Signal Control Method for Urban Road Networks Based on Dynamic Identification of Critical Nodes

Abstract

Ensuring the reliable operation of all nodes in a road network is often challenging due to the influence of managed resources and other dynamic factors. This study proposes a method for identifying critical nodes based on multi-attribute decision-making, aimed at enhancing traffic efficiency and reliability. By utilizing dynamic traffic flow data and real-time ranking of node criticality, an adaptive signal optimization approach was developed to establish a collaborative control method for road network signals. First, a quantitative analysis was conducted to evaluate the impact of road network topology, traffic volume, and travel time reliability, enabling a comprehensive ranking of critical nodes. Subsequently, based on real-time traffic flow and critical node rankings, a signal collaborative control method was established to optimize travel time reliability while mitigating congestion and resource inefficiencies. Case analysis revealed that nodes with higher OD (origin–destination) pairs do not necessarily exhibit high traffic flow or criticality, underscoring the importance of targeted signal control strategies. The results demonstrate that the proposed optimization method effectively improves the dynamic reliability and operational efficiency of road networks while contributing to sustainable transportation by enhancing adaptability to traffic fluctuations. This study provides theoretical and practical references for advancing sustainable traffic management and supporting the transition to smarter transportation systems.

1. Introduction

Following the completion of new urban road network infrastructures, current research is focused on the optimization of traffic signal control in urban intelligent transportation. To improve the operation efficiency of the road network traffic system, a lot of research has been conducted to develop various traffic signal control methods [1,2]. One of the commonly used strategies is to learn and obtain the optimal policy function through intelligent algorithms [3,4,5]. However, in large-scale urban road network areas, because of the dynamic and random traffic state and increase in the number of nodes in the controlled areas, the difficulty of solving this traffic signal control method expands exponentially, resulting in unstable control effects of the road network system.

In addition to the aforementioned constraints, in practice, owing to the influence of management resources and other factors, it is often impossible to ensure that all nodes and road sections in the road network operate in a reliable state [6,7,8]. By prioritizing management and controlling critical nodes that have a significant impact on road network reliability, the travel time reliability of the entire road network can be maintained at an appropriate level. Studies have revealed the universality of intermediate faults in road networks [4]. These studies suggest that when critical nodes in the road network fail, multiple intersections fail owing to connection relationships, resulting in the paralysis of the entire road network [9,10]. A previous study showed that if 5% or 10% of important nodes fail, the whole network will be paralyzed [11]. Therefore, it is crucial to scientifically and reasonably define and identify critical nodes and then apply control strategies to achieve a preferable performance of the road network.

Critical nodes refer to nodes that have a significant impact on the structure, function, and reliability of a network [12]. The failure of a small number of critical nodes in a transportation network can greatly reduce the operational efficiency and reliability of the network. Currently, sustainable development is one of the key objectives in modern urban planning and management. As a critical component of the transportation network, the optimization and management of critical nodes in the road network play a vital role in traffic flow regulation, resource allocation optimization, and the enhancement of the reliability of transportation systems. Prioritizing the management of critical nodes can alleviate traffic congestion, reduce energy consumption, and minimize pollutant emissions, thereby providing significant support for achieving the sustainable development of transportation systems. Therefore, achieving collaborative control between critical nodes and other nodes is expected to make the operation of the road network better adapted to the complex and variable traffic environment. It is crucial to identify critical nodes and analyze their mechanisms that affect the overall operation of the network.

Under dynamic and unpredictable traffic flow conditions, traditional static management methods are no longer adequate to cope with the complexity of urban traffic, necessitating the development of dynamic and adaptive traffic control strategies. This study introduces an adaptive collaborative control method for road networks based on the dynamic identification of critical nodes, aimed at enhancing the operational efficiency and reliability of urban traffic systems. By integrating a multi-attribute decision-making framework and real-time traffic data into traffic signal control, this method seeks to improve travel time reliability and adapt to the typical variable traffic conditions in urban environments. The research contributions include (i) a quantitative analysis of the impact of the topology structure, node flow, and reliability indicators of the road network on the identification of critical nodes and establishment of a dynamic identification method for critical nodes in the road network based on multi-attribute decision-making, (ii) achieving collaborative optimization of road network signals based on dynamic traffic flow data and real-time ranking of node criticality, and (iii) establishing simulation experiments to verify that the proposed adaptive control method improves the travel time reliability of the road network.

