Abstract
Previous research studies of traffic networks are mainly based on planar networks and less considered the influence of multilayer networks, which illustrate and represent different appropriate urban traffic modes. Development of rail and road networks is inseparable from the development of a prosperous urban area; thus, research on multilayer networks has scientific potential and fulfils a real need. In this paper, a framework of complex network based integrated multilayer urban growth and optimisation model (CNIMUGOM) is proposed, to analyse the complex relationships between the traffic network structure, the population growth, and the urban land-use. The innovation of this paper is the combination of the traffic complex multilayer networks and the “Four Step Model” (which stands for trip generation, trip distribution, model split, and traffic assignment steps). With the multiobjective, multilayer network coevolution and optimisation model, a more efficient traffic network layout was generated based on different land-use, population density, and travel speed scenarios. Then, this paper has proved that the proposed CNIMUGOM can save the traffic network construction investment, reduce the travel cost, make the urban traffic network more efficient, and decrease the total traffic flow amount. This research has connected the recent complex multilayer network related study and traditional urban economic model based study. The findings of the study afford to improve the current land-use and traffic integrated models and can provide traffic network planning suggestions for urban agglomeration development.
1. Introduction
Many developing countries still face rapid urbanisation process, with numerous scholars focusing on the prompt urban traffic network growth and coevolution process [1–3]. To measure the growth and coevolution process of an urban traffic network, research concentrated on two aspects: the first is modelling and simulation, and the second is its natural course [4–6]. The study of the coevolution process can clearly analyse the interrelationships between urban land-use and traffic networks [7, 8]. Focusing on the coevolution model of the 19th and 20th century in London, the group of Levinson and Xie found that population distribution and network density are positively correlated. They validated a simulation model to fit the empirical evidence better and noted that evolution is an iterative process of interaction, investment, and divestment. Moreover, they illustrated how surface traffic networks could grow and decline spontaneously over time, providing further evidence for the property of self-organisation [5, 9–11]. Similarly, a new dynamic model based on the logistic equation, which captures the dynamic characteristics of the coevolution process between the street surface and urban traffic structure, was developed [12]. After that, the coevolution of urban traffic growth grounded in the case of Beijing is described [13], and a coevolution model is suggested with stability analysis and numerical simulation. In another way, Rui [14] used the multiagent based model to discuss the complex traffic network growth and land-use coevolution process, and centrality indices from different aspects were thoroughly analysed with the study of Stockholm. Recently, Li et al. [15] proposed an optimal urban expressway system model, which considered the interaction equilibrium of transportation and land-use.
Although those previous investigations are essential and necessary for further research of urban traffic network and land-use, they all treat networks as purely planar. Some of them considered the network growth and variation, but none of them considered the influence of multilayer networks, which is the cause of attention to multilayer networks related analysis.
Development of rail and street networks is inseparable from the development of a prosperous urban area; thus, research on multilayer networks has scientific potential and fulfils a real need. To bridge the gap of the multilayer network representation of real-world networks, Kurant and Thiran [16] first proposed a general multilayer model that facilitated the description and analysis of multilayer networks. The authors examined three transportation networks and found that a small error on a multilayer network could cause cascading failures. They also investigated the relationships between degree, betweenness, and real loads and found that, as opposed to the commonly acknowledged view, the correlations in their dataset between the three factors were not that apparent. More recently, multilayer networks earned more attention, as traffic dynamics in two-layer complex networks were considered by Ma et al. [17], and Albert et al. [18] introduced a standardised model to simulate the elements navigating those networks and analysed congestion in multilayer transportation networks. Furthermore, the ratio of speeds of coupling different modes, network accessibility, mobility, and behaviour of different layers were considered [18–23]. These researches bring a new broad of perspective, investigating the aspects of multilayer networks to consider the interrelationship and cooperation of different traffic layers and modes [22, 23].
