Evaluating the Connectivity and Imbalance Contribution of New Sections towards Highway Network: A Complex Network Perspective

. Te evaluation of the impacts of new sections on the highway network is an essential aspect of the feasibility study. Existing studies predominantly concentrated on engineering-oriented feasibility assessments, often overlooking their potential efects on parallel sections and the overall network. In this research, we present an evaluation model for new sections based on complex networks, focusing on the connectivity and imbalance of transportation networks. Tis model serves as a supplementary approach for enhancing the feasibility analysis of new highway projects. Te model comprises three distinct modules, namely, complex network, eigenvalue, and evaluation. Terein, the complex network provides diverse attributes for sections with the dynamic edge weights. Moreover, probability betweenness centrality and volume betweenness centrality have been presented as an eigenvalue of sections based on the multilayer complex network. Furthermore, the connectivity evaluation based on the eigenvalue and the imbalance evaluation based on the entropy and Gini coefcient are conducted. Trough the case study, the results of the model demonstrate the connectivity and imbalance contribution of new sections and provide a novel perspective for the feasibility study.


Introduction
Te planning, construction, and operation of highways have resulted in the emergence of regionalization and networking in the transportation systems engineering [1], as there has been an improvement in the connectivity of infrastructure.However, the increasingly complex highway network has exhibited a spatial-temporal imbalance that may have an adverse impact on trafc, leading to the waste of the highway's resources.Te new sections can redistribute trafc assignment in the network impacting the associated sections and the overall network.However, quantitative measure of the impacts on the partial and overall network imbalance is lacking, and the dynamic features of trafc on the road sections and network are not precisely assessed in the feasibility study of new road sections [2].Terefore, the evaluation of the impact of new sections on the network and the associated sections is necessary.In our study, we present an evaluation model that evaluates the impact of new sections on the connectivity and imbalance using the complex network methodology.Te connectivity refers to the ability of the road section to facilitate transportation that refects the signifcance of sections in the network, and the imbalance represents the distribution of property in the network that refects the network stability.Te connectivity and imbalance contribution of new sections can be evaluated with the model.
Te remainder of this paper is organized as follows."Related Work" (Section 2) ofers the comprehensive overview of the current literature related to the research.In "Methodology" (Section 3), we introduce the evaluation model for new sections in details. "Case Study" (Section 4) then performs the detailed analysis of new section's impacts for the case study.Finally, conclusions and future directions are presented in "Conclusions and Future Directions" (Section 5).

Related Work
In the existing researches, the complex network methodology has been adopted to study the highway network which transforms highway elements and their relationships into nodes and edges in networks, which can be used to describe the behavior of the system and the relationships of elements in the network [3][4][5].Villas Boas et al. [6] established a complex network of highways with cities as nodes.Xiao [7] analyzed the complexity of the highway network structure by establishing a road network structure based on logical relationships of elements.Xu et al. [8] established a directed network from the data of the signal control system of a city trafc system and evaluated the performance of key nodes.Costa et al. [9] used indicators such as the node degree, average shortest distance, and betweenness centrality for performance measurement.On one hand, the impacts of new sections can be assessed and some methods have been proposed for evaluating the connectivity, such as roadway capacity and the level of service as outlined in the highway manual [10], travel time analysis by Schrank and Lomax [11], and user satisfaction-based evaluation by Levinson and Lomax [12].Moreover, Schrank and Lomax [13] and Brown et al. [14] used congestion indicator based on the trafc volume, travel distance, and travel time to assess the degree of congestion.Shim and Yeo [15] used indicators such as in degree, out degree, and betweenness centrality to evaluate the connectivity.Feng [16] proposed the connectivity based on the network stability evaluation.Tian et al. [17] and Ando et al. [18] used the section length, capacity, and efciency to evaluate the road network connectivity.Liu and Yan [19] used current-fow efciency to evaluate the network connectivity performance.On the other hand, the imbalance is another part of the evaluation.Zheng et al. [20] used the node degree, point intensity, intensity distribution and average shortest path in the network, and clustering coefcient, etc., for balance analysis.Li et al. [21], Sun et al. [22], and Yu et al. [23] analyzed the imbalance feature of spatial-temporal distribution of expressways at the microaspect by considering factors such as travel time and occupancy rate.Deng [24] analyzed the imbalanced spatialtemporal distribution of trafc volume of a single section or route of the highway.For the imbalance analysis, indicators such as road network area density and road transportation density have been selected by He et al. [25] and Fang [26].Dai [27] studied the imbalance of trafc.Furthermore, in the feasibility study of newly constructed road sections [2,28], the four-stage prediction method was utilized to analyze and predict the trafc volume, in terms of annual average daily trafc volume, to determine the capacity of new road sections.However, the analysis of trafc of-peaks and peaks has been lacking in detail.
Drawing on these research studies, it can be concluded that the evaluation of road networks is typically predicated on the one aspect, i.e., network's structure or trafc.Each approach reveals unique features of the network from diverse perspectives.Nevertheless, the common challenge arises in accurately accounting for the impacts of new sections on parallel sections and network structure.In details, the connectivity of parallel sections and the imbalance of network are afected by the new sections and accurately assessing these impacts remains problematic.