This paper is structured as follows. Section 2 provides a literature review, focusing on the identification of critical nodes in the road network and dynamic signal control methods while taking into consideration the influence of critical nodes, thus laying the theoretical foundation for this study. Section 3 describes the research problem and defines the parameters. Section 4 introduces the optimization model, which includes a dynamic identification method for critical nodes and a signal collaborative control method for the road network. The subsequent section presents the experimental validation and analysis. Finally, the paper concludes with a summary of the main findings and contributions, along with a discussion of potential directions for future research.

2. Brief Review of Previous Research

2.1. Identification of Critical Nodes in the Road Network

Previous studies have often used static methods to rank the criticality of nodes, such as node connectivity [13,14], node betweenness centrality [15], and node aggregation coefficient [16,17]. However, these indicators did not consider the dynamic characteristics of traffic flow in the network. Therefore, some scholars proposed using indicators such as peak-hour traffic flow (the hourly traffic volume of a day when traffic volume is at its peak) [18], traffic load degree (the ratio of actual traffic density on the road to the density under maximum capacity) [19], and saturation degree (the ratio of maximum traffic volume to maximum capacity) [20] to characterize the traffic operation characteristics of road networks. Due to the difficulty of accurately reflecting the importance of road network nodes as a whole using a single indicator [21,22], multi-indicator comprehensive evaluation methods have gradually emerged to reflect the importance of road network nodes from different perspectives [21,23]. A comprehensive analysis of multiple attributes of networks, combined with the low complexity of local attribute index calculation and the high accuracy of global index calculation, establishes methods for identifying critical nodes in networks, such as the grey relational analysis method (GRA) [24,25], principal component analysis (PCA) [26], cosine similarity method (CSM) [27], TOPSIS method [28], etc. Therefore, it is reasonable to establish a method for identifying critical nodes based on multi-attribute metrics. At the same time, optimizing the management of the road network by dynamically identifying critical nodes will further improve the management efficiency of the urban road network.

2.2. Dynamic Signal Control Methods

Traffic signal control optimization has always been an important area of research in intelligent transportation systems. A substantial body of research has sought to improve traffic signal control strategies and thereby enhance traffic operational efficiency through advanced traffic signal control methods. There are three primary methods of traffic signal control: pre-timed signal control, vehicle detection, and adaptive control. The pre-timed control method, proposed in the 1950s, remains the most widely used approach today, with the drawback of not being able to adjust dynamically according to traffic conditions [29,30,31]. The vehicle detection method uses sensors to detect whether there are vehicles passing through the intersection, which then triggers a switch to another signal phase. It has some dynamic characteristics but cannot adapt to complex traffic scenarios [32,33,34]. Adaptive control captures the traffic state at intersections (traffic flow, delay, signal phases, etc.) dynamically, learns from these characteristics through a model, and adjusts some parameters of the traffic signal dynamically [35,36]. With the advancement of intelligent transportation technologies, research on traffic adaptive control methods based on reinforcement learning has recently received widespread attention. Based on the dynamic identification of critical components of the road network, scholars have employed various adaptive control methods to study the intelligent management and control of the road network. A comparison and summary of the related research are presented in

Table 1. Comparison and summary of the related research.