With the novel study of multilayer networks, the relationships between the urban traffic networks and land-use of surrounding areas might be discussed. With the analysis of some new indicators, their complex relationships become measurable. This fills the research gap of recent urban traffic network structure based studies; with the study of network coevolution process, the complex dynamics growth process can be partly studied. The influence of multilayer networks can be also measured, which connect the growth of upper-layer and lower-layer network and the accessibility change of surrounding areas related to urban land-use [24]. Based on these works, in this paper, the framework of complex network based integrated multilayer urban growth and optimisation model (CNIMUGOM) will be proposed first. Considering the traffic network structure, with the population growth rate and its affection of urban land-use model (the change of accessibility of employment and population ), and the coupling features of multilayer networks, the study identifies issues regarding the “Four Step Model” (FSM). For the travel demand model, the number of trips generated and attracted ; the number of trips from traffic zone i to zone j, ; the generalised travel cost ; the travel time on the link a, ; and the traffic flow will be calculated. For the street investment model and traffic network growth model, the collected revenue of this link , the measurement of multilayer network structure status , the overall spending function , and the general investment of iteration will be calculated in Matlab. With the multiobjective, multilayer network coevolution and optimisation model, the more efficient network layout can be generated based on some different land-use (different population density) scenarios. Based on the traffic network and urban land-use coevolution process, the proposed CNIMUGOM can save the traffic network construction investment, reduce the travel cost, and make the urban traffic network more efficient. Based on the simulation, the proposed network can increase network efficiency and decrease the total traffic flow amount.
2. Methodology
This study refers to the general urban transportation system as a multilayer network. The upper-layer network represents the public rail network, which may include rapid transit, LRT, monorail, MRT, and subway. Here, this study does not distinguish between rail transit modes. The lower-layer network represents urban street and road networks.
2.1. Single-Layer Network Representation Method
This method is used by many scholars, and related studies of different cities were widely accepted [25–28]. According to the primal approach, the single-layer network representation method, as shown in Figure 1, the black lines on the right act as streets or roads, the nodes represent the street intersections, and the grey spots are buildings.
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With this method, the different undirected or directed networks can be used to represent the urban transportation networks:where is the set of nodes, N is the number of nodes whenE is the unordered pairs or edges of elements of and is denoted by , and is the weight of each edge; in some functions, it can be denoted as , and the weight can be treated as the length of the edge or lanes.
The number of edges is denoted as M. The adjacency matrix of networks isrepresenting the connection between nodes and , which is defined aswhere to remove any self-connections. In addition, is symmetrical and nonnegative.
We define D as the network diameter, with
Here, is the sum of edge numbers between node pair i and j.
The degree centrality was proposed by Freeman [29]; the function is
The number of link incidents on a node can reflect the importance of the node in relation to spatial geography, which indicates that a node with more neighbours is more important in a network.
The closeness centrality was proposed by Marchiori and Latora [30]; the function is
It is denoted as the reciprocal of the average distance between each node pair . This index means that if a node is closer to the other remaining nodes, it is more important in the network; it describes the relative location of a node.
The betweenness centrality was proposed by Freeman [29]; the function is
It is defined as the total number of shortest paths between two separate nodes and passing through node ; it reflects the load on node and can be alternately understood as the controllability of the node. On this basis, centrality can be clarified asand the normalisation of betweenness is described as the function shows, where is the maximum possible value.
The average shortest path length was proposed by Albert and Barabási [31]; the function is
It is defined as the average number of steps along the shortest paths for all possible pairs of network nodes.
The network efficiency was proposed by [32]; the function is
It shows the average efficiency of transit flow or information between nodes in the network.
2.2. Multilayer Network Representation Method
Gu et al. [33] used this method to study airline networks and rail networks. The formula of the undirected multilayer network (see Figure 2) can be represented asas the set of different layers; here, the superscript U is used to define the upper-layer network and superscript L to set the lower-layer [33, 34].