Methodology
3.1.Problem Formulation.Te highway system, characterized by the bidirectional trafc fow and fully enclosed operation, is represented with the directed weighted network based on neighboring nodes, and this paper evaluates the impacts of new sections from the following two aspects.Defnition 1. Connectivity is the ratio of property between the section and network that refects the importance of the section in the network.Te formula is expressed as follows: where C e represents the connectivity of section, e is the road section in the network, G is the network, and f(e) and f(G) are the functions of property on the section e and network G, respectively.
Defnition 2. Imbalance is the sum of the equilibrium function of elements that refects the degree of unequal distribution in the network.Te formula is expressed as follows: where I represents the imbalance, c is the property of sections, and f(c) represents the equilibrium function.
Te impact of new sections on parallel sections can be analyzed by assessing their connectivity, which refects the change of importance of both new and parallel sections.In addition, network imbalance can provide insight into the trend of the distribution of network property.

Te Evaluation Model for New
Sections.Te evaluation model for new sections encompasses three interrelated modules, namely, complex network, eigenvalue, and evaluation.Te module "complex network" encompasses the multilayer network, which is further composed of the structural network, weighted network, and origin-destination (OD) network.In the module "eigenvalue," two betweenness centralities are conducted based on the module "complex network" to capture both structural and trafc features of road sections.Finally, in the module "evaluation," the imbalance analysis and connectivity analysis are implemented.Figure 1 visually illustrates the composition of the model and intricate relationship between these modules.

Complex Network of the Highway.
Te structural network denotes as directed by the graph G � (V, E, A), which represents the connection relationships of elements in the highway system.Tese elements include nodes and edges.Nodes correspond to entrances and exits of highway, 2 Journal of Advanced Transportation which are described as a set containing m nodes, denoted as Te directed edges are represented as E � e 1 , e 2 , ...e n  , which is the set of the road section between nodes.Te adjacency matrix for nodes, denoted as A ∈ R n×n , signifes the existence of the connection as follows: if the road section e ij � (v i , v j ) exists, then a ij � 1; otherwise, a ij � 0.
In addition, in the weighted network, the weight is the generalized cost function of the edge, defned as w ij that consists of the road toll, vehicle operating cost, and time value cost.Due to trafc varying over time, the dynamic edge weights based on the trafc volume are constructed to capture the dynamic properties accurately.Te mathematical expression of the generalized cost function is presented in the following equation: where w ij represents the general cost of the section e ij ; K ηij is the road toll of the vehicle type η passing the section e ij ; δ η is the operation cost of the vehicle type η (including fuel, tyre wear, and car maintenance) [29]; L ij is the length of the section e ij ; φ η is the time cost of the user; T ij is the travel time of vehicles in free-fow condition; v ij is the trafc volume; C ij is the road capacity; and α and β are hyperparameters of the travel time.Furthermore, moving on to the OD network, each trip contains this information, such as vehicle's plate number, vehicle type, origin, destination, departure time, and arrival time.Te collection of all trips forms the OD network.Finally, with the multilayer network, the evaluation model can analyze the highway systems accurately.