Critical Components	Signal Control Methods for Road Networks	Optimized Performance Metrics	Experimental Platform/Applicable Scenarios
Critical Intersections (Critical Nodes)	Adaptive control based on deep recurrent Q-networks (DRQNs) [37,38] Adaptive control based on deep reinforcement learning (DRL) [39,40] Adaptive signal based on the reinforcement learning algorithm of advantage actor critic (A2C) [41]	Average delay Travel time Average waiting time Average queue length Average CO2 emissions Efficiency of key intersections	Interactive simulation experimental environment Experiments based on SUMO (Simulation of Urban Mobility) Network with large events by SUMO
Critical Road Sections or Critical Paths	Active perimeter control based on the macroscopic fundamental diagram [42] Real-time control optimization method of a supersaturated intersection group based on bobbin combination [43] Regional signal coordination control method based on the importance of depth search of intersections [44] Coordinated green-wave control method on arterial roads considering critical path sequence [45]	Maximum critical route capacity Equilibrium saturation of approaches Average delay of networks from a global perspective Maximum weighted sum of green-wave-bandwidth of critical path sequences	Road networks with recurrent congestion point Experiments based on SUMO and Synchro 7 VISSIM simulation with actual road networks
This Research
Real-Time Critical Nodes	Scheme selection adaptive control based on optimization theory	Uniform delay Incremental delay Travel time reliability	Experiments based on Dynameq 2.13 and Matlab R2018a

. The research on critical nodes within regional road networks predominantly utilizes limited traffic data to optimize signal management, aiming to enhance the operational performance of these networks. Investigations into critical paths or sections employ dynamic recognition-based signal optimization control to boost traffic efficiency across intersection groups or subnetwork areas, thereby reducing vehicle delays. This study adopts a novel perspective by considering traffic efficiency and travel time reliability within the road network. It leverages a ranking system for critical nodes and implements collaborative signal control strategies to improve the operational performance of the transportation system.

3. Problem Description

Because of the influence of managed resources and other factors, it is often impossible to ensure that all nodes and links in the road network operate in a reliable state. Nodes are a critical component of road networks and a frequent source of vehicle congestion and blockage. By prioritizing the management of critical nodes that have a significant impact on the operation of the road network, it may be possible to ensure that the road network operates in a better state. Therefore, this study employed multi-attribute decision-making theory to identify critical nodes in a random environment by determining the criticality of all nodes in real time, thereby achieving efficient control of the road network.

The research framework is shown in Figure 1. The left side of the figure is a schematic diagram of the road network. The right side of Figure 1 shows the workflow of the two main works in this study. The first one is the dynamic identification of critical nodes, which begins by establishing the critical node evaluation matrix based on the attribute indicators that affect the criticality of nodes in the road network. This study examines the critical nodes from three perspectives: the topological structure of nodes within the network, the traffic flow managed by nodes, and the influence of nodes on the road network’s travel time reliability. Since different evaluation attributes have different meanings and dimensions, a multi-attribute decision-making method was used to standardize and normalize the evaluation matrix, ultimately ranking the criticality of nodes. Another one is to achieve collaborative control of road network signals. Utilizing node origin–destination (OD) pairs, real-time traffic flow, average node delay, and the ratio of delay to travel time, real-time criticality ranking of nodes was derived. Subsequently, corresponding signal control schemes were applied to optimize road network management and achieve collaborative control between critical nodes and other nodes. Finally, the changes in road network reliability before and after optimization were compared, and the effectiveness and applicability of the proposed optimization method were analyzed.

A full list of the variables used in this manuscript, along with their definitions and role in the model, can be found in

Table 2. Parameter description table.