The connected network can be used to represent the rail network and urban street network as
The weight is denoted by in the function [35]. Red nodes represent rail stations, and blue nodes represent street intersections; solid lines represent their connections, and dotted lines represent cross links (cooperative relationship) between different layers. The rail network station is connecting with the nearest street network intersection and its weight is equal to 1 [23]. To simplify the transfer process, when measuring the accessibility, this study sets the transfer time between the two layers to 3 minutes [36]. The multimodal transportation and relations between the vehicle-based network and the pedestrian network can be gained from the multilayer network.
The multilayer network model of urban traffic networks; the upper-layer represents rail network topology, and the lower-layer represents the street network topology or OD zones.
In this definition, is the set of the street network topology, and N is the node number of lower-layer whenand E represents the unordered pairs of edges and elements of and is denoted by
Similarly, the definition of the multilayer network is
The adjacency matrix of networks is symmetrical and nonnegative, representing the connection between zones i and j, wherewhere is the Euclidean distance. Define to theoretically remove any self-connections to exclude the impact of the network element itself. Then, the adjacency matrix of multilayer networks is
2.3. Analysing the Coevolution Process of Multilayer Network
2.3.1. The Inner Connections of Multilayer Networks
For a big city, building all of these traffic links as railways is a waste of money and less effective, and only constructing streets without any massive rapid transit will cause a heavy traffic jam. The specific structure of the traffic multilayer networks is the result of the combination of natural and historical conditions, location characteristics, economic development and conditions, urban space layout, and even traffic customers. The network structures determine the relationships of different layers; they are highly related but have apparent differences. One of them is the complementary element of another, using their advantages to make up for the deficiency of others. The structural change of one layer will cause the change of another layer. Their combination completes the urban traffic networks functions
The traffic distribution is more focused on the choice of routes between origin-destination OD zones. It is an iterated process which compares the trips assigned with the link capacity to find an equilibrium status and show the number of travellers on each route and link in a given transportation network. The classic User Equilibrium (UE) is widely approved and used to describe the user’s route behaviour. Once the users believe that they find the smallest impedance (perceptual impedance), they will not change their routes, although the routes are not the smallest route and the impedance may not be the smallest route impedance. Thus, an equilibrium is reached. Related functions will be provided in later sections.
With the population movement and growth, the growth of upper-layer and lower-layer networks is determined; considering the UE problem, we use the Frank–Wolfe BPR method and set the travel cost on the congested links as infinite. The Frank–Wolfe algorithm is “an iterative first-order optimisation algorithm for constrained convex optimisation” [37, 38], which is widely used to deal with the traffic equilibrium problem such as UE. After that, the network structure data, the cost of each link, the OD demand data, and the capacity of each link are obtained; then, we initialise the optimisation aims, construct the impedance function, and find the initial impedance. Then, we use the All or Nothing Assignment technique [39] to assign the traffic flow and update the data, and finally the optimal function value converging to 5% of estimation error range can be obtained. This technique is widely used and well accepted by many traffic network researchers [38, 40–42].
2.3.2. The Coevolution Relationships between Traffic Multilayer Network and Urban Land-Use
Many related types of research have discussed their connections and inner relationships [43, 44]. Because the traffic capacity and travel speed are different, the impacts of network structure change to land-use typically can be measured by the accessibility [45, 46]. The spatial connection means that urban land-use growth is characterised by continuity in a planar space [47], and then it is meaningful for the measurement of the population diffusion and urban growth process. Urban land-use change process also has a positive correlation with the variation of the population [48].