Eigenvalue of the Road Section.
Te eigenvalue of the road section consists of probability betweenness centrality and volume betweenness centrality that are involved with the road structure and trafc, respectively, and refect the signifcance of each road section in transportation supply and demand [30].
(i) Probability betweenness centrality of road section based on the multinomial logit.
In transportation networks, probability betweenness centrality is the probability that the road section is contained in the paths of OD pairs in the network.Tis measure serves as an indicator of the road section's connectivity in the transportation supply.Te higher the probability betweenness centrality is, the greater the connectivity of the road section towards the network will be.Te mathematic formula is expressed as in the following equation: where pb e ij is the probability betweenness centrality for the section e ij ; p st,m represents the probability that the m th path of the OD (s, t) pair is chosen; is a variant of 0 or 1; if the m th path consists of the section e ij , then ξ st,m � 0; N is the number of nodes; and M is the number of available paths of the OD (s, t) pair.To obtain the path choice probability, the study adopts the multinomial logit (MNL) model that is a general tool most widely used to model the travel behavior of transportation system users.Tis approach calculates the probability based on their weights; the formula is expressed as in the following equation: where p st,m is the probability that the m th path is chosen; θ and ϕ are discrete parameters; w denotes the weight of each path as systematic utility; and M is the number of available paths of each OD (s, t) pair.(ii) Volume betweenness centrality of the road section.
Volume betweenness centrality is the ratio of the trafc volume on the section to the total trafc Journal of Advanced Transportation volume in the network over a statistical period.Tis measure serves as an indicator of the road section's connectivity in transportation demand.Volume betweenness centrality is afected by the spatialtemporal trafc; the higher the value is, the greater the connectivity of the section is.
where vb e ij is volume betweenness centrality of the section e ij ; T is the statistical period; τ is the conversion coefcient; η is the vehicle type; q η st is the volume of η type vehicle for the OD (s, t) pair; p η st,m represents the ratio that the m th path of the OD (s, t) pair is chosen for μ e ij st,m is 0 or 1; if the m th path contains the section e ij , then μ e ij st,m � 1; otherwise, μ e ij st,m � 0; and M is the number of available paths of the OD (s, t) pair.

Evaluation on the Connectivity and Imbalance of New
Sections.Te connectivity and imbalance contributions of new sections towards highway network can be analyzed with these eigenvalues.Te following will explain the evaluation in details: (1) Connectivity Analysis.Connectivity attribution contains trafc supply and demand aspects, specifcally denoted as probability betweenness centrality and volume betweenness centrality in this study.By calculating the connectivity of road sections prior to and following changes in the road network structure, the impact of new road sections on the parallel sections can be evaluated.
(2) Imbalance Analysis Based on Entropy and Gini Coefcient.To evaluate the imbalance, the concept of the entropy and Gini coefcient are adopted as indicators.By analyzing the temporal and spatial imbalance on the probability betweenness centrality, the infuence of new sections on the imbalance is evaluated for the feasibility study.
Entropy is a measure of the degree of intrinsic properties change within a system.Te mathematical formula can be expressed as in the following equation: where S represents the entropy value, and a i represents the property value.Te Gini coefcient, a widely used measure of the inequality of a distribution, is utilized to evaluate the imbalance of system properties.Specifcally, it enables the assessment of the imbalance in the allocation of resources.Te Gini coefcient increases as the degree of imbalance rises.Te following equation provides the mathematical expression: where Gini represents the Gini coefcient; d n is the value of the property; and N denotes the number of road sections.
Tese concepts can be clarifed by some examples.Table 1 demonstrates four examples with diferent conditions for the imbalance analysis.In example 1, the before and after distributions are even distribution; the entropy increases while the Gini keeps the same.In example 2, the condition is increasing an invalid section, i.e., "3": 0.0; the entropy keeps the same while the Gini increases.In example 3, the condition is changing the distribution order; both the entropy and the Gini keep the same.In example 4, the condition is closer to even distribution; the entropy increases and the Gini decreases.From these examples, we can conclude that the entropy increases or the Gini decreases means more balance and both the entropy and the Gini coefcient are adopted together for accurately evaluating the imbalance.