$j$:	The node $j$ in the road network, $j=\mathrm{1,2},3\dots \dots J$;
$k$:	The criterion $k$ for identifying the critical node, $k=\mathrm{1,2},\mathrm{3,4}$;
$a$:	Link $a$ in the road network, $a=\mathrm{1,2},3\dots \dots A$;
$f^t(j,a)$:	The traffic flow on link $a$ connecting node $j$ during time window $t$, pcu/h;
$U\left(t\right)$:	The evaluation matrix of critical nodes during time window $t$;
$u_{j1}$:	The number of OD pairs through node $j$ during time window $t$;
$u_{j2}$:	The traffic volume through node $j$ during time window $t$, pcu/h;
$u_{j3}$:	The average delay of vehicles through node $j$ during time window $t$, s;
$u_{j4}$:	The ratio of average delay to average travel time of vehicles through node $j$ during time window $t$;
$E_1$:	The weight coefficient of OD pairs;
$E_2$:	The weight coefficient of the traffic volume;
$E_3$:	The weight coefficient of the average delay;
$E_4$:	The weight coefficient of the ratio of average delay to average travel time;
${CR}_j\left(t\right)$:	The standardized evaluation matrix of nodes during time window $t$;
${co}_{jk}$:	The contribution degree of node $j$ under criterion $k$;
$E_k$:	The weighting coefficient of criterion $k$;
$G_j\left(t\right)$:	The criticality of node $j$ during time window $t$;
$C_{opt}$:	The optimal signal cycle, s;
$L$:	The total lost time, s;
$Y$:	The sum of the maximum flow ratios for all signal phases within a cycle;
$C$:	The cycle length, $C=C_{opt}$, s;
$\lambda$:	The green time ratio;
$x$:	The degree of saturation;
$e$:	The correction factor for node signal control types, with a value of 0.5 for pre-timed signal controls;
$Q$:	The capacity of lanes, pcu/h.

.

4. Optimization Model

4.1. Identification of Critical Nodes Based on Multi-Attribute Decision-Making

We identified critical nodes from three aspects: the location of nodes in the road network topology, traffic volume through nodes, and the impact of nodes on the reliability of the road network. The critical nodes in this study include nodes that occupy important positions in the road network, undertake significant traffic volume, and have a significant impact on travel time reliability.

Occupying an important position in the road network was quantified using the number of OD pairs and paths through the node. Each OD pair is connected by multiple paths. A signal control node has multiple different paths passing through, among which different paths may have the same OD pair. This study uses “the OD pairs of nodes” to represent the number of OD pairs for all paths passing through each signal control node, which can also be understood as the number of paths that a signal control node passes through. Undertaking significant traffic volume was quantified by the traffic volume through the node within a given time. The significant impact on travel time reliability refers to the negative impact on the network’s travel time reliability if the travel time reliability of the node is reduced, resulting in a decrease in the overall operational efficiency of the road network.

The definition of travel time reliability in this study is as follows: if the ratio of delay to travel time for a trip is less than a specified threshold, the trip is considered reliable [46]. It is believed that the ratio of delay to travel time is a critical attribute affecting the travel time reliability of the road network in this study. Taking an intersection as an example, during a certain time window, if the ratio of delay to travel time and average delay for all vehicles leaving the intersection is lower, the probability of being less than a pre-set threshold is higher. This indicates that the travel time reliability is higher, the fluctuation of travel time is smaller, and the service level of the intersection is higher. Therefore, the average delay and the ratio of delay to travel time are considered critical indicators reflecting the travel time reliability of the road network. Since delay and the ratio of delay to travel time are important indicators reflecting the travel time reliability, minimizing the delay control as the optimization objective has a positive effect on improving the travel time reliability of the road network.

Based on the four aforementioned evaluation indicators and multi-attribute decision-making theory, the importance ranking of all nodes in the road network mainly involves three steps:

• Standardization of the Critical Node Evaluation Matrix

The number of nodes in the network is $J$, and the evaluation index of critical nodes are the OD pairs is $u_{j1}$. The traffic volume is $u_{j2}$, the average delay at the node is $u_{j3}$, and the ratio of delay to travel time is $u_{j4}$. The evaluation matrix $U$ for nodes during the time window $t$ is calculated as follows:

$$U(t)=\left[\begin{array}{ccc}{u}_{11}& \cdots & u_{14}\\ ⋮& \ddots & ⋮\\ u_{J1}& \cdots & u_{J4}\end{array}\right]$$

(1)

The evaluation matrix $U$ was normalized by normalizing the attribute indicator values of different categories and units into comparable values according to the following formula:

$$U{cr}_j\left(t\right)={\displaystyle \frac{u_{jk}}{b_{jmax}}} k=\mathrm{1,2},\mathrm{3,4},j=\mathrm{1,2},3,\dots \dots J$$

(2)

where $b_{j max}$ = $\underset{j}{\max }{u}_{jk}$. The standardized evaluation matrix is calculated as follows:

$${CR}_j\left(t\right)=\left[\begin{array}{ccc}{cr}_{11}& \cdots & {cr}_{14}\\ ⋮& \ddots & ⋮\\ {cr}_{J1}& \cdots & {cr}_{J4}\end{array}\right]$$

(3)

ii.

