# EtHgSC: Eigen Trick-Based Hypergraph Stable Clustering Algorithm in VANET

Authors: Mays Kareem Jabbar; Hafedh Trabelsi
Journal: Journal of Electrical and Computer Engineering
 (2023)

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## Abstract

A smart city’s vehicular communication strategy is important. A significant problem with vehicular communication is scalability. Clustering can help with vehicular ad hoc network (VANET) problems; however, clustering in VANET faces stability problems because of the rapid mobility of the vehicles. To achieve high stability for the VANET, this paper presents a new efficient Eigen-trick-based hypergraph stable clustering algorithm (EtHgSC). This algorithm has a twofold scheme for stable CH selection. In the first part of the proposed scheme, the cluster generation is handled using an improved hypergraph-based spectral clustering algorithm using the Eigen-trick method. The “Eigen-trick” method is used to partition both vertices and hyperedges, which provides an approach for reducing the computational complexity of the clustering. The cluster head (CH) is chosen in the second part, taking into account the requirements for keeping a stable connection with most neighbors. In addition to relative speed, neighboring degree, and eccentricity that are used to select the CH, the vehicle time to leave metric is introduced to increase the CH stability. The grey relational analysis model is used to find each vehicle’s score, and the CH is selected based on the maximum vehicle’s score. The results show the supremacy of our proposed scheme in terms of CH lifetime, cluster member (CM) lifetime, and the change rate of CH. Also, the proposed scheme achieves a considerable reduction in terms of packet delay.


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## Body

## 1. Introduction

An intelligent transportation system (ITS) has emerged as a popular research area in recent years because of the ongoing development of wireless communication technologies and embedded systems. An essential component of the ITS is the vehicular ad hoc network (VANET). VANET is a subset of mobile ad hoc networks (MANET) [1]. Additionally, the majority of the VANET clustering algorithms were evolved from the earlier MANET clustering algorithms, such as the mobility-based metric for clustering (MOBIC) in [2] and the weighted clustering algorithm (WCA) in [3]. In a VANET, vehicles communicate with each other and fixed infrastructure via vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I), respectively. Clustering the vehicles can address problems with scalability, broadcast storms, hidden terminal problems, and quality of service that VANETs experience because of the high mobility of the vehicles [4]. Vehicles group together to form clusters according to similar factors, including the vehicles’ speed, direction, number of cars within the transmission range, and position. Each cluster has a cluster head (CH) that is responsible for the cluster and the cluster members (CMs) [5]. Many clustering algorithms have been introduced, but their effectiveness declines as the number of vehicles in a city environment increases. Because of the high mobility of vehicles, cluster stability is reduced, so VANET necessitates a special clustering technique that considers mobility-related characteristics to prevent frequent cluster breaking. The communication link weakness is also one of the major issues in the urban environment.The method used for cluster formation and CH selection in any clustering algorithm is very important to achieve stability. In a graph representation of a network of vehicles, a vehicle node is connected to two other vehicle nodes [6–8]. This graphical representation might work well in areas with low population density, such as highways or sparsely populated cities. In contrast, graph theory does not apply in dense urban environments where a vehicle is always connected to more than two other vehicles. A good way to represent dense vehicle networks is with hypergraphs.A hypergraph is frequently used to describe multiple interactions between items in a cohesive way, and spectral clustering is one of the best methods for dividing those objects (vertices) into various communities [9]. Hypergraph spectral clustering has been used to cluster the vehicles in the cluster formation phase. The authors in [10] proposed hypergraph partitioning using the tensor trace maximization (TTM) method.This paper presents the Eigen-trick-based hypergraph stable clustering algorithm (EtHgSC), which has a twofold scheme for stable clustering. In the first part of the proposed scheme, the cluster generation is handled using an improved hypergraph-based spectral clustering algorithm using the Eigen-trick method. “Eigen-trick” method is used to partition both vertices and hyperedges, which provides an approach for reducing the computational complexity of the clustering. To improve clustering, the Eigen trick is used to calculate the modified Laplacian value in the TTM [10]. The CH is chosen in the second part, considering the requirements for keeping a stable connection with most neighbors. In addition to relative speed, neighboring degree, and eccentricity that are used to select the CH, the vehicle time to leave metric is introduced to increase the clustering stability. The grey relational analysis model is used to find each vehicle’s score. The vehicle with a high score is selected as a CH for each cluster at each instant. The contributions of this paper are listed as follows:(i)
The hypergraph-based spectral clustering algorithm is introduced for cluster formation, and the Laplacian value of the TTM is modified using the Eigen-trick method. It considers the transformable connection between the vertex Laplacian and the hyperedge Laplacian, which can speed up the solution of eigenproblems without losing information.(ii)
The fast-moving vehicle may leave the cluster at any time and go out of the CH’s transmission range. In that case, the packet drop would be higher. It is better to know the leaving time of the vehicle in the cluster. Time to leaveTleave is an important parameter to consider when selecting a stable CH.(iii)
A vehicle may change its direction, so estimating the next position of the vehicle beforehand is necessary to know. Thus, the predicted vehicle direction at the next instant is calculated with the help of the predicted vehicle position by the predictive directional greedy routing protocol (PGRP). It helps to select the lane and its length to get the necessary estimated time to leaveTleave′. This parameter reduces the frequent cluster breakage at junctions.(iv)
A relational analysis is developed using grey relational analysis (GRA) between four CH selection parameters instead of knowledge-based weightage. GRA is a decision-making model that uses for a few data (when we don’t have too much data). The scheme for selecting a CH is used by the GRA to find each vehicle’s score for selecting the CH. By using this scheme, strong connectivity and a stable link lifetime are obtained.The rest of this article is organized as follows. The related works are introduced in Section2; Section 3 presents our proposed EtHgSC model with its parts (cluster generation and CH selection phases). Simulation results and discussion are presented in Section 4. Finally, the conclusion and future work are introduced in Section 5.