Their coevolution relationships can be simply represented in Figure 3, in which the traffic generation and attraction are determined by the urban land-use spatial distribution; with the increase of population density, land-use intensity, and sprawl or generation of new land-use parcels, traffic demand is increasing with different scales, which naturally needs more traffic infrastructures to satisfy it. In turn, the increase of the investment of multilayer traffic networks, which make the traffic networks grow densely, increases the accessibility of land parcels; as more streets pass the area, people can easily access this previously less developed area. The abilities of networks can also be improved, as the investment is always used to increase the capacities and the topological network functions of each layer. Based on the location theory and the scarcity of urban land resources, the attraction of these land parcels connected with traffic networks will increase. Then, more people are attracted, and the coevolution process emerges. Here, the land-use coevolution model is developed to reflect the relationship of accessibility with population growth and movement, and traffic network growth. Other factors, such as house price and related policies, are excluded to keep this relationship succinct and clear. Hence, the land-use model is highly simplified, mainly focused on the accessibility, and represented as the distribution of population and employment.
To represent the competition characteristic of neighbourhood interactions, the land-use coevolution model also includes both centrifugal and centripetal forces [49, 50]. Assume that people always want to live near to the location of jobs to save travel cost, but far away from other people such as potential contestants. However, on the other hand, employment wants to be accessible both to other businesses (to save cargo transportation cost) and to people (who are the suppliers of labour and customers). Both of them want to live with a higher traffic network structure service grade. Related functions will be provided in Section 3.
3. The CNIMUGOM
3.1. The Framework of CNIMUGOM
The content of the model is suggested, as shown in Figure 4, and different blocks stand for different models, with the relationships between those models illustrated as arrows. The most famous models are the travel demand model, street investment model, traffic network growth model, population growth model, urban land-use growth model, and network optimisation model.
3.2. Traffic Demand Model
The demographic, socioeconomic, land-use, and online hailing data were incorporated to reveal the root causing that influence traffic demand [51–53]. When considering the traffic network growth, an important part that must be focused on is that this kind of growth is always constrained and stimulated by traffic demand. Based on the initial traffic network data, the classic “Four Step Model” (FSM) can be generated and used, considering the multilayer networks and the network coevolution process. The travel demand model in an initial network is predetermined by population and employment data and related to the topology of the multilayer network.
3.2.1. Trip Generation Model
A simple traffic generation model was given by Levinson and Zhu [49] and Xie and Levinson [54], where trip generation and trip attraction of different traffic zones were calculated; with respect to the population and employment and their natural carriers, households and companies, the simple linear relationships were proposed, and the assumption formula iswhere and stand for the number of trips generated and attracted; E and stand for the amount of the employment and population of traffic zone i, respectively; and , , , , , and are adjustable coefficients of this simple linear equation set, and their values are 0, 0.5, 1, 0, 1, and 0.5, respectively.
3.2.2. Trip Distribution Model
For the general distribution process, the mostly used doubly constrained trip distribution model is adopted, that is, a gravity-based model, to match both trip generation and attraction of locations based on a negative exponential function that assumes that the interactions of zones decrease with the travel time between them [50, 54]:where is the number of trips from zone i to zone j, and its table can be gained as the OD matrix; is the number of trips attracted in zone i; is the number of trips attracted by zone j; and is the travel cost. are the balancing coefficients which were precalibrated [55, 56].
3.2.3. Measurement of the Network Structure Related to Generalised Travel Cost
The generalised travel cost as normal travel cost is measured by travel time, a generalised concept considering the congested travel cost, zonal accessibility, and so forth from zone i to zone j and considering the intrazonal and interzonal costs. Hence, the equation can be written aswhere is the interzonal travel time which is the summation of travel costs along the shortest path between i and j, and represents the generalised travel time that vehicle spends on the link a; if this link exactly belongs to the shortest path between zone i and zone j, ; otherwise, . The embodies the generalised intrazonal travel time in zone i; this is a simplified measure method also for the convenient calculation. Sometimes, the intrazonal costs are neglected for convenient calculation, but they will be counted especially when the zone has higher land-use intensity and density; they capture a variety of costs incurred on trips associated with land-use and traffic network structure differences.
The travel time on the link a is measured as ; normally, it can be calculated as the ratio between the street length and the speed ; however, it is always constrained by the traffic flow and the collected revenue of this link :Here, is the balance parameter, which represents the average value of time.