Determination of the Weight of the Evaluation Index

After obtaining the evaluation matrix of critical nodes, the information entropy of each evaluation index was determined using the entropy method, and the weight coefficients were then obtained. If the information entropy of the evaluation index is small, it indicates that it contains fast information, and therefore, it has a significant weight coefficient. However, if the information entropy is high, the weight coefficient is small.

The contribution degree of node $j$ for the evaluation index $k$ is ${co}_{jk}$, which is calculated as follows:

$${co}_{jk}={\displaystyle \frac{{cr}_{jk}}{\sum _{j=1}^J{cr}_{jk}}}$$

(4)

where ${co}_{jk}$ represents the information of evaluation indicator $k$. The total contribution of all nodes is the weight coefficient $E_k$ of the evaluation indicator $k$, calculated as follows:

$$E_k=-\gamma \sum _{j=1}^J{co}_{jk}\ln {co}_{jk}$$

(5)

where $\gamma$ is a constant, $\gamma ={\displaystyle \frac{1}{\ln J}}$, and $0\le E_k\le 1$. If ${co}_{1k}=\cdots ={co}_{jk}=\cdots ={co}_{Jk}$, $E_k\to 1$ and $E_k=0$.

iii.

Comprehensive Sorting of Critical Nodes

After obtaining the weight coefficient $E_k$ of each evaluation indicator $k$, the real-time criticality of each node can be obtained based on the four evaluation indicator values, as follows:

$$G_j\left(t\right)=E_1\times u_{j1}+E_2\times u_{j2}+E_3\times u_{j3}+E_4\times u_{j4}$$

(6)

4.2. Signal Collaborative Control Process for the Road Network

The crux of the performance of adaptive signal control relies on whether the detected flow truly reflects the current state of traffic flow. If the detected flow is consistent with the actual situation, the corresponding signal control scheme can handle the current traffic demand. If the flow rate does not match the actual traffic state, the signal control scheme may not be able to adapt to the current traffic demand and, thus, cannot achieve the expected control performance. In particular, in situations with high traffic flow, the probability of signal control schemes failing to meet traffic demand will significantly increase owing to vehicle congestion and road network imbalance.

In this study, based on the dynamic identification of critical nodes in the road network and combined with real-time traffic flow information, a scheme selection adaptive control method was adopted to achieve collaborative control of node signals in the road network. During each time window, it was first determined whether the real-time traffic flow exceeded the pre-set threshold. If it did, a change in the signal timing plan was required. Before changing the signal timing plan, the critical nodes of the road network were identified. If these nodes were consistent with the historical nodes, the signal timing plans for these historical–critical nodes were optimized. If they were not consistent, the signal timing plans for the newly identified critical nodes were optimized. Specifically, the optimal signal cycle for all nodes in the road network was determined according to Equation [7,47]. Then, with the optimization objective of minimizing the sum of uniform delay and incremental delay in controlled delay, as shown in Equation [8,9,48], the signal timing scheme under different signal cycles was determined.

$$C_{opt}\left(L, Y\right)=(1.5L+5)/(1-Y)$$

(7)

$$d_1=0.5C{\left(1-\lambda \right)}^2/\left[1-\min \left(1,x\right)\lambda \right]$$

(8)

$$d_2=900C[\left(x-1\right)+\sqrt{{\left(x-1\right)}^2+({\displaystyle \frac{8ex}{QC}})}]$$

(9)

$$\min f\left(C, \lambda ,x, e,Q\right)=d_1+d_2$$

(10)

Based on the dynamically identifying critical nodes, the signal collaborative control method for the road network was established. The process of optimization is shown in Figure 2.