## 2. Related Works

Different clustering algorithms have been introduced for VANET, which their primary goal is to increase cluster stability; we present some of them in this section.Affinity propagation algorithm (APROVE) was presented in [11]. APROVE algorithm was proposed using the affinity propagation algorithm in a distributed manner. The mobility metrics are used in CH selection. The authors asserted the existence of clusters with excellent stability. With the same concept of mobility metric, another algorithm was proposed. A mobility-based scheme for dynamic clustering in VANETs (MoDyC) was presented in [12] to improve clustering stability.Vehicular multihop algorithm for stable clustering (VMaSC) was proposed in [13], and it is used in VANET to build stable multihop clusters with a small number of CHs. The CH is selected based on the node with the least mobility via multihops, which is determined as a function of the speed differential between neighboring nodes. Also, the VMaSC improved in [14] by integrating IEEE 802.11p-based multihop clustering and the long-term evolution (LTE). This algorithm has high citations; it was used as a reference in many articles.Based on the concept of vehicular mobility in a highway environment, Arkian et al. [15] presented a new clustering scheme based on multi-hop.Some algorithms used other metrics for the selection of the CH. Passive multihop clustering (PMC) was presented in [16], and it uses the priority neighbor following strategy for selecting the CH. This strategy improves clustering stability and reduces the cost of clustering effectively.The authors in [17, 18] used the fuzzy logic scheme to improve the network’s reliability and stability.Other algorithms are considered a VANET as a graph. Clustering techniques in graph theory are divided into two categories: hierarchical clustering and partitioned clustering. On the basis of the existing cluster structure, hierarchical algorithms construct clusters. These algorithms build partitions sequentially, whereas partitioning methods only use one partition to separate nodes into clusters.Some methods employed a K-mean approach and combined it with other algorithms to build stable clusters, such as K-mean and Floyd-Warshall algorithms (KMFW) in [19], the K-means clustering method, a new routing protocol (KMRP) in [20], and a betweenness centrality based clustering (BCBC) in [21]. The K-mean method is one of the simplest partitioned clustering methods. The K-mean method is not capable of disconnecting the nonlinearly distinguishable clusters. To resolve this, spectral clustering is used based on eigenvectors of the matrix [10].The authors in [22] introduced position-based prioritized clustering (PPC), which generates clusters based on geographic location information and vehicle priorities. The CH election process is similar to the computation of minimum dominating sets used in graph theory. Other algorithms used heuristic algorithms for cluster formation [23]. Hybrid clustering algorithm based on roadside (HCAR) was presented, and this algorithm is a centralized approach with road side units (RSUs) distribution.However, a cluster-based VANET-oriented evolving graph (CVoEG) [6] is an evolving graph that uses spectral clustering approach for cluster formation. Another algorithm proposed by Khan et al. [7] is also based on graph theory and uses a heuristic algorithm for cluster formation. The authors selected CHs based on connectivity and determined its eccentricity. The CH was chosen as the highest eccentric vehicle in a cluster. This algorithm forms stable clusters and increases the probability of connectivity of the elected route. Table 1 presents clustering algorithms in the literature in terms of the CH selection metrics, and simulator tools used.Table 1
State-of-the-art comparison.
ReferenceAlgorithmCH selection metricSimulator tools[11]APROVEDistance and speedNS2[12]MoDyCRelative position, moving direction, and link lifetimeNS2, SUMO[19]KMFWAverage distanceNS2[13]VMaSCAverage speedNS3, SUMO[14]VMaSC-LTEAverage speedNS3, SUMO[15]Arkian et al. [15]Speed, neighborOMNET++, SUMO[16]PMCSpeed, neighbors, link lifetime, and positionNS2, vanetMobiSim[22]PPCTravel time, ID, and relative velocityNS2[23]HCARLowest IDNS2, VANET MobiSim[6]CVoEGLink lifetime, position, and relative speedMATLAB, SUMO, MOVE[18]FCMSRelative speed, degree, security, and trustworthiness—[24]JCVMovement at the junction, relative position, time, and degree of a nodeSUMO, CVANETSIM, JAVA[7]Khan et al. [7]Link connectivityMATLAB, SUMO, MOVE[20]KMRPVelocity, free buffer size, and node degreeNS2[10]HGCMRelative speed, neighboring, trust scoreMATLAB, SUMO[21]BCBCBetweenness centrality methodMATLAB, SUMOAs we said, the graphical depiction might work well in sparsely populated areas like highways or small cities, but not in crowded urban scenarios. Graph theory, on the other hand, does not work in urban dense scenarios where a vehicle is always connected to more than two other vehicles. A good representation of dense vehicle networks is a hypergraph.Also, we see that many clustering algorithms are based on mobility and speed measures to select the CH, but these measures are lost in the urban scenario, especially when the vehicle density is high because the vehicle speed will be low. So, it is better to search for new metrics for selecting the CH.So, our primary goal is to propose a new clustering algorithm that aims to increase the clusters’ stability. Our proposed model has two main parts: the cluster formation is presented using a hypergraph spectral clustering algorithm. The hypergraph spectral clustering algorithm is used with a modified Laplacian value in the TTM by using the Eigen-trick method to improve the clustering. The Eigen-trick method improves clustering efficiency by using higher-order information in eigenvalues. It considers the transformable connection between the vertex Laplacian and the hyperedge Laplacian, which can speed up the solution of eigenproblems without losing information. The second part is the CH selection phase; in addition to relative speed, neighbor degree, and eccentricity, the vehicle’s time to leave parameter is used in the selection of the CH to increase the CH stability. The vehicle’s direction is unpredictable at the road junction, which reduces the CH’s stability [24]. So, our proposed scheme has solved the problem of CH stability at the junction using the estimated next time to leave parameter. Strong connection and robust link lifetime are ensured by selecting a suitable CH using the GRA model.


## 3. The Proposed EtHgSC Model

A new clustering algorithm for a VANET structure in an urban scenario is presented in this article. The proposed approach (EtHgSC) includes two main parts: the cluster generation part, which uses the improved hypergraph spectral clustering method using the Eigen-trick method for generating the clusters. The second part is the CH selection. CH stability is governed by using four factors: inclusive of relative speed, neighboring degree, eccentricity, and the vehicle’s time to leave with the estimated next vehicle’s position. These two parts are explained in detail in this section. The flow diagram of our proposed model is shown in Figure1.Figure 1
The EtHgSC flow diagram.


### 3.1. Clusters Generation of EtHgSC

In this proposed model, a multilane road structure is taken into account in an urban scenario. The real map incorporates the total number of carsN together with their speeds and positions. The transmission range Trv is the same for all on-board unit (OBU) equipment. The RSUs communicate with the same communication module, with the transmission range TrRSU. The OBU contains both IEEE 802.11p radio technology and a global positioning system (GPS) unit. Every car in the network functions as a node V) acting as a source, destination, or router.In the process of spectral clustering, information loss is evident [25]. To prevent it, the TTM clustering has been proposed for spectral clustering of vehicles. It is called TTM because the hypergraph partition is equivalent to tensor decomposition into a lower rank. Despite the fact that the TTM was designed to further partition the hypergraph into lower nondecomposable graphs, the information loss can be further minimized with the help of the Eigen trick in TTM hypergraph partitioning. In this section, a weighted hypergraph TTM is proposed for partitioning, with its improvement using the Eigen-trick method for the large-scale VANET for clustering. Figure 2 presents an example of our proposed model.Figure 2
Proposed model.