The BPR function is used to reflect the relationship of the free flow time and congested speed on link a, is the link capacity, and and are correction factors equal to 0.15 and 4, respectively:
For example, in a zone with higher land-use intensity and density, maybe the jobs provided and population amount are much more than the average level; thus, the travel cost will increase correspondingly considering the traffic congestion levels, longer elevator wait time in skyscrapers, and greater difficulty of finding parking. On the other hand, influenced by the traffic network structure, within most of the situations (without considering Braess’s paradox), with higher traffic network structure service level, it will alleviate the traffic congestion and increase the accessibility, and usually the cost will be reduced. Based on these assumptions it can be measured aswhere is a specified base intrazonal travel cost for all zones, is a changeable parameter to balance the impact of network structure, and . is the number of activities (the employment and population) in zone i; the function is , connecting with the trip generation model. is the average number of activities for all zones. is the measurement of network structure status; generally for a zone with higher traffic infrastructure level, the time used to pass this zone will lower; hence, considering those fundamental traffic network indicators and MCA indicators, this study chooses the node degree centrality , the betweenness , and the closeness centrality as related indicators because of the previous analysis, and the function iswhere , , and are parameters equal to 1000, 1, and 10000, respectively, and is the average status of network structure for all zones.
3.2.4. Mode Choice in a Multilayer Network
In this research, mode choice is used to connect the travel demand with multilayer networks. Over this process, the trips between a given origin and destination are split into trips using transit and automobile for simplicity. The most frequently used function is Binary Logit Model [57], also called “econometric formulation,” which merely refers a log ratio of the possibility of choosing a mode () to the possibility of not choosing this mode , and the function iswhere are related parameters, and are the travel costs of using the mode of street and rail transit (the measurement of only considers the situation of travel using the street system, but considers using street and rail systems and the cooperation parts), and are the travel times for using the mode of street and rail transit, stands for income, and is the number of travellers. This function has deterministic meaning; that is, when faced with the same options, they will always make the same choices. Hereby considering the multilayer relationship, a function set is proposed; related parameters refer to the basic multilayer network indicators, combined with the BPR function and the real traffic loads on the link a. This function set iswhere and are the length of automobile and train transit routes, is the average speed of rail transit, is the rail traffic flow, and is the collected revenue of this rail link. The part also means the transfer and wait time cost; for a higher investment rail line, the transfer and wait time are usually smaller.
3.2.5. Traffic Assignment Model
Here, User Equilibrium (UE) model is chosen; this model is proposed by Sheffi [42] and has been well recognised and validated by the traffic researchers; here, it will not be discussed further, and these conditions can be referred to in Sheffi [42].
Then, the lower-level function is shown as
Here, is the equilibrium flows of link a, is travel time, and the function is purposed to get the lowest travel costs. The is the path, is the traffic flow of OD pair i and j, is the trips between i and j, is a definitional constraint, and is the set of links in the network.
3.3. Link Investment Model for Multilayer Network
The network growth process from the network structure parts is discussed previously, but in fact, this process is constrained by the investment inputs too. As mentioned before, this study is mainly considering a close network system; here, this study also treats the growth process as autonomous investment process. The link investment models are following the definition of Levinson et al. [49] and Xie and Levinson [11], where they describe the economic decisions of individual links autonomously; that is, the price for using the street during a given period, which users should pay for this autonomous agent, is dependent on the general traffic flow and the total length of these links. After a period, all the revenues are used to improve the network structure and link capacity. In each period, it is assumed that the collected revenues of toll from links by a general toll rate , and this is an iteration process:where if the study treats the network as a directed graph, and the total traffic flow in a given period k can be given as and , or for an undirected graph as , then the connection between the collected revenues and the travel time on the link a can be discussed; that is, is determined by the . is the link length of iteration .