Step 1: Real-time detection of relevant parameters of nodes, including OD pairs $u_{j1}$, traffic volume through node $u_{j2}$, average delay $u_{j3}$, the ratio of delay to travel time $u_{j4}$. Obtain the signal control scheme for the road network, including cycle length $C$, green time ratio $\lambda$. Dynamically obtain traffic flow saturation $x$.

Step 2: Determine whether the traffic flow of the current road network is greater than the pre-set value.

Step 3: The signal control scheme remains unchanged when the current traffic flow is less than the pre-set value and returns to Step 1. When the traffic flow is greater than the pre-set value, use the critical node identification method proposed in the previous section to identify the critical nodes of the current road network.

Step 4: If the current critical node is consistent with historical–critical nodes, optimize the signal timing of the historical–critical nodes using Formulas (7)–(10), $C=C_h^{\prime }$, $\lambda ={\lambda }_h^{\prime }$. If otherwise, optimize signal timing for new critical nodes, $C=C_r^{\prime }$, $\lambda ={\lambda }_r^{\prime }$.

Step 5: Continue to detect real-time traffic data and update the road network signal control scheme.

In the process of signal collaborative control of the road network, the optimization objective is to minimize the sum of uniform delay and incremental delay at critical nodes for signal optimization control. The travel time is mainly composed of driving time and delay. If the driving time remains constant and the delay decreases, the ratio of delay to travel time will also decrease. The probability of the delay to travel time ratio being less than a predetermined threshold increases. According to the definition of travel reliability in this study, the travel time reliability of critical nodes is improved; therefore, the travel time reliability of the road network is optimized.

5. Experiment and Analysis

5.1. Experimental Road Network

The road network established is shown in Figure 3. Two signal control methods were used to optimize the travel time reliability of the case road network, namely, (1) only optimizing critical nodes and (2) simultaneously optimizing critical nodes and the overall signal strategy of the road network. Both control methods were based on the dynamic identification of critical nodes in the road network.

The traffic simulation software Dynameq 2.13 was used to conduct the simulation experiment. Nodes 1–16 are signal-controlled nodes, while nodes 22–25 are nodes that connect the virtual road section of four centroids (OD points) with the actual road section. The default simulation time of Dynameq is twice that of the required time to ensure that the path travel time information used by the path selection algorithm is sufficiently complete. When the simulation is completed, dynamic allocation enables the road network to reach a homogeneous state.

The traffic demand period set in the simulation experiment was 3 h. At the end of the demand period, the traffic flow of the road network reached the highest state. After 3 h, the road network was controlled using the default dynamic allocation of Dynameq. The physical range of each node in the road network was pre-marked, and this range was not composed of intersection parking lines. During the experiment, the arrival time of all vehicles entering the node through the starting position was recorded, and when they left the node’s physical range, the departure time was recorded. Then, the travel time of all vehicles at each node was calculated during a time window.

The iteration process of each simulation was 50 times. Following the completion of the experiments, data required for the identification of critical nodes, such as traffic volume through nodes, the average delay of nodes, the ratio of delay to travel time of nodes, and the number of OD pairs, were synchronized and saved. Subsequently, programs for the identification of critical nodes and the calculation of travel time reliability were developed by MATLAB R2018a. By inputting the experimental output data and running the program, a ranking of the criticality of all nodes in the network and the travel time reliability before and after optimization were derived. Based on the identified critical nodes, signal optimization control was applied to these nodes, thereby achieving coordinated optimization of the network signals.

In the experiment, the optimized signal timing schemes for critical nodes were determined by minimizing control delay. In the process of identifying critical nodes, the traffic volumes of all nodes were based on historical data. However, in practical applications, critical nodes are identified based on real-time traffic volume.

The signal timing plans for the three control strategies are shown in

Table 3. Control scheme for the experimental road network.