#### 3.1.1. Cluster Formation as a Hypergraph

In spectral clustering, the most common criterion method is the radio cut method. For a set of clustersV1,V2,…,Vk satisfying V1∩…∩Vk=∅ and V1∪…∪Vk=V. And as we mentioned before, L is a spectral clustering major tool.In weighted hypergraphH=V,E,W, V is a vertices, V∈v1,v2,v3,……,vn is the vehicles (nodes), E is the edges E∈e1,e2,e3,…….,em, it is the links among vehicles cij, it depends on the distance between vehicles d, and W is the weight of the edges, W ⟶ [0, 1].(1)cij=1,dij≤Trv,0,dij>Trv.The problem is partitioning the weighted hypergraphV into k disjoint sets, V1,……Vk. The followings are the steps for partitioning the hypergraph with the modified TTM using the Eigen-trick method:(1)
The degree of any node is the number of vehicular nodes connected to other nodesv∈V, degv=∑e∈Ewe. The vertex diagonal matrix (degree matrix) is Digv∈RN×N. The other important term associated is a volume, which is coined as volV1=∑vϵV1degv, which is the number of nodes incident on node V1 such that V1⊆V. The associativity of V1 is defined as the sum of incident edges’ weights as assocV1=∑We. The normalized associativity of these separate partitions is given as follows:(2)N−assocV1,……Vk=∑i=1kassocVivolVi.This problem can be presented as the tensor problem as discussed in [25], and this is the reason it is termed TTM. The clustering of the tensors is formulated as the problem of maximizing the diagonal terms, and edge weights are the diagonal elements in the VANET hypergraph. So, the problem is coined the tensor maximization problem. The solution to this problem is searched by spectral relaxation by computing the k eigenvectors of the normalized adjacency matrix [25].(2)
The defined adjacency matrix is used for spectral clustering. It is computed using the following formula, which is specified here for the tensor (orderm):(3)Ai1,i2,……im=wi1,i2……..im,ifi1,i2,………,imaredistinct,0,otherwise.Also, the incidence matrixI is defined, I∈RN×m. It represents the connection between vertices and edges. iv,e=1, if there is a connection between a vertex v and an edge e, and otherwise, iv,e=0. The edges’ diagonal matrix Dige∈Rm×m is obtained as follows:(4)dege=∑v∈Viv,e.In our work, the weighted incidence matrix is used, and it is calculated as follows:(5)iv,e=we,ifconnected,0,otherwise.(3)
The Fiedler vector-based unnormalized Laplacian matrix was used in the TTM approach in [25]:(6)Lv=Digv−1/2ADigv−1/2.In our work, by utilizing higher-order information in eigenvalues, the Eigen trick improves clustering efficiency. It considers the transformable connection between the vertex LaplacianLv and the hyperedge Laplacian Le, which can speed up the solution of eigenproblems without losing information. During this time, the partitions for vertices and hyperedges are both obtained simultaneously [26].The hyperedge LaplacianLe, Le∈Rm×m, is calculated as follows [26]:(7)Le=Dige−1/2W1/2ITDigv−1IW1/2Dige−1/2.In actuality, there is a link between the symmetry ofLv and Le. In “Eigen trick,” an interesting attribute of two Laplacians is that the k trailing eigenvectors of the vertex Laplacian Uv∈RN×k can be computed from the corresponding eigenvectors of the hyperedge Laplacian Ue∈Rm×k and vice versa. Ue=eigLe.(4)
Computed theUv∈RN×k from the hyperedge eigenvectors Ue.(8)Uv=Digv−1/2IDige−1/2W1/2Ue.(5)
View each row ofV as a vector for a node, and cluster the N nodes into k clusters V1,V2,…,Vk through k-means clustering.(6)
The optimal set of clusters and the clustering efficiency are evaluated using the Calinski–Harabasz indexs. Using this index, we can see how closely vehicles are clustered together in a cluster, as well as how far apart all clusters are. The s index is calculated as follows [27]:(9)s=trBktrZk×Vehinum−kk−1.This index evaluates how closely each cluster’s vehicles are spaced out. In this case,k stands for the number of clusters, while Vehinum is the size of each cluster. The dispersion between clusters is denoted by trBk, and the dispersion among cars inside a cluster is denoted by trzk. Equations (10) and (11) are used to calculate these two factors.(10)Zk=∑q=1k∑x∈Cqx−cqx−cqT,(11)Bk=∑q=1knqcq−cEcq−cET.A set of points in clusterq are shown here as cq, and the cluster center is x. The clusters with nq points in them have a center called cE.The maximum value ofs is used to select an ideal set of clusters from the pool of generated clusters using this index. Coptimal=Cnum:∀maxs.The complete cluster formation algorithm is as in Algorithm1.Algorithm 1: Cluster formation using Eigen-trick method.
Input: maximum vehicles: Vehimax, vehicle’s location: VehiLoc(1)
Choose the time periodt during which there is the largest number of vehicles Vehimax(2)
A hypergraph formationH=V,E,W(3)
The similarity matrixA is calculated based on the distance proximity between the vehicles A∈RN×N, and also, an incidence matrix I is generated, its size of I∈RN×m(4)
A diagonal matrix for vehiclesDigv∈RN×N, and for edges Dige∈Rm×m(5)
Find the hyperedge LaplacianLe=Dige−1/2W1/2ITDigv−1IW1/2Dige−1/2(6)
Thek dominant eigenvector of Le is calculated Ue∈Rm×k(7)
Computed theUv∈RN×k from the hyperedge eigenvectors Ue(8)
Normalize each row ofUv¯=Uv(9)
Run k-means on the rows ofUv¯(10)
Cnum is obtained through k-means partition Par=V1,……Vk(11)
Compute the Calinski–Harabaszs index for each cluster Cnum(12)
Coptimal=Cnum:∀maxs is the optimal number of clustersOutput: Coptimal, partitioning vehicles in clusters


#### 3.1.2. RSUs Deployment

VANET’s weaknesses as well as network performance are enhanced by RSUs. The RSUs can successfully meet increased reliability, vehicular density, and decreased overhead delay, particularly in urban environments. When connecting to one or more vehicle clusters, the RSU functions as a gateway inside a router. It can also connect with RSUs using wireless or optical networks through an inclusive deployment [28]. Few studies in the literature addressed RSU deployment, but the RSU is a component of VANET and is critical, as we have developed in our work. The actual VANET network needs RSU communication too [29].So, after Algorithm1 has produced the ideal set of clusters, the fetching location for RSU starts. Using betweenness centrality, a traffic graph’s dynamic structure is created. A network is considered as a graph G=V,E, where V vehicles’ connections are connected by a certain set of edges E. A graph’s centrality matrix serves as a measure of its compactness [30]. The graph’s most frequently visited vertex is determined by centrality. For a vehicle v, it can be calculated as follows:(12)CBv=∑s≠v≠u∈Vσstvσst.Here, the number of shortest paths from nodes to node u is shown here as σst, and the number of pathways that pass through v is σstv. The center of the cluster is determined by the vehicle with the highest centrality value Vehi=maxCB. This is where the RSU will be installed.Now, the cluster generation phase for our scheme is complete.


### 3.2. CH Selection Measures

The next step is the selection of the CH, which is the vehicular node in a cluster that coordinates or heads the cluster. Continuous communication between the cluster and RSU, security, and upkeep of the routing path all fall under the purview of a CH. The stability of the CH in VANET is the main motive for designing any clustering algorithm. The proposed methodology for the CH selection is based on four parameters; relative speed, neighboring degree, eccentricity, and the vehicle’s time to leave with the estimated next vehicle’s position. These parameters are calculated for each vehicle at each instantt. The four parameters are discussed in the following.


#### 3.2.1. Relative Speed Scoreψvehi

It is characterized as a shift in a node’s relative average vehicle speed with respect to its neighbors’ speeds. The long-term velocity of vehicles is taken into consideration by a reward function that is developed. The speed of each vehicleVvehi is calculated. As a result, their speed receives an absolute value δ that can be used to reward or penalize them. As a result, the relative average speed is increased or decreased, and ψvehi is calculated as follows [15]:(13)ψvehit+1=ψvehit+δ;Vvehi−Vavg≤Sthr,ψvehit+1=ψvehit−δ;Vvehi−Vavg>Sthr,where Sthr is the variable that makes sure the vehicle going with velocity is nearly moving at the same speed as that of the neighbors. Using the traffic control interface (TraCI) parameters, the initial value of ψvehi is calculated, and δ is set to 0.01.


#### 3.2.2. Neighboring Degreend

The two vehicular nodesvi and vj’s connection state at time t in the cluster created Coptimal with vehicle density Vehinum are defined as follows [31]:(14)nd=∑j=1Vehinumcij;∀1.Sincend is high, the CH won’t be dynamic for a very long period. The total number of nearby vehicles is the neighborhood degree. A vehicle’s neighbors are those that are within its transmission range. If the distance between two cars at time t is less than or equal to Trv, then cij is 1.


#### 3.2.3. EccentricityEcc

Due to the vehicles’ high speed in real time, communication links break more frequently. An evolving cluster model is necessary for maintaining the link. Reclustering will typically be unavoidable after the CH resigns or no longer meets the criteria to remain a CH. Eccentricity is a concept that is introduced to ensure stability. Here, spectral clustering is used to create a dynamic graph-based model.It is intended for a vehicular graph topology to be hypergraphH=V,E,W, and the concept of eccentricity Ecc is introduced. Ecc is the mean/average eigenscore λi of each vehicle in a cluster; it is calculated as follows [6]:(15)Ecc=1Vehinum∑λiϵVehinumλi.


#### 3.2.4. Vehicle’s Time to LeaveTleave

It is the amount of time needed for a vehicle to reach the last section of the lane. This parameter guarantees to choose a CH with enough time remaining to finish the lane that causes it to head for a longer period. It is determined using the length of the laneL, the distance covered by a vehicle on the road segment D, and the vehicle speed at that time v:(16)Tleave=L−Dv.The possibility of the vehicle changing direction changes the lane length. Figure3 can give a picture of the problem. Tleave fails to estimate the road junctions. The unpredictable change in the direction of the vehicle at the junction can abruptly change the prediction. The lane lengths may differ, and so, the vehicle’s estimated next position must be known ahead of time. It helps to select the lane and its length to get the necessary Tleave.Figure 3
Problem inTleave without the next position estimation.


#### 3.2.5. Estimated Next Position

To predict the next vehicle position at timet+1, firstly the new vehicle coordinates should be calculated from the current coordinates X2,Y2, distance d=v2t2, and vehicle current direction θ2, and it is calculated as follows [32]:(17)X2′=X2+v2t2cosθ2,Y2′=Y2+v2t2sinθ2.Then, a vehicle new direction using the predicted positionX2′,Y2′ is calculated by using(18)θ2′=tan−1Y2−Y2′X2−X2′.From the vehicle’s new directionθ2′, the next lane length L and D are known now, and then, we can calculate the Tleave′ with the predicted next vehicle’s position using equation (16).After calculating the four measures for each vehicle at each instant of time, the GRA model is used to find the vehicles’ score. The GRA is explained in the next subsection.