Case Study
4.1.Data Information.Te evaluation model is validated with the highway network of the Guangdong-Hong Kong-Macao Greater Bay Area (the Greater Bay Area, GBA) in China (see Figure 2).Te dataset of the study area comprises road network structure data and vehicle trip data that were collected over a period of seven days from July 5, 2021, to July 11, 2021.Te data format is illustrated in details in Tables 2-4 that contains node information, road section information and their connection relationship, and trip information.
Te Pearl River Estuary cross-river highway consists of six sections, namely, A, B, C, D, E, and F, as shown in Figure 3.At present, sections A, B, and D bear the cross-river trafc that the average daily trafc volume is 337 k passenger car unit (PCU) and increases over time, so these roads are facing great trafc pressure.Fortunately, sections C and E are currently under construction and section E is an eightlane bidirectional highway designed for the speed limit of 100 km/h, scheduled to be opened to trafc in 2024.Section C is a sixteen-lane bidirectional highway designed for the speed limit of 100 km/h and is expected to be opened to trafc in 2027.Te evaluation model was adopted for assessing the impacts of the new sections C and E on the network and the parallel cross-river roads, i.e., sections A, B, D, and F.

Establishment of Complex Network.
For the highway network in the Greater Bay Area, the network has been established with 764 nodes and 1730 edges, and the edges of the network are assigned dynamic weights.Te network is depicted in Figure 3.

Connectivity Analysis.
Tere are three parameters that need to be determined, i.e., θ, ϕ, and M in equation (5).Te larger the value of θ, the more familiar the road users are with the road network.Its value depends on θ 2 � π 2 /6δ 2 , where δ 2 is the variance of random residuals in the random utility theory.By calculating with samples, θ � −4.0; ϕ is the compensation parameter, here ϕ � 0. Considering the road network's scale, the deviation of the probability betweenness centrality, and the calculation cost, M is taken as 3, i.e., M � 3.

Static Connectivity Analysis.
According to whether the network includes the new section C or E, the network could be named as the existing network (without new sections) and the future network (with new sections).Figure 5(a) illustrates the distribution of the probability betweenness centrality in the existing network; as the value varies, the chromaticity shifts accordingly, and higher values yield a shift towards the red end of the spectrum and lower values result in a shift towards the blue end.To calculate the volume betweenness centrality, the average volume value during the period of 7:00-19:00 from July 5 to 11, 2021, was utilized.Gray sections in Figure 5(b) denote missing data or not yet opened.
In Figure 5, we can conclude that a minority of sections plays a pivotal role in the connectivity, and the distribution of probability betweenness centrality and volume betweenness centrality is relatively consistent that shows approximate equilibrium between transportation supply and demand.However, it should be noted that there is no positive correlation between probability betweenness centrality and volume betweenness centrality for some sections, as illustrated in the red ellipse in Figure 6; each point represents a distinct road section, and these sections exhibit lower probability betweenness centrality but higher volume betweenness centrality.As such, the assessment of the connectivity should be the combination of both probability betweenness centrality and volume betweenness centrality.
For the cross-river sections, Table 5 shows the changes of the connectivity in the existing and future networks, where pb i b and pb i a denote probability betweenness centrality of    From Table 5, it can be concluded that  pb i b � 0.1046 for section ABDF and  pb i a � 0.1095 for section ABC-DEF; both values are approximately equal to each other and that means the new sections do not signifcantly increase the overall probability betweenness centrality of the cross-river roads and only redistribute the trafc on these sections.Furthermore, the analysis shows that section D in the existing network has the highest probability betweenness centrality, while section C in the future network has the highest probability betweenness centrality; the new sections have a signifcant impact on the connectivity of sections B and D, while sections A and F are minimally afected.To summarize, the new sections have an impact on the connectivity of the cross-river roads when sharing the trafc pressure of sections B and D but have little impact on the overall network.Notably, due to the trafc prohibition of large buses and all trucks, section D has higher probability betweenness centrality but lower volume betweenness centrality compared to sections A and B.