Control Strategy	Control Scheme
	Applicable Objects		Signal Timing
			Cycle/s	Green Light/s	Red Light/s	Yellow Light/s	Full Red/s
Pre-timed Control	All Nodes		70 s	32	32	3	3
Pre-timed Control Based on Critical Node Identification	Critical Nodes		100	47	47	3	3
	Other Nodes		70 s	32	32	3	3
Scheme Selection Adaptive Control	Control Scheme 1	Critical Nodes	100	47	47	3	3
		Other Nodes	70 s	32	32	3	3
	Control Scheme 2	All Nodes	100	47	47	3	3

. The first and second control strategies were signal timing control, and the third control strategy was a scheme selective adaptive control. The first control strategy used the same signal timing plan for all nodes. The second control strategy utilized separate timing plans for critical nodes and the other nodes. Regarding the third control strategy, when the traffic volume is low, the signal timing scheme with a high green signal ratio was used to control critical nodes, while the signal timing scheme with a low green signal ratio was used to control the other nodes. When the traffic volume exceeded the pre-set value, the high green signal ratio was used for all nodes. In the experiment, the statistical period of traffic flow was measured in units of time windows, with a time window of 180 s.

5.2. Dynamic Identification of Critical Nodes

Webster’s method [44] and the HCM (Highway Capacity Manual) method [45] were used to determine the timing control scheme for the road network. The method proposed in Section 4.1 was then used to dynamically identify critical nodes. The criticality ranking of 16 signalized nodes in the road network was calculated, as shown in Figure 4.

The left side of Figure 4 depicts the experimental road network, with the identified critical nodes 3, 10, 13, and 16 prominently marked with red numbers. The right side of Figure 4 illustrates the variation in the criticality of the 16 nodes during the simulation period. When the ranking of criticality is higher, the node is more critical in the road network. The criticality variation of each node is represented by a line graph. The horizontal axis represents the sequence number of the time window, with each time window being 180 s long. The vertical axis represents the ranking of the node’s criticality within each time window. It is evident that nodes 3, 10, 13, and 16 have a significantly higher criticality ranking compared to other nodes, which validates the effectiveness of the dynamic critical node identification method.

This study primarily identifies critical nodes in the road network based on attributes such as the location of nodes in the road network topology, traffic volume through nodes, and the impact of nodes on the reliability of the road network. Figure 5 illustrates the variation in these attribute indicators for all nodes in the road network during the simulation. Figure 5a displays the changes in the average delay of all nodes. Figure 5b shows the changes in the ratio of delay to travel time. The variations in these two indicators represent the evolution of the travel time reliability of all nodes. The red solid line box in the figure indicates that nodes 3, 10, 13, and 16 had significantly higher delay and ratio of delay to travel time compared to other nodes, suggesting that these nodes were critical in influencing the travel time reliability of the road network. Accordingly, optimizing the travel time reliability at nodes 3, 10, 13, and 16 could enhance the overall travel time reliability of the road network.

Figure 5c depicts the variation in traffic flow across the 16 nodes, reflecting the topological position of each node in the road network. The red solid line box in Figure 5c indicates that the traffic flow passing through nodes 3, 10, 13, and 16 was significantly higher than that of other nodes, suggesting that these nodes occupy pivotal positions in the topological structure of the road network.

Table 4. OD pairs of 16 nodes.

Number of Nodes	OD Pairs	Number of Nodes	OD Pairs
1	11	9	10
2	11	10	6
3	19	11	10
4	12	12	10
5	9	13	17
$\underset{\dots }{6}$	$\underset{\dots }{18}$	14	11
7	11	15	13
8	7	16	15

presents the number of origin–destination (OD) pairs for all paths passing through each of the 16 signal control nodes in the experimental road network. When the number of paths through a node is high, the node within the network’s topological structure is more important. The black solid line in the table highlights that nodes 3, 13, and 16 had a high number of paths passing through them. It is interesting to note that node 10 had a relatively low number of paths passing through it, while node 6 had a relatively high number of paths (as indicated by the black dashed line). However, node 10 was identified as a critical node, whereas node 6 was not. As shown by the red dashed line boxes in Figure 5a–c, node 6 had lower average delay, ratios of delay to travel time, and traffic volumes compared to the other identified critical nodes. Therefore, it can be concluded that when the number of OD pairs for nodes is high, their criticality may not necessarily be high.