### 3.3. Grey Relational Analysis Model

GRA is a component of grey system theory, which is useful for resolving problems involving complex interrelationships between multiple factors and variables. The basic step of GRA is to convert the performance of each option into a comparability sequence, which is denoted by the lettersXij. The term for this step is “grey relational generating” [33]. A reference sequence is defined in accordance with these sequences. Following that, the grey relational coefficient between the reference sequence and all comparability sequences is determined γX0j,Xij. The grey relational grade between each comparability sequence and the reference sequence X0 is then determined using these grey relational coefficients. If a comparability sequence translated from an alternative has the highest grey relational grade between the reference sequence and itself, that alternative will be the best choice. The grade gv is calculated as follows [34]:(19)gv=∑j=1nwjγX0j,Xij,fori=1,2,……,Vehinum,where n=4 (the number of CH selection parameters in our work), i is a number of alternatives. γX0j,Xij is the grey relational coefficient, how near Xij is to X0j is determined using it. wj is the weight of the four parameters, where ∑j=1nwj=1.The grade for each vehicle in a cluster is calculated using the GRA. The vehicle with the maximum grade is selected as a CH for each cluster at each instant of timet.The pseudocode for the CH selection phase is introduced in Algorithm2.Algorithm 2: The pseudocode for CH selection using the GRA model.
Input: number of clusters, number of vehicles in each cluster at each instant of time, vehicles coordinators (X,Y), current lane length (L), current direction θFort = 1: instant of timeFori = 1: clustersForj = 1: vehicles in clusterCalculate the CH parameters:nd, ψvehi, and EccPredicted the next vehicle’s coordinators (X′, Y′)Find the new vehicle’s direction from predicted coordinatorsθ′CalculateTleave′ using predicted vehicle’s positionEnd forInput the four matrices to GRA model with their weights, where∑j=1nwj=1Generate a matrixX of Vehinum×4Normalize the matrixXFori=1:Vehinum(a)
Forj=1:nCalculate the grey relational coefficientγX0j,Xij(b)
end forend forGenerate a graph object from grey relational coefficientγX0j,XijCalculate the Vehicle’s gradegv with the help of γX0j,XijHighest-grade vehicle is selected as a CHCH=maxgvEnd forEnd forOutput: CH vehicle


### 3.4. Time Complexity of the EtHgSC Scheme

The total time complexity of the EtHgSC scheme is expressed as follows:(20)OTOT=OCF+ORSU+OCH,where OCF stands for the time complexity of cluster formations, ORSU for RSU deployment, and OCH for CH selection. The formation of a hypergraph, the application of Laplacian, and the Eigen-trick technique to solve eigenproblems, followed by K-means to get the ideal set of clusters, are the key steps in the cluster formation process.In hypergraph, the quotient of similarity is taken between each of the vehicles, and this is calculated by constructing ad nearest neighbour graph with complexity ON2d. By using the Eigen-trick method, the hyperedge Laplacian Le is processed instead of vertex Laplacian Lv, and the time complexity for Le is ONNZI2/N.“Eigen trick” provides us a way to calculateUv with lower time and space complexity. The time complexity for solving the eigenproblem of the vertex is ON3, while the corresponding cost to hyperedge is Om3. Instead of solving the eigenproblem of Lv directly, the Eigen-trick method is proposed to solve it using the eigenproblem of Le. It will reduce the computational complexity from ON3, to Om3 significantly m<N. Another benefit is that the conversion between Ue and Uv is accurate without any information loss [26]. The last is the K-means complexity which is dependent on the OCoptimal=OτNCnum. So, OCF will be(21)OCF=ON2d+ONNZI2N+Om3+OτNCnum.The complexity is reduced by removing the terms of less computational power(22)OCF=ON2d+Om3.Due to the use of a graph in the RSU deployment, the computational complexity is as follows:(23)ORSU=ON2d.The CH selection parameters are relative speed scoreψvehi, neighbourhood degree η, eccentricity E, and time to leave estimation Tleave′.(24)OCH=Oψvehi+Oη+OE+OTleave′.The time complexity forψvehi, η, and E is calculated as follows:(25)Oψvehi=ON,Oη=OlogN2.The relative speed is calculated using a straightforward threshold function based on the vehicle’s speed. The neighborhood is the next, and it is determined by thecij affinity matrix for neighboring vehicles.The affinity matrix and eigenvalue decomposition of spectral clustering methods are used to determine the eccentricity:(26)OE=ON2d+ON3.The complexity of time to leave is calculated for all the vehicles in the network(27)OTleave′=ON.Thus, the complete time complexity is reduced to moving all the terms with less complexity than cubic and quadratic terms(28)OCH=OlogN2+ON3.The overall complexity is primarily dependent on the hypergraph(29)OTOT=2ON2d+Om3+OlogN2+ON3.


## 3.1. Clusters Generation of EtHgSC

In this proposed model, a multilane road structure is taken into account in an urban scenario. The real map incorporates the total number of carsN together with their speeds and positions. The transmission range Trv is the same for all on-board unit (OBU) equipment. The RSUs communicate with the same communication module, with the transmission range TrRSU. The OBU contains both IEEE 802.11p radio technology and a global positioning system (GPS) unit. Every car in the network functions as a node V) acting as a source, destination, or router.In the process of spectral clustering, information loss is evident [25]. To prevent it, the TTM clustering has been proposed for spectral clustering of vehicles. It is called TTM because the hypergraph partition is equivalent to tensor decomposition into a lower rank. Despite the fact that the TTM was designed to further partition the hypergraph into lower nondecomposable graphs, the information loss can be further minimized with the help of the Eigen trick in TTM hypergraph partitioning. In this section, a weighted hypergraph TTM is proposed for partitioning, with its improvement using the Eigen-trick method for the large-scale VANET for clustering. Figure 2 presents an example of our proposed model.Figure 2
Proposed model.


