is made from the perspective of passenger flow direction,field theory,and the actual physical significance. The source and the gathering point of passenger flow are shown in Table 1.
The source point within the stations serves as the main source of passenger flow, the size of which will determine the strength of gravity field in a direct way. Also, the distribution of the gathering points will directly determine the direction, scale and space distribution of passenger flow. The distribution characteristics and field space scale of passenger flow are what arise from the interaction between the source and gathering points.
2. Passenger flow field and its characteristic quantity 2.1 The concept of passenger flow field. Field in "Cihai" is defined "The field of physics is an interactive existence of substance in the whole space." The field has the following basic elements [42]. Firstly,it has a specific zone. Secondly,many elements,here referred to the physical quantity,are in this zone. Thirdly, all the elements,with their own specific distribution laws, interact with each other.
Definition 5 The gravity field of passenger flow: Passenger flow refers to how passengers move in a space zone. The gravity field of passenger flow means a space field formed in a space zone with all kinds of elements (including flow and direction) interacting each other. It is like the physical field and established with stations as the source and gathering points of passenger flow, and all the operation lines as the track of movement. Table 1. The interpretation of the source point and the gathering point.

Node Type
The Interpretation of Field Theory The Actual Physical Significance

The Source Point
Passenger flow go from the source point to all directions along radial straight lines.
Subway stations with residential area around in morning rush hours and with offices around in evening rush hours.

The Gathering point
Passenger flow go from all directions to the gathering point along radial straight lines.
Subway stations with residential area around in evening rush hours and with offices around in morning rush hours.
All the elements in the gravity field of passenger flow change with the time. This gravity field is a multidimensional space and a vector field composed of such variables as time and space.  Table 1. The expression of the field strength is as follows. Here The sum of column vector in a matrix is the total positive field strength at Station v P .
Similarly,the negative matrix of field strength is M À E !.

2.3
The potential energy of the gravity field of passenger flow. Definition 7 The field potential energy of passenger flow: The potential energy function means the function of energy aggregation. In other words, it is the circulation capacity of passenger flow between the source point and any point in the field. It is used to describe the scalar of work regarding the field force. Regarding the passenger flow, there is a motional tendency of movement from a high potential energy position to a lower one. The potential energy is the definite integral function of the field strength along the path,see Fig 2. Here is the expression of potential energy,as follows.
There are many source points in the network. Set ψ oi as the potential energy of the points v i and v o , the positive matrix of field energy M þ c oi are as follows.
The matrix M þ c oi falls into the symmetry type, i.e., ψ + 12 = ψ + 21 . In accordance with the principle of the potential energy of physical field,the potential energy at any point v P is equal to the algebraic sum of potential energy in an independent way, i.e., the sum of column vector regarding the matrix. So is the negative matrix of potential energy M À c oi .

The computation of the potential energy
As the function of potential energy is definite integral of field strength, the function of field strength of passenger flow is not continuous, has no primary function. Thus,the Formula of Newton-Leibniz cannot be applied here. The formula of mechanical quadrature of À X n k¼0 A k E ! ðr k Þ is adopted in this paper to approximate the value of definite integral.
As for the solution to the potential energy function, the following computation Condition 1 and Condition 2 must be met, both of which are equivalent.

Condition 1:
In the quadrature formula of the potential energy, E ! ðr k Þ ¼ E e k þ d k is recorded, with δ k as the error. As for any small positive number ε > 0, and in the case of 9δ > with the stabilization of the quadrature formula.

Condition 2:
In the case of coefficient A k > 0 (k = 0,1,Á Á Á,n) in the approximate sequence of number, the approximate formula is stable.
The following is a simple verification.
With any small positive number ε > 0 chosen,and in the case of d ¼ ε Making the integral interval of [0,r oi ] into n equal divisions and in accordance with Condition 2, the Formula of Newton-Cotes is not stable in the case of exponent number n ! 8. In the URT network model,there are usually at least 8 stations in every line, making the Formula of Newton-Cotes inapplicable.
For improving the computation accuracy of quadrature, we have adopted Composite Simpson's Rule. n divisions are made in the interval of [0,r oi ], with the Simpson Formula adopted in every sub-interval and the following expressions are acquired.
Since we cannot acquire the primary function,we calculate this quadrature according to the list of one-to-one projection value regarding the existing field strength value and distance in this paper with a classified discussion of n.
1. If r oi is an even number,we will have n ¼ r oi 2 . With r k+1/2 as an integral number in the computation process,the list of one-to-one projection value regarding the field strength and distance can be employed to get E ! ðr kþ1=2 Þ.
2. If r oi is an odd number,we will have n ¼ r oi À 1 2 . With r k+1/2 not as an integral number in the computation process,we will,with the estimation of polynomial interpolating fitting, acquire E ! ðr kþ1=2 Þ according to the list of one-to-one projection value regarding the existing field strength.

Data analysis 1.The source and background of the data
Here we use passenger flow data of No. 10 Line of Beijing Subway in time of 07:00~08:00 in October 2013 for an analysis and discussion, see Table 2.
The construction of the topological diagram of Beijing Subway is shown in Fig 3.