On the other hand, the maintenance cost function is based on the toll function; considering the link capacity and average speed , , for an ordinary street or rail line, the capacities in two directions are normally equal to each other, and the overall spending function is
The function of this profit part is
Here, it is assumed that is the profit of iteration , also equal to the general investment of iteration , then connect with the network growth models and network optimisation models in the next sections.
The investment model considers the general revenues and expenditure of the system. Here, as per these previously proposed models, if the gathered revenue exceeds the maintenance cost, the profit part will be spent at the end of a time period without saving it for the future, with all being used for the capacity increase of each line, with the functionwhere . The free flow speed of new links with new capacity can be measured by a log-linear relationship calibrated by Zhang and Levinson [58] as .
3.4. The Population Growth Model
The coevolution model has illustrated some basic points of the population movements and their impacts; they will be further discussed here. When dealing with the urban expansion process, especially the land-use change process and the evolution of the urban street network, the population is treated as the most frequently considered mechanism and has the most significant consequence [59]. Typically, the population evolution model includes two parts: the population movement and the natural increase. The natural increase of population will change the amount of the total population, whereas the population movement will change the population distribution [60].
For simplicity, it can be assumed that the natural increase rate of the population is a constant, and then a simple exponential function can be limitedly used, as follows:where is the population increment of iteration , is the population amount of iteration , and is the increase rate, as per iteration [61].
3.5. Urban Land-Use Change Model
For the evolution models, inspired by the research of Levinson et al. [49] and Xie and Levinson [11], accessibility was used to reflect the desirability of a land parcel by calculating the accessibility of opportunities and activities [62], showing that smaller opportunities provide diminishing influences, also called gravity-based measurement, assuming that it fits a negative exponential cost function:where is the accessibility of node i to all opportunities in node j if the total number of traffic analysis nodes is J. The is the travel cost sensitivity parameter to the node which also indicates how the accessibility of a node declines with the increase of travel time, and is the travel cost from i to j.
The only factors which affect location choice of households and companies are accessibility and population density. Then, for the accessibility of employment and population in a land parcel, the function can be expanded aswhere is the accessibility to employment from node i while is the accessibility to population. In the multilayer model, ; hence, the accessibility considered different travel modes. This potential measure method has its specific practical advantage; that is, it can be easily calculated by the existing land-use and transport data.
3.6. Network Structure, Land-Use, and Population Coevolution Model
The desirability function sets are developed to illustrate the dynamics of employment accessibility and population accessibility based on independent decisions concerning their locations and network structure service grade . For the people desirability, the parameters are all equal to 1:and, for the dynamics of employment desirability, are equal to 0.9, −0.9 and 0.9, respectively:where, for the independent decisions made by people, the competition between employment accessibility and population accessibility acted as centrifugal forces. Conversely, for the independent decisions made by businesses, the employment accessibility and population accessibility both strengthen the employment desirability and serve as centripetal forces. are the related parameters, and these relationships between parameters were discussed in Xie and Levinson [10] as follows: “… accessibility to jobs did show a statistically significant positive effect on home sale values, while accessibility to resident workers did show a statistically significant negative effect …” (p.164).
The increase in employment and population in zone i is proportional to the single node’s desirability difference quantity; here, and are the average desirability at iteration k, considering the natural increase rate r, and ; then, we have the function
3.7. The Multilayer Network Growth Process
The multilayer network coevolution process is combined with the coevolution model. The design purpose of the rail network is to serve more people and increase convenience. Hence, the first objective is to maximise the total population covered. The second objective is to minimise the total travel costs—or to obtain a better network structure. For the upper-layer, a new node is randomly selected from these lower-layer nodes which have the highest population, and a new scale-free network is generated. For the lower-layer, a random number of new nodes (less than 10) will be added in, and a new scale-free network is generated. The network user will choose to use the rail networks for saving travel cost, and then, based on the limited capacity of rail networks, it has the upgraded distribution of traffic flow. Based on this, the population will be redistributed.