### 3.1.1. Cluster Formation as a Hypergraph

In spectral clustering, the most common criterion method is the radio cut method. For a set of clustersV1,V2,…,Vk satisfying V1∩…∩Vk=∅ and V1∪…∪Vk=V. And as we mentioned before, L is a spectral clustering major tool.In weighted hypergraphH=V,E,W, V is a vertices, V∈v1,v2,v3,……,vn is the vehicles (nodes), E is the edges E∈e1,e2,e3,…….,em, it is the links among vehicles cij, it depends on the distance between vehicles d, and W is the weight of the edges, W ⟶ [0, 1].(1)cij=1,dij≤Trv,0,dij>Trv.The problem is partitioning the weighted hypergraphV into k disjoint sets, V1,……Vk. The followings are the steps for partitioning the hypergraph with the modified TTM using the Eigen-trick method:(1)
The degree of any node is the number of vehicular nodes connected to other nodesv∈V, degv=∑e∈Ewe. The vertex diagonal matrix (degree matrix) is Digv∈RN×N. The other important term associated is a volume, which is coined as volV1=∑vϵV1degv, which is the number of nodes incident on node V1 such that V1⊆V. The associativity of V1 is defined as the sum of incident edges’ weights as assocV1=∑We. The normalized associativity of these separate partitions is given as follows:(2)N−assocV1,……Vk=∑i=1kassocVivolVi.This problem can be presented as the tensor problem as discussed in [25], and this is the reason it is termed TTM. The clustering of the tensors is formulated as the problem of maximizing the diagonal terms, and edge weights are the diagonal elements in the VANET hypergraph. So, the problem is coined the tensor maximization problem. The solution to this problem is searched by spectral relaxation by computing the k eigenvectors of the normalized adjacency matrix [25].(2)
The defined adjacency matrix is used for spectral clustering. It is computed using the following formula, which is specified here for the tensor (orderm):(3)Ai1,i2,……im=wi1,i2……..im,ifi1,i2,………,imaredistinct,0,otherwise.Also, the incidence matrixI is defined, I∈RN×m. It represents the connection between vertices and edges. iv,e=1, if there is a connection between a vertex v and an edge e, and otherwise, iv,e=0. The edges’ diagonal matrix Dige∈Rm×m is obtained as follows:(4)dege=∑v∈Viv,e.In our work, the weighted incidence matrix is used, and it is calculated as follows:(5)iv,e=we,ifconnected,0,otherwise.(3)
The Fiedler vector-based unnormalized Laplacian matrix was used in the TTM approach in [25]:(6)Lv=Digv−1/2ADigv−1/2.In our work, by utilizing higher-order information in eigenvalues, the Eigen trick improves clustering efficiency. It considers the transformable connection between the vertex LaplacianLv and the hyperedge Laplacian Le, which can speed up the solution of eigenproblems without losing information. During this time, the partitions for vertices and hyperedges are both obtained simultaneously [26].The hyperedge LaplacianLe, Le∈Rm×m, is calculated as follows [26]:(7)Le=Dige−1/2W1/2ITDigv−1IW1/2Dige−1/2.In actuality, there is a link between the symmetry ofLv and Le. In “Eigen trick,” an interesting attribute of two Laplacians is that the k trailing eigenvectors of the vertex Laplacian Uv∈RN×k can be computed from the corresponding eigenvectors of the hyperedge Laplacian Ue∈Rm×k and vice versa. Ue=eigLe.(4)
Computed theUv∈RN×k from the hyperedge eigenvectors Ue.(8)Uv=Digv−1/2IDige−1/2W1/2Ue.(5)
View each row ofV as a vector for a node, and cluster the N nodes into k clusters V1,V2,…,Vk through k-means clustering.(6)
The optimal set of clusters and the clustering efficiency are evaluated using the Calinski–Harabasz indexs. Using this index, we can see how closely vehicles are clustered together in a cluster, as well as how far apart all clusters are. The s index is calculated as follows [27]:(9)s=trBktrZk×Vehinum−kk−1.This index evaluates how closely each cluster’s vehicles are spaced out. In this case,k stands for the number of clusters, while Vehinum is the size of each cluster. The dispersion between clusters is denoted by trBk, and the dispersion among cars inside a cluster is denoted by trzk. Equations (10) and (11) are used to calculate these two factors.(10)Zk=∑q=1k∑x∈Cqx−cqx−cqT,(11)Bk=∑q=1knqcq−cEcq−cET.A set of points in clusterq are shown here as cq, and the cluster center is x. The clusters with nq points in them have a center called cE.The maximum value ofs is used to select an ideal set of clusters from the pool of generated clusters using this index. Coptimal=Cnum:∀maxs.The complete cluster formation algorithm is as in Algorithm1.Algorithm 1: Cluster formation using Eigen-trick method.
Input: maximum vehicles: Vehimax, vehicle’s location: VehiLoc(1)
Choose the time periodt during which there is the largest number of vehicles Vehimax(2)
A hypergraph formationH=V,E,W(3)
The similarity matrixA is calculated based on the distance proximity between the vehicles A∈RN×N, and also, an incidence matrix I is generated, its size of I∈RN×m(4)
A diagonal matrix for vehiclesDigv∈RN×N, and for edges Dige∈Rm×m(5)
Find the hyperedge LaplacianLe=Dige−1/2W1/2ITDigv−1IW1/2Dige−1/2(6)
Thek dominant eigenvector of Le is calculated Ue∈Rm×k(7)
Computed theUv∈RN×k from the hyperedge eigenvectors Ue(8)
Normalize each row ofUv¯=Uv(9)
Run k-means on the rows ofUv¯(10)
Cnum is obtained through k-means partition Par=V1,……Vk(11)
Compute the Calinski–Harabaszs index for each cluster Cnum(12)
Coptimal=Cnum:∀maxs is the optimal number of clustersOutput: Coptimal, partitioning vehicles in clusters


### 3.1.2. RSUs Deployment

VANET’s weaknesses as well as network performance are enhanced by RSUs. The RSUs can successfully meet increased reliability, vehicular density, and decreased overhead delay, particularly in urban environments. When connecting to one or more vehicle clusters, the RSU functions as a gateway inside a router. It can also connect with RSUs using wireless or optical networks through an inclusive deployment [28]. Few studies in the literature addressed RSU deployment, but the RSU is a component of VANET and is critical, as we have developed in our work. The actual VANET network needs RSU communication too [29].So, after Algorithm1 has produced the ideal set of clusters, the fetching location for RSU starts. Using betweenness centrality, a traffic graph’s dynamic structure is created. A network is considered as a graph G=V,E, where V vehicles’ connections are connected by a certain set of edges E. A graph’s centrality matrix serves as a measure of its compactness [30]. The graph’s most frequently visited vertex is determined by centrality. For a vehicle v, it can be calculated as follows:(12)CBv=∑s≠v≠u∈Vσstvσst.Here, the number of shortest paths from nodes to node u is shown here as σst, and the number of pathways that pass through v is σstv. The center of the cluster is determined by the vehicle with the highest centrality value Vehi=maxCB. This is where the RSU will be installed.Now, the cluster generation phase for our scheme is complete.


## 3.1.1. Cluster Formation as a Hypergraph

In spectral clustering, the most common criterion method is the radio cut method. For a set of clustersV1,V2,…,Vk satisfying V1∩…∩Vk=∅ and V1∪…∪Vk=V. And as we mentioned before, L is a spectral clustering major tool.In weighted hypergraphH=V,E,W, V is a vertices, V∈v1,v2,v3,……,vn is the vehicles (nodes), E is the edges E∈e1,e2,e3,…….,em, it is the links among vehicles cij, it depends on the distance between vehicles d, and W is the weight of the edges, W ⟶ [0, 1].(1)cij=1,dij≤Trv,0,dij>Trv.The problem is partitioning the weighted hypergraphV into k disjoint sets, V1,……Vk. The followings are the steps for partitioning the hypergraph with the modified TTM using the Eigen-trick method:(1)
The degree of any node is the number of vehicular nodes connected to other nodesv∈V, degv=∑e∈Ewe. The vertex diagonal matrix (degree matrix) is Digv∈RN×N. The other important term associated is a volume, which is coined as volV1=∑vϵV1degv, which is the number of nodes incident on node V1 such that V1⊆V. The associativity of V1 is defined as the sum of incident edges’ weights as assocV1=∑We. The normalized associativity of these separate partitions is given as follows:(2)N−assocV1,……Vk=∑i=1kassocVivolVi.This problem can be presented as the tensor problem as discussed in [25], and this is the reason it is termed TTM. The clustering of the tensors is formulated as the problem of maximizing the diagonal terms, and edge weights are the diagonal elements in the VANET hypergraph. So, the problem is coined the tensor maximization problem. The solution to this problem is searched by spectral relaxation by computing the k eigenvectors of the normalized adjacency matrix [25].(2)
The defined adjacency matrix is used for spectral clustering. It is computed using the following formula, which is specified here for the tensor (orderm):(3)Ai1,i2,……im=wi1,i2……..im,ifi1,i2,………,imaredistinct,0,otherwise.Also, the incidence matrixI is defined, I∈RN×m. It represents the connection between vertices and edges. iv,e=1, if there is a connection between a vertex v and an edge e, and otherwise, iv,e=0. The edges’ diagonal matrix Dige∈Rm×m is obtained as follows:(4)dege=∑v∈Viv,e.In our work, the weighted incidence matrix is used, and it is calculated as follows:(5)iv,e=we,ifconnected,0,otherwise.(3)
The Fiedler vector-based unnormalized Laplacian matrix was used in the TTM approach in [25]:(6)Lv=Digv−1/2ADigv−1/2.In our work, by utilizing higher-order information in eigenvalues, the Eigen trick improves clustering efficiency. It considers the transformable connection between the vertex LaplacianLv and the hyperedge Laplacian Le, which can speed up the solution of eigenproblems without losing information. During this time, the partitions for vertices and hyperedges are both obtained simultaneously [26].The hyperedge LaplacianLe, Le∈Rm×m, is calculated as follows [26]:(7)Le=Dige−1/2W1/2ITDigv−1IW1/2Dige−1/2.In actuality, there is a link between the symmetry ofLv and Le. In “Eigen trick,” an interesting attribute of two Laplacians is that the k trailing eigenvectors of the vertex Laplacian Uv∈RN×k can be computed from the corresponding eigenvectors of the hyperedge Laplacian Ue∈Rm×k and vice versa. Ue=eigLe.(4)
Computed theUv∈RN×k from the hyperedge eigenvectors Ue.(8)Uv=Digv−1/2IDige−1/2W1/2Ue.(5)
View each row ofV as a vector for a node, and cluster the N nodes into k clusters V1,V2,…,Vk through k-means clustering.(6)
The optimal set of clusters and the clustering efficiency are evaluated using the Calinski–Harabasz indexs. Using this index, we can see how closely vehicles are clustered together in a cluster, as well as how far apart all clusters are. The s index is calculated as follows [27]:(9)s=trBktrZk×Vehinum−kk−1.This index evaluates how closely each cluster’s vehicles are spaced out. In this case,k stands for the number of clusters, while Vehinum is the size of each cluster. The dispersion between clusters is denoted by trBk, and the dispersion among cars inside a cluster is denoted by trzk. Equations (10) and (11) are used to calculate these two factors.(10)Zk=∑q=1k∑x∈Cqx−cqx−cqT,(11)Bk=∑q=1knqcq−cEcq−cET.A set of points in clusterq are shown here as cq, and the cluster center is x. The clusters with nq points in them have a center called cE.The maximum value ofs is used to select an ideal set of clusters from the pool of generated clusters using this index. Coptimal=Cnum:∀maxs.The complete cluster formation algorithm is as in Algorithm1.Algorithm 1: Cluster formation using Eigen-trick method.
Input: maximum vehicles: Vehimax, vehicle’s location: VehiLoc(1)
Choose the time periodt during which there is the largest number of vehicles Vehimax(2)
A hypergraph formationH=V,E,W(3)
The similarity matrixA is calculated based on the distance proximity between the vehicles A∈RN×N, and also, an incidence matrix I is generated, its size of I∈RN×m(4)
A diagonal matrix for vehiclesDigv∈RN×N, and for edges Dige∈Rm×m(5)
Find the hyperedge LaplacianLe=Dige−1/2W1/2ITDigv−1IW1/2Dige−1/2(6)
Thek dominant eigenvector of Le is calculated Ue∈Rm×k(7)
Computed theUv∈RN×k from the hyperedge eigenvectors Ue(8)
Normalize each row ofUv¯=Uv(9)
Run k-means on the rows ofUv¯(10)
Cnum is obtained through k-means partition Par=V1,……Vk(11)
Compute the Calinski–Harabaszs index for each cluster Cnum(12)
Coptimal=Cnum:∀maxs is the optimal number of clustersOutput: Coptimal, partitioning vehicles in clusters