The verification of the parameters of characteristic quantity
If the gravity field of passenger flow is composed of one single line, the parameter of λ is the average number of travel stations along this line. If we take the whole URT network as the passenger flow field, the parameter of λ is the average number of travel stations in the network. The case study and background in this paper are aimed at the topological diagram of the whole network in Beijing URT, with the choice of λ = 11. At the same time, the values of the parameter of D in the formula of the field of passenger flow are composed of two parts. Firstly, for passenger flow with one traveling line, D is the total length of this line. Secondly, in regard to passenger flow of transfer, D is the maximum diameter distance in the network.
According to Definition 2, we have firstly conducted a verification of the optimum path based on passenger flow data, the method of which is shown in Fig 4. With every cross-section passenger flow introduced into the network and based on the OD matrix, we weight every edge according to the optimum path. And it turns out when κ = 1.531 the model has the best effect, get more details in chapter 5 of Reference [43]. The analysis has established that, the size of field strength is directly related to the average traveling stations, the incoming passenger flow and leaving out of station and the betweenness centrality of the node. A stronger strength will lead to a bigger attraction of passenger flow between two stations. The following can be known from Fig 5. Firstly, taking Zhichunlu Station (TS/L13S2/L10S3) and Zhichunli Station(L10S2) for example, as a transfer station, Zhichunlu Station has a bigger load of passenger flow, and a larger betweenness centrality value, so its values of positive and negative field strength obviously exceed Zhichunli Station. Secondly, as the average traveling distance is λ = 7 at No. 10 Line, the field strength of E max turns out to be at the station whose distance is R oi = 7 from the source point; passengers are more likely to get off at this station from the perspective of probability.
We can get the following conclusions after an analysis of Figs 6 and 7. Firstly, as for different stations with the same size of incoming passenger flow, a bigger betweenness centrality will lead to stronger field strength. Taking Shiliuzhuang (L10S26) and Jiaomendong (L10S28) as example, the betweenness centrality of the L10S26 is 0.09704 and L10S28 is 0.09655, and the field strength of the former evidently exceeds the latter with the same incoming and leavingout size. Secondly, regarding the same stations, different incoming size will lead to different positive and negative field strength. Taking the transfer station of Zhichunlu(TS/L13S2/L10S3) as an example, the transfer-in size into No. 10 Line from other lines (No. 13 Line to No. 10 Line) obviously exceeds the transfer-out size (No. 10 Line to No. 13 Line), showing that the    Attraction characteristics of passenger flow in URT network example, the passengers entering in Guomao Station are more likely to get off at the zones in Table 3.
A denser contour line of field strength will lead to greater changes in its value, with the biggest value in the central point. Greater field strength means a stronger attraction of passenger flow between two stations, with a relatively big probability of existence of the largest cross-section passenger flow in the path. With an analysis of several typical zones in the table and the comparison of the actual traveling data and that of cross-section passenger flow, we can establish that passengers in those zones are more likely to get off at Guomao Station in morning rush hours. We know that the potential energy describes the aggregation energy of passenger flow at any point in the field.
An analysis of the diagram of contour lines has helped us conclude that passenger flow and its aggregation capacity in the two zones of high potential energy will reach the maximum value. What's more, as the potential energy is the integral of the field strength along the path, a longer path between the two zones will lead to bigger potential energy. Here is a description of several typical couple zones of potential energy.
1. According to Fig 8, there are seven typical couple zones of high potential energy in Line 10, i.e., are shown in Table 4. A longer path with zones of higher field strength between two zones in the network means a stronger attraction of passenger flow. For example, we have the zones of high field strength of "Mudanyuan-Sanyuanqiao" and "Songjiazhuang-Jijiamiao". The analysis of the actual physic al significance is as follows. Firstly, from the perspective of incoming passengers, some big stations with a big incoming size are at the main path of passenger flow, e.g., Dawanglu, Sanyuanqiao, Songjiazhuang, and Jiaomenxi Station. The gathering capacity of the passenger flow during this time interval reaches the maximum, which verifies the definition of high potential energy. Secondly, in combination with the actual data of cross-section passenger flow, several high sections all exist at the main optimum path, e.g., Shuangjing-Guomao Section, Jintaixizhao -Hujialou Section, Zhichunli-Zhichunlu Section, which verifies the actual significance of zones of high potential energy.
2. There are also some couple zones of low potential energy in the Line 10, i.e., such as Suzhoujie Station (L10S0) to Zhichunlu Station(TS/L13S2/L10S3) and Chedaogou Station (L10S41) to Bagou Station(L10S44) and so on in Fig 8. A shorter path means a weaker aggregation capacity for passenger flow, with the consequently smallest potential energy among the neighborhoods, resulting in the "couple zones" of low potential energy.

Conclusions
Based on the construction of URT network and regarding the field theory, we have conducted a quantitative analysis of passenger flow with regard to space-time distribution and its potential characteristics in the network. With the concept of the gravity field of passenger flow, the field strength and potential energy are proposed. The key parameters have been offered, including the definition of betweenness centrality based on the optimum path, and the employment of Composite Simpson's Rule. With a simulation and verification of the proposed model, there are some results. Firstly, the analysis of "force" helps us know that a denser field strength means great changes in its values, with that the center of field strength contour reaching relative maximum value and a https://doi.org/10.1371/journal.pone.0184131.g008 Table 4. The zones of high potential energy in the line 10.
Although this paper has achieved some meaningful results, some parameters of the proposed model is still need to be improved in the future. Firstly, we will improve the computation of potential energy by adding the influence factor of passenger flow changes. Secondly, Gamma function, the continuation of factorial, can be employed to error estimation in factorial of potential energy.