4. Solution Process and Related Simulation Scenarios
To solve the model, this section proposed a solution process given here, and the related simulation scenarios are discussed.
4.1. Solution Process
Here is the solution process by steps: Step 1. Set the initial amount of population and employment (different land-use situations) and the travel speeds, and distribute them in a square area with 100 km of side length and 100 nodes. Here, the random network is used to discuss the growth model, and the population is equal to the employment. Step 2. Based on the functions of traffic demand model, at each step, it can have the initial OD matrix and the initial travel costs, and the traffic flows can be measured based on the traffic assignment model by User Equilibrium (UE) model and Frank–Wolfe algorithm and converge to 5% of estimation error range. Step 3.The collected revenues and overall expenditure can be calculated by the traffic flows. Generally, consider the spending of the total profit for the network growth, and the multilayer network growth process introduced here. With the growth of multilayer networks, the network growth will be based on them with optimised APL, D, and travel cost. After the network growth and considering these flows on the rail networks, the free flow speed of new links, and link capacity, traffic flows on different links including upper-layers can be updated, and an equilibrium status can be reached again. Step 4. Considering the change of population growth and network structures, land-use accessibility and desirability are changed, which causes the redistribution of population and employment, and then the OD matrix and traffic flows change. Then, with the new distribution of population and employment, the new travel speeds and network structure can be obtained; return to Step 1.
4.2. Related Simulation Scenarios
As per Step 3 and the initial status of the simulated city proposed, some scenarios will be considered here as in Table 1. For simplicity, here, we merely tested the model with the regular network not growing, and then more focus was given to the multilayer network growth-related models. Here, this study considers 8 different simulation scenarios; for example, RRR stands for the initial population and employment, travel speeds are randomly distributed (see Figure 5), and so on. As the relationship between land-use and population density could be modelled well [48], the discussion of different land-use scenarios can turn to the discussion of the distribution of population and employment.
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5. Simulation and Validation
In the same way as the simulated city, suppose that a square urban area with length equal to meters has an initial number of nodes (randomly and regularly distributed when the network is not growing to test the proposed model, and randomly distributed when the network is growing) , with an initial population of 500 thousand, and the natural increase rate of population is 1.5% per iteration. Then, the model validation part is proposed at the end of this section.
5.1. When the Network Is Not Growing
The total profits are used for the capacity increase of each line, which means that although the network is not growing, the capacities are improving. Because the network structure is not changed, it cannot measure the change of APL, D, EG, and so on; hence, the study will more focus on the total travel times which not only show how easily one citizen can arrive one area but also describe the traffic status of the whole network. With the initial traffic flows on each street, the total travel time cost through these congested streets is also shown in Figure 6. With iteration, we can see that this time cost of different simulation scenarios decreases sharply until reaching a stable state (Figure 7) (results come from 10 times of simulation), which means that the street capacity is optimised during the population redistribution process. No matter what kind of simulation scenarios the network initially chose, at last, they have similar trends. The population is redistributed following the network travel cost change, and some areas with higher population also emerge because of the change of street capacity and travel speed.
Here, the red lines are the travel times of each street, and the width stands for their values. The black lines are the traffic flows on these streets. Here is the performance of the random network and regular network in 6 iterations.
5.2. When the Network Is Growing
With the growth of networks, based on a random network shown in Figure 8, in line with previous researches, the change trends of APL and D can be obtained. Considering these different scenarios, we can see that, with the increase of multilayer networks, the APL and D are ultimately decreased. With the growth of the rail network, it is clear that more traffic flows are attracted by the rail network, and after a few steps, the traffic flow becomes more evenly distributed.