## 3.1.2. RSUs Deployment

VANET’s weaknesses as well as network performance are enhanced by RSUs. The RSUs can successfully meet increased reliability, vehicular density, and decreased overhead delay, particularly in urban environments. When connecting to one or more vehicle clusters, the RSU functions as a gateway inside a router. It can also connect with RSUs using wireless or optical networks through an inclusive deployment [28]. Few studies in the literature addressed RSU deployment, but the RSU is a component of VANET and is critical, as we have developed in our work. The actual VANET network needs RSU communication too [29].So, after Algorithm1 has produced the ideal set of clusters, the fetching location for RSU starts. Using betweenness centrality, a traffic graph’s dynamic structure is created. A network is considered as a graph G=V,E, where V vehicles’ connections are connected by a certain set of edges E. A graph’s centrality matrix serves as a measure of its compactness [30]. The graph’s most frequently visited vertex is determined by centrality. For a vehicle v, it can be calculated as follows:(12)CBv=∑s≠v≠u∈Vσstvσst.Here, the number of shortest paths from nodes to node u is shown here as σst, and the number of pathways that pass through v is σstv. The center of the cluster is determined by the vehicle with the highest centrality value Vehi=maxCB. This is where the RSU will be installed.Now, the cluster generation phase for our scheme is complete.


## 3.2. CH Selection Measures

The next step is the selection of the CH, which is the vehicular node in a cluster that coordinates or heads the cluster. Continuous communication between the cluster and RSU, security, and upkeep of the routing path all fall under the purview of a CH. The stability of the CH in VANET is the main motive for designing any clustering algorithm. The proposed methodology for the CH selection is based on four parameters; relative speed, neighboring degree, eccentricity, and the vehicle’s time to leave with the estimated next vehicle’s position. These parameters are calculated for each vehicle at each instantt. The four parameters are discussed in the following.


### 3.2.1. Relative Speed Scoreψvehi

It is characterized as a shift in a node’s relative average vehicle speed with respect to its neighbors’ speeds. The long-term velocity of vehicles is taken into consideration by a reward function that is developed. The speed of each vehicleVvehi is calculated. As a result, their speed receives an absolute value δ that can be used to reward or penalize them. As a result, the relative average speed is increased or decreased, and ψvehi is calculated as follows [15]:(13)ψvehit+1=ψvehit+δ;Vvehi−Vavg≤Sthr,ψvehit+1=ψvehit−δ;Vvehi−Vavg>Sthr,where Sthr is the variable that makes sure the vehicle going with velocity is nearly moving at the same speed as that of the neighbors. Using the traffic control interface (TraCI) parameters, the initial value of ψvehi is calculated, and δ is set to 0.01.


### 3.2.2. Neighboring Degreend

The two vehicular nodesvi and vj’s connection state at time t in the cluster created Coptimal with vehicle density Vehinum are defined as follows [31]:(14)nd=∑j=1Vehinumcij;∀1.Sincend is high, the CH won’t be dynamic for a very long period. The total number of nearby vehicles is the neighborhood degree. A vehicle’s neighbors are those that are within its transmission range. If the distance between two cars at time t is less than or equal to Trv, then cij is 1.


### 3.2.3. EccentricityEcc

Due to the vehicles’ high speed in real time, communication links break more frequently. An evolving cluster model is necessary for maintaining the link. Reclustering will typically be unavoidable after the CH resigns or no longer meets the criteria to remain a CH. Eccentricity is a concept that is introduced to ensure stability. Here, spectral clustering is used to create a dynamic graph-based model.It is intended for a vehicular graph topology to be hypergraphH=V,E,W, and the concept of eccentricity Ecc is introduced. Ecc is the mean/average eigenscore λi of each vehicle in a cluster; it is calculated as follows [6]:(15)Ecc=1Vehinum∑λiϵVehinumλi.


### 3.2.4. Vehicle’s Time to LeaveTleave

It is the amount of time needed for a vehicle to reach the last section of the lane. This parameter guarantees to choose a CH with enough time remaining to finish the lane that causes it to head for a longer period. It is determined using the length of the laneL, the distance covered by a vehicle on the road segment D, and the vehicle speed at that time v:(16)Tleave=L−Dv.The possibility of the vehicle changing direction changes the lane length. Figure3 can give a picture of the problem. Tleave fails to estimate the road junctions. The unpredictable change in the direction of the vehicle at the junction can abruptly change the prediction. The lane lengths may differ, and so, the vehicle’s estimated next position must be known ahead of time. It helps to select the lane and its length to get the necessary Tleave.Figure 3
Problem inTleave without the next position estimation.


### 3.2.5. Estimated Next Position

To predict the next vehicle position at timet+1, firstly the new vehicle coordinates should be calculated from the current coordinates X2,Y2, distance d=v2t2, and vehicle current direction θ2, and it is calculated as follows [32]:(17)X2′=X2+v2t2cosθ2,Y2′=Y2+v2t2sinθ2.Then, a vehicle new direction using the predicted positionX2′,Y2′ is calculated by using(18)θ2′=tan−1Y2−Y2′X2−X2′.From the vehicle’s new directionθ2′, the next lane length L and D are known now, and then, we can calculate the Tleave′ with the predicted next vehicle’s position using equation (16).After calculating the four measures for each vehicle at each instant of time, the GRA model is used to find the vehicles’ score. The GRA is explained in the next subsection.