Dynamic Connectivity Analysis.
Te edge weight and path selection are afected by the spatial-temporal trafc volume; meanwhile, the probability betweenness centrality and volume betweenness centrality change with the trafc.To investigate this relationship, the temporal features of connectivity in sections A, B, and D of the cross-river roads were analyzed.Te dynamic characteristics of the trafc volume, probability betweenness centrality, and volume betweenness centrality for these sections are shown in Figure 7.
Te trafc status of the sections can be classifed with the variation tendency of the indicators in the dynamic analysis from Figure 7. Terein, section A showed a slight decrease in probability betweenness centrality and an increase in volume betweenness centrality, indicating the increase in congestion but still in the acceptable range for the users.Moreover, section B experienced the decrease of both the probability and volume betweenness centrality, indicating congestion and out of the acceptable range for the users and the connectivity of the section is decreasing.Conversely, section D had an increase in both indicators and the connectivity of the section is increasing, indicating there is no serious congestion.

Imbalance Analysis.
Tis study evaluates the network imbalance with the probability betweenness centrality that involves the road structure and trafc.In the static imbalance analysis, the road structure is considered, while in the dynamic imbalance analysis, the efect of the trafc is taken into account.

Static Imbalance Analysis.
Te impacts of new sections that contain the imbalance of the cross-river roads and the network are evaluated in the static imbalance analysis.Te probability betweenness centrality is adopted as the indicator for assessing the static imbalance of network with new sections.Te values of the entropy and Gini coefcient of the probability betweenness centrality are presented in Tables 6 and 7, respectively.
In Table 6, signifcantly increased 40.23% was observed on the entropy of the cross-river roads with new sections compared with the cross-river roads without new sections and that indicates an improvement in the road structure of the cross-river roads.While, due to the high trafc capacity of section C, the Gini coefcient increased by 3.24%, and there are some efcient policies which can be considered for decreasing the Gini coefcient, such as restricting vehicle entry and raising tolls.Furthermore, the entropy and Gini coefcient remain basically unchanged for the existing and future networks as in Table 7, indicating that no impact of the new sections on the network imbalance was demonstrated in the presented evaluation.

Dynamic Imbalance Analysis.
Te impact of dynamic trafc volume on the imbalance of the network is investigated.Te analysis of the imbalance is performed separately for the cross-river roads and the existing network.
Te results are presented in Figure 8.Our investigation reveals an inverse relationship between the entropy and Gini coefcient in dynamic imbalance analysis.A higher entropy value corresponds to a lower Gini coefcient, and conversely, as depicted in Figure 8.As the trafc volume grows, the entropy decreases and the Gini coefcient increases for the cross-river roads, indicating increased imbalance in this context.In contrast, the entropy value increases as the Gini coefcient decreases for the overall network, signifying a reduction in imbalance across the entire network.As can be seen, the imbalance of the cross-river roads should get more attention during trafc peak hours and not the overall network.

4.5.
Discussion.An initial objective of the new section project was to identify its feasibility.In the context of redistributing trafc assignment due to the integration of new sections into the existing road network, it is imperative to assess the infuence of these new sections on both the network itself and the associated sections.In the present investigation, we conducted evaluations with a focus on connectivity and imbalance, using the metrics of probability betweenness centrality and volume betweenness centrality, which respectively signify trafc supply and demand.Tese evaluations are expected to serve as valuable tools for the optimization of various parameters related to new sections,    Connectivity within a transportation network is intricately linked to the physical structure of roads and the volume of trafc they accommodate [15,25].To evaluate the extent of the contribution to connectivity, we conducted the comprehensive analysis of both the new sections and their interconnected counterparts in the network, and this analysis hinged on the principles of probability betweenness centrality and volume betweenness centrality grounded in the domain of the complex network theory.In addition, we performed the imbalance analysis, employing a dual eigenvalue approach, to evaluate the imbalance with the entropy and Gini coefcient; this composite index was devised to address the limitations of individual metrics.
Drawing upon these meticulously devised metrics, we formulated the evaluation model within the framework of the complex network theory.Te present results are signifcant in at least three major respects.First, the quantifcation of the infuence of new sections on the connectivity and imbalance of parallel sections is achieved through the innovative utilization of probability betweenness centrality and volume betweenness centrality which are rooted in trafc supply and demand considerations.Tis approach introduces a supplementary means for enhancing feasibility studies and provides a pathway to expedite and refne assessments.Second, the dynamic interaction between these two betweenness centrality metrics afords insight into the trafc state of road sections and their temporal behavior.Tird, the combination of the entropy and Gini coefcient for measuring imbalance demonstrates enhanced precision in the imbalance of the parallel sections and the network, thereby establishing a robust theoretical foundation for efcient trafc management strategies.
In keeping with prior research, our study underscores the signifcance of comprehensive impact analyses pertaining to new road sections, encompassing connectivity and imbalance evaluations focusing on trafc supply and demand.Notably, the adoption of complex network-based models facilitates more expeditious analyses.Nonetheless, certain limitations warrant acknowledgment.First, the challenges associated with data collection and processing are pronounced, particularly in regard to trafc data, where data quality and its potential impacts on the model's results require further investigation.Second, the treatment of specialized road sections, including those with restrictions on heavy vehicles, presents a distinctive challenge; accordingly, the model calls for refnement, possibly through the vehicle type classifcations in the eigenvalue calculations as a viable solution to this issue.