APL decreased from more than 5.9 × 103 to around 4 × 103, a decrease of about 32% (see Figure 9), and D decreased from 15 × 104 to 8 × 104, a decrease of about 46% (see Figure 10). With the network growth process, the increase of network efficiency (Figure 11) by around 27% to 44% can be seen, with the URU having the lowest change rate and RUU having the highest change rate, which means that, with the multilayer network growth model, the proposed network structure became optimised. The total traffic flow is optimised by around 30% of decrease (Figure 12) by the process of redistribution of population and employment and land-use change, which reduce the movement between two nodes and make the population and employment reach the balance in an area, and intrazonal movement increases. Obviously, the total travel cost is also decreased; this also means the improvement of network efficiency. Some nodes with higher population have emerged with their advanced traffic accessibility, and “new town” or “developing area” also emerged, as more and more new nodes are located in the right middle area.
5.3. Model Validation
The change of average travel speed ratio can influence the model choice and the growth of networks [23]. To validate the simulation results, another quantitative index is proposed here as beta, the average travel speed ratio between upper-layer and lower-layer networks, and the function iswhere is the average travel speed of the street network and is the average travel speed of the rail network. Here, we set the average travel speed of the rail network as a constant; then, as the value of beta increases from 0 to 1, this means that the average travel speed of the street network is increasing.
Here, to validate the model, we have run the CNIMUGOM 100 times with UUU simulation scenario with the same network and have the average value of the APL (see Figure 13) and traffic flow (see Figure 14) with different beta values. It is shown that, with the value of beta belonging to 0.2 to 0.3, the APL has the lowest value. While the situation changes a little bit when the traffic flow is considered, showing that the traffic flow has the lowest value when the value of beta belongs to 0.1 to 0.2. Furthermore, the CNIMUGOM works pretty stably with different beta values and simulation scenarios.
6. Conclusion
Research about coevolution models and multilayer network models remains somewhat superficial; the combinations of coevolution models and multilayer network models with land-use applications are relatively rare. While research in this area is resolutely forward-looking and fulfils the requirements of mathematical ability, as with previous city models, it lacks a systematic and comprehensive economic description and combination with urban economic theory.
The proposed CNIMUGOM has expanded the traditional planar model to a multilayer network model, which can better fit the real situations. This research also considered the multilayer network coevolution process further and considered the self-organisation properties of networks. This model has proposed an efficient way to combine these different traffic modes, well considering the complex evolution and coevolution relationships between network structure, land-use, and population. CNIMUGOM can be used to optimise the APL and D, to make the network have smaller total traffic flow, and save the total travel cost; in other words, we can have a more efficient network structure. The inner relationships between submodels such as the travel demand model, street investment model, traffic network growth model, population growth model, urban land-use growth model, and network optimisation model were analysed and discussed.
The proposed CNIMUGOM has some superior properties compared to other models, as it is a universal model and can be used in many cities for their modelling optimal planning schemes and for the large-scale investments. However, some parts can be further improved. For example, the growth rate of the population is based on the exponential distribution for short-term analysis, which may not be very close to the real situation, and the logistic population growth model can be used for long-term analysis to adjust it. The study of economic models is still open and can be combined with big data; with the study of CNIMUGOM, the population distribution acquisition method of big data can be adopted; for example, the cellphone signalling data and GPS data can be used to represent the distribution of the urban population. Next, we proposed that all the revenues are used to improve the network structure and link capacity; however, in reality, the incomes can hardly be all used for the improvement of the networks. Thirdly, the redistribution of the population, for now, is relatively hard, which has to consider the place attachment for the population and the difficulty of its relocation, but from the experiences of Beijing, based on the public rental housing projects, it can be partly done. For now, the CNIMUGOM is based on the simulated city data; for a real city, the model can be easily applied with road network data, GPS data, and population data.
Data Availability
The simulated data used to support the findings of this study are included within the article, and related codes are available from the corresponding author upon request.
Disclosure
This paper is modified and expanded based on Section 4.4 of Rui Ding’s doctoral dissertation.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was funded by National Natural Science Foundation of China (No. 72001053) and Science and Technology Planning Project of Guizhou Province of China (No. Qian ke he ji chu [2020]1Y283). Universiti Putra Malaysia is hereby acknowledged.