## 3.2.1. Relative Speed Scoreψvehi

It is characterized as a shift in a node’s relative average vehicle speed with respect to its neighbors’ speeds. The long-term velocity of vehicles is taken into consideration by a reward function that is developed. The speed of each vehicleVvehi is calculated. As a result, their speed receives an absolute value δ that can be used to reward or penalize them. As a result, the relative average speed is increased or decreased, and ψvehi is calculated as follows [15]:(13)ψvehit+1=ψvehit+δ;Vvehi−Vavg≤Sthr,ψvehit+1=ψvehit−δ;Vvehi−Vavg>Sthr,where Sthr is the variable that makes sure the vehicle going with velocity is nearly moving at the same speed as that of the neighbors. Using the traffic control interface (TraCI) parameters, the initial value of ψvehi is calculated, and δ is set to 0.01.


## 3.2.2. Neighboring Degreend

The two vehicular nodesvi and vj’s connection state at time t in the cluster created Coptimal with vehicle density Vehinum are defined as follows [31]:(14)nd=∑j=1Vehinumcij;∀1.Sincend is high, the CH won’t be dynamic for a very long period. The total number of nearby vehicles is the neighborhood degree. A vehicle’s neighbors are those that are within its transmission range. If the distance between two cars at time t is less than or equal to Trv, then cij is 1.


## 3.2.3. EccentricityEcc

Due to the vehicles’ high speed in real time, communication links break more frequently. An evolving cluster model is necessary for maintaining the link. Reclustering will typically be unavoidable after the CH resigns or no longer meets the criteria to remain a CH. Eccentricity is a concept that is introduced to ensure stability. Here, spectral clustering is used to create a dynamic graph-based model.It is intended for a vehicular graph topology to be hypergraphH=V,E,W, and the concept of eccentricity Ecc is introduced. Ecc is the mean/average eigenscore λi of each vehicle in a cluster; it is calculated as follows [6]:(15)Ecc=1Vehinum∑λiϵVehinumλi.


## 3.2.4. Vehicle’s Time to LeaveTleave

It is the amount of time needed for a vehicle to reach the last section of the lane. This parameter guarantees to choose a CH with enough time remaining to finish the lane that causes it to head for a longer period. It is determined using the length of the laneL, the distance covered by a vehicle on the road segment D, and the vehicle speed at that time v:(16)Tleave=L−Dv.The possibility of the vehicle changing direction changes the lane length. Figure3 can give a picture of the problem. Tleave fails to estimate the road junctions. The unpredictable change in the direction of the vehicle at the junction can abruptly change the prediction. The lane lengths may differ, and so, the vehicle’s estimated next position must be known ahead of time. It helps to select the lane and its length to get the necessary Tleave.Figure 3
Problem inTleave without the next position estimation.


## 3.2.5. Estimated Next Position

To predict the next vehicle position at timet+1, firstly the new vehicle coordinates should be calculated from the current coordinates X2,Y2, distance d=v2t2, and vehicle current direction θ2, and it is calculated as follows [32]:(17)X2′=X2+v2t2cosθ2,Y2′=Y2+v2t2sinθ2.Then, a vehicle new direction using the predicted positionX2′,Y2′ is calculated by using(18)θ2′=tan−1Y2−Y2′X2−X2′.From the vehicle’s new directionθ2′, the next lane length L and D are known now, and then, we can calculate the Tleave′ with the predicted next vehicle’s position using equation (16).After calculating the four measures for each vehicle at each instant of time, the GRA model is used to find the vehicles’ score. The GRA is explained in the next subsection.


## 3.3. Grey Relational Analysis Model

GRA is a component of grey system theory, which is useful for resolving problems involving complex interrelationships between multiple factors and variables. The basic step of GRA is to convert the performance of each option into a comparability sequence, which is denoted by the lettersXij. The term for this step is “grey relational generating” [33]. A reference sequence is defined in accordance with these sequences. Following that, the grey relational coefficient between the reference sequence and all comparability sequences is determined γX0j,Xij. The grey relational grade between each comparability sequence and the reference sequence X0 is then determined using these grey relational coefficients. If a comparability sequence translated from an alternative has the highest grey relational grade between the reference sequence and itself, that alternative will be the best choice. The grade gv is calculated as follows [34]:(19)gv=∑j=1nwjγX0j,Xij,fori=1,2,……,Vehinum,where n=4 (the number of CH selection parameters in our work), i is a number of alternatives. γX0j,Xij is the grey relational coefficient, how near Xij is to X0j is determined using it. wj is the weight of the four parameters, where ∑j=1nwj=1.The grade for each vehicle in a cluster is calculated using the GRA. The vehicle with the maximum grade is selected as a CH for each cluster at each instant of timet.The pseudocode for the CH selection phase is introduced in Algorithm2.Algorithm 2: The pseudocode for CH selection using the GRA model.
Input: number of clusters, number of vehicles in each cluster at each instant of time, vehicles coordinators (X,Y), current lane length (L), current direction θFort = 1: instant of timeFori = 1: clustersForj = 1: vehicles in clusterCalculate the CH parameters:nd, ψvehi, and EccPredicted the next vehicle’s coordinators (X′, Y′)Find the new vehicle’s direction from predicted coordinatorsθ′CalculateTleave′ using predicted vehicle’s positionEnd forInput the four matrices to GRA model with their weights, where∑j=1nwj=1Generate a matrixX of Vehinum×4Normalize the matrixXFori=1:Vehinum(a)
Forj=1:nCalculate the grey relational coefficientγX0j,Xij(b)
end forend forGenerate a graph object from grey relational coefficientγX0j,XijCalculate the Vehicle’s gradegv with the help of γX0j,XijHighest-grade vehicle is selected as a CHCH=maxgvEnd forEnd forOutput: CH vehicle


## 3.4. Time Complexity of the EtHgSC Scheme

The total time complexity of the EtHgSC scheme is expressed as follows:(20)OTOT=OCF+ORSU+OCH,where OCF stands for the time complexity of cluster formations, ORSU for RSU deployment, and OCH for CH selection. The formation of a hypergraph, the application of Laplacian, and the Eigen-trick technique to solve eigenproblems, followed by K-means to get the ideal set of clusters, are the key steps in the cluster formation process.In hypergraph, the quotient of similarity is taken between each of the vehicles, and this is calculated by constructing ad nearest neighbour graph with complexity ON2d. By using the Eigen-trick method, the hyperedge Laplacian Le is processed instead of vertex Laplacian Lv, and the time complexity for Le is ONNZI2/N.“Eigen trick” provides us a way to calculateUv with lower time and space complexity. The time complexity for solving the eigenproblem of the vertex is ON3, while the corresponding cost to hyperedge is Om3. Instead of solving the eigenproblem of Lv directly, the Eigen-trick method is proposed to solve it using the eigenproblem of Le. It will reduce the computational complexity from ON3, to Om3 significantly m<N. Another benefit is that the conversion between Ue and Uv is accurate without any information loss [26]. The last is the K-means complexity which is dependent on the OCoptimal=OτNCnum. So, OCF will be(21)OCF=ON2d+ONNZI2N+Om3+OτNCnum.The complexity is reduced by removing the terms of less computational power(22)OCF=ON2d+Om3.Due to the use of a graph in the RSU deployment, the computational complexity is as follows:(23)ORSU=ON2d.The CH selection parameters are relative speed scoreψvehi, neighbourhood degree η, eccentricity E, and time to leave estimation Tleave′.(24)OCH=Oψvehi+Oη+OE+OTleave′.The time complexity forψvehi, η, and E is calculated as follows:(25)Oψvehi=ON,Oη=OlogN2.The relative speed is calculated using a straightforward threshold function based on the vehicle’s speed. The neighborhood is the next, and it is determined by thecij affinity matrix for neighboring vehicles.The affinity matrix and eigenvalue decomposition of spectral clustering methods are used to determine the eccentricity:(26)OE=ON2d+ON3.The complexity of time to leave is calculated for all the vehicles in the network(27)OTleave′=ON.Thus, the complete time complexity is reduced to moving all the terms with less complexity than cubic and quadratic terms(28)OCH=OlogN2+ON3.The overall complexity is primarily dependent on the hypergraph(29)OTOT=2ON2d+Om3+OlogN2+ON3.