Conclusion and Future Directions
Tis is important for evaluating the impact of new sections because the connectivity and imbalance contribution of new sections can be predicted in the feasibility study; thus, designed guidance can be provided to generate more accurate road design parameters, such as lane number, speed limit, and location.Tis study presents the evaluation model of new sections based on complex networks, focusing on the connectivity and imbalance contribution of new sections; the model can be used as the supplementary approach for the feasibility study of new highway.Te model comprises three modules, i.e., complex network, eigenvalue, and evaluation.In "complex network," the multilayer complex network has been established that provides multiple attributes of sections and incorporates the dynamic edge weight for accurately calculating the transportation costs; meanwhile, the dynamic weight is the foundation of dynamic analysis.In "eigenvalue," probability betweenness centrality and volume betweenness centrality have been established which represent the attribution of sections on trafc supply and demand, respectively.Terein, the MNL model is adopted for the trafc assignment for the probability betweenness centrality and the beneft is that it is consistent with the actual routing selection almost.In "evaluation," on one hand, the connectivity evaluation of road sections is based on probability and volume betweenness centrality.Te case study is conducted with the highway network in the Greater Bay Area to validate the evaluation model.Te result shows that the new sections are efective in sharing the trafc pressure of sections B and D by 65.1%, and the status of trafc in sections can be judged by the dynamic analysis; On the other hand, the imbalance evaluation is based on the combination of the entropy and Gini coefcient.By adding the new sections C and E, the entropy of the probability betweenness centrality on the cross-river roads increases by 40.23%, indicating an improvement in its structure, while the Gini coefcient increases by 3.24% for the high trafc capacity of section C; meanwhile, the imbalance analysis shows that local sections should be given more attention during trafc peak hours rather than the overall network.To summarize, by evaluating the connectivity and imbalance contribution of new sections, the results of the model can provide guidance for feasibility study of new road sections.
In the further research, it is anticipated that the changes of the road structure will lead to diferent trafc assignments, and in order to improve the accuracy of connectivity and imbalance analysis, it is necessary that trafc demand prediction should be employed for trafc properties analysis.

Figure 1 :
Figure 1: Te evaluation model for new sections based on the complex network.

Figure 2 :
Figure 2: Te Greater Bay Area in maps.

Figure 3 :Figure 4 :Figure 5 :
Figure 3: Te network for the Greater Bay Area.

Figure 7 :
Figure 7: Dynamic characteristics of sections A, B, and D in the existing network.(a) Trafc volume.(b) Probability betweenness centrality.(c) Volume betweenness centrality.

Figure 8 :
Figure 8: Dynamic characteristics of the imbalance with probability betweenness centrality.(a) Te cross-river roads.(b) Te existing network.

Table 1 :
Examples for the imbalance analysis.

Table 3 :
Road information and connection relationship.

Table 6 :
Te imbalance of the cross-river roads.

Table 7 :
Te imbalance of the existing and future networks.