In summary, the analysis of the attribute indicators such as the topological structure of all nodes in the road network, the traffic flow passing through the nodes, and the impact of the nodes on the travel time reliability of the network leads to the conclusion that the dynamic identification of key nodes in the road network based on multi-attribute decision-making theory is justified.

5.3. Road Network Control Based on Critical Node Identification

Section 5.2 validates the rationality of critical node identification. The experimental findings indicate that nodes 3, 10, 13, and 16 consistently rank in the top four in terms of criticality, leading to the conclusion that these nodes are pivotal within the road network.

In this section, further analysis was conducted on the travel time reliability of road networks based on critical node identification under three different control schemes, which include pre-timed signal control, pre-timed signal control based on critical node identification, and scheme selection adaptive control. The performance of signal control is shown in Figure 6 and Figure 7.

Figure 6 illustrates the evolution of travel time reliability in the road network under three different signal control strategies. The horizontal axis represents the number of time windows, with each time window being 180 s. When the 60th time window had elapsed, the traffic demand in the network peaked. Subsequently, the simulation software employed the default dynamic assignment to control the network. The vertical axis represents travel time reliability. The scatter points in three different colors represent the travel time reliability under the three control strategies.

When the simulation was started, the road network was gradually loaded with traffic flow, and the travel time reliability gradually decreased from 1. Frames 1, 2, and 3 represent the change in the characteristics of travel time reliability during low and high network flow and stable network operation, respectively. Frame 1 shows the travel time reliability during medium-low traffic flow. In this phase, the travel time reliability under signal control based on critical node identification and adaptive control strategies was significantly higher than the travel time reliability under pre-timed signal control. As shown in Frame 2, as traffic flow increased to a saturated state, travel time reliability further decreased. During this period, the travel time reliability under the scheme selection adaptive control strategy was significantly higher than that under pre-timed signal control. Frame 3 shows that when the network’s traffic loading ended and the software adopted the default dynamic assignment to control traffic flow, the travel time reliability of the network basically stabilized, and the scatter points exhibited a clustering characteristic.

Under the three control strategies, the travel time reliability of the network showed a trend of rapid decrease followed by stabilization, which was related to the network’s operation characteristics presented by traffic flow transitioning from free flow to congested flow, and the trend of reliability changes was in line with actual conditions.

As shown in Figure 7, signal timing control based on the identification of critical nodes and scheme selection adaptive control improved the travel time reliability of the road network compared to signal timing control. When traffic flow is low, signal timing control based on the identification of critical nodes can obtain better performance, and the reliability of travel time is twice that of signal timing control. In the later stage of increased traffic flow, scheme selection adaptive control achieved better travel time reliability.

6. Conclusions

This paper establishes a dynamic identification method for critical nodes based on multi-attribute decision-making theory and proposes a collaborative control method for road network signals, with the aim of optimizing travel time reliability. The conclusions drawn from the simulation case analysis are as follows:

• The multi-attribute decision-making approach allows for a reasonable identification of critical nodes in dynamic road networks. When identifying critical nodes within a road network, it is essential to consider multiple attributes. Traffic flow at nodes serving multiple origin–destination (OD) pairs may not always be high.
• The road network signal cooperative control strategy developed in this study, which is based on the dynamic identification of critical nodes, enhances travel time reliability. The signal control strategy that focuses on critical node recognition proved to be effective, particularly in low-traffic scenarios, doubling travel time reliability compared to the fixed-time control strategy. Additionally, the adaptive control scheme is better suited to accommodate fluctuations in traffic flow within the road network.

This study employed Dynameq to construct an experimental road network for facilitating the analysis of the proposed method’s effectiveness and applicability. Future research endeavors will incorporate real-time traffic data from actual road networks and aim to conduct field experiments within these networks to the greatest extent possible. Additionally, the solution method utilized in this study will be benchmarked against intelligence optimization techniques or reinforcement learning methods to further explore the feasibility and effectiveness of signal control strategies that rely on the dynamic identification of critical nodes.