## 4. Performance Evaluation and Simulation Results

The simulation is done using MATLAB as a network simulator, SUMO as a traffic simulator [35], and with the help of TraCI [36]. The area considered for the study is the real map of Iraq’s capital, Baghdad, with latitude = 33.3730°N and longitude = 44.3960°W; it is extracted from OSM, Figure 4. The goal of this study is to evaluate cluster stability at low and high vehicle densities (100 and 1000, respectively), as well as in a dynamic scenario. Also, our proposed model is tested at different vehicle speeds (10, 15, 20, and 25 m/s). Table 2 presents the simulation parameters.Figure 4
Map of Baghdad in SUMO.Table 2
Simulation parameters.
ParametersValuesNetwork parametersIEEE 802.11pScenarioUrbanLanes458 lanesVehicle density100, 1000Simulation time (t)1500 sTv200 mTRSU350 mMaximum speed of vehicle10–25 m/sThirteen clusters are generated using Algorithm1. Each cluster is depicted in a different color. RSUs serve as a cluster center; they are portrayed in a triangle shape in Figure 5. The discussion of the results is divided into two stages: the effect of the predicted next vehicle’s position on the cluster’s stability at different vehicle speeds and for the worst-case scenario (1000 vehicles) and a state-of-the-art comparison in terms of stability using different cluster performance parameters.Figure 5
Clusters formation along with RSU deployment with maximum vehicle density.The following four cluster performance parameters are used to evaluate the performance of our EtHgSC scheme [1]:(1)
CH lifetime: It is the time between a vehicle becoming a CH and subsequently changing to another state. The stability of a cluster depends highly on this metric. If the CH has a long lifetime, the necessity of creating a new cluster will be low.(2)
CM lifetime: It refers to the period beginning with the vehicle’s connection to the cluster and becoming one of its members and ending with its departure.(3)
CH change rate: It is the number of state transitions from CH to another state per unit of time. A lower CH change rate is preferable for the stability of VANET clustering.(4)
Clusters number: The number of clusters that form while the network is in use is intended. When there are fewer cluster numbers, the clustering method performs better.


### 4.1. Effect of Predicted Next Vehicle’s Position on the Stability at Different Vehicle Speeds

Figures6–8 demonstrate the effect of using Tleave′ with predicted next vehicle’s position and without predicted next vehicle’s position (based on current position) on the CH lifetime, CM lifetime, and the change rate of CH, respectively. The proposed scheme is tested at different vehicle speeds for the worst-case scenario of 1000 vehicles. In a comparison with our proposed model using Tleave without predicated, we see that our proposed model using Tleave′ with the predicted next position achieves the highest CH lifetime, CM lifetime, and lowest CH change rate, especially at high speed (25 m/s). The Tleave′ with predicated next vehicle’s position increases the CH lifetime by 22% and the CM lifetime by 12% and decreases the CH change rate by 31% at all vehicle speeds in a comparison with our EtHgSC using the current vehicle’s position Tleave. The predicted next vehicle position increases the cluster stability because it solves the problem of changing the CH at junctions. So, it helps to select the vehicle as a CH, which has a high ability to stay in the cluster as long as possible.Figure 6
CH lifetimes for EtHgSC with and without predicted next vehicle’s position.Figure 7
CM lifetimes for EtHgSC with and without predicted next vehicle’s position.Figure 8
CH change rate for EtHgSC with and without predicted next vehicle’s position.


### 4.2. State-of-the-Art Comparison

Table3 compares the number of clusters and average CH lifetime for EtHgSC with other techniques from the literature. We compare with the CVoEG [6], Arkian et al. [15], JCV [24], VMaSC [14], and PMC [16]. The VMaSC and PMC are the most-cited clustering algorithms in the literature. This comparison occurs at low traffic density (100 vehicles), and the maximum vehicles speed is set to 25 m/s, as well as Tv is set to 200 m for all algorithms.Table 3
Comparative analysis of different algorithms at low traffic density.
AlgorithmAv. CH lifetime (%)Cluster numberEtHgSC815CVoEG65.520JCV7616Arkian et al. [15]5855VMaSC6617PMC41—The number of clusters that formed over time was another indicator of the algorithm’s success. The quality of the created clusters can be assessed using these numbers. Few clusters of limited mobility vehicles are able to connect effectively and maintain stable clustering. On the other hand, increased clustering eventually results in increased overhead and mergers. At low and high traffic levels, the EtHgSC constructs 5 and 13 clusters, respectively.Khan et al. [6] presented the CVoEG algorithm. They evaluated a graph spectral clustering technique on a network of highways. According to the I-5 highway analysis of the California environment, CVoEG [6] produces 20 clusters with low traffic density along a road length of 12 km. It is anticipated to reach a stability level of 65.5%. Vehicle speed is employed in this study to simulate graph edges. As a result, with low variance, low cluster formation results from the eigenvalues being nearly identical due to the almost identical speeds of the cars.Our proposed model has raised the concern of the CH stability at the junction, as the vehicle’s direction is unpredictable at the road junction. A similar problem is highlighted by Mukhtaruzzaman and Atiquzzaman [24] in their work. In the junction-based clustering in VANET (JCV) [24], the vehicle direction and transmission range are considered for the clustering, and the relative position, direction at the junction, time spent on the road, and node’s degree are considered as the CH selection parameters. With 16 clusters constructed, the JCV achieves 76% stability.In the method proposed by Arkian et al. [15], for a highway length of 3000 m, a large number of dynamic clusters are projected with a low variance of only 90 vehicles using two-lane analysis. This method uses neighborhood analysis, so when there are just 90 vehicles, there must be a lot of clusters to cover all the vehicles in a sparse region. With low traffic flow and a large number of clusters, the CH stability is 58%.The average CH lifetime of the VMaSC algorithm is 66%, with 17 clusters generated for 100 vehicles. This algorithm considers the relevant mobility metric as the CH selection factor. Because of the rapid movement of the vehicle nodes in VANET, the duration of each CH is extremely short.The PMC algorithm achieves 41% stability. This ratio is the lowest compared with all other algorithms although the CH selection is based on the neighbor following strategy. The authors did not mention the number of constructed clusters, such that Table3 lacks that.We can see that there is a significant effect of the number of constructed clusters on stability; a few clusters’ numbers improve the clustering stability. The highest stability is achieved using our EtHgSC. So, we can conclude that the formation of VANET as a hypergraph improves the clustering efficiency compared with other algorithms. Also, the use of the Eigen-trick method to improve the hypergraph algorithm as well as the CH selection scheme of our EtHgSC algorithm helps to achieve high stability.The results in Figures9–12 evaluate the clustering stability using average CH lifetime, average CM lifetime, CH change rate, and delay. Our EtHgSC is evaluated at different speeds (10, 15, 20, 25, 30 m/s) and under low traffic level (100 vehicles) and compared with the JCV, VMaSC, and PMC algorithms to show its supremacy.Figure 9
Average CH lifetime at different speeds.Figure 10
Average CM lifetime at different speeds.Figure 11
CH change rate at different speeds.Figure 12
Average delay at different speeds.


#### 4.2.1. Average CH Lifetime

We can see that the average CH lifetime indicates a decreasing trend as vehicle speed increases. The network’s architecture significantly changes as a result of the speed increase, breaking the connection. From Figure9, the average CH lifetime for JCV is higher than that of VMaSC and PMC because the JCV algorithm takes into consideration the change of vehicle route at the junctions when selecting the CH, so it achieves a high CH duration. We can see that our EtHgSC has the highest duration time of CH in a comparison with JCV, VMaSC, and PMC. This achievement is due to our intelligent cluster formation algorithm and the CH selection scheme, which helps to select the vehicle that stays in the cluster as long as possible based on different measures. Also, our proposed model solved the problem of unpredicted changes for CH at the junctions, which led to an increase in the stability of the CH.


#### 4.2.2. Average CM Lifetime

Figure10 shows the lifetime of CM for different speeds. The PMC algorithm has a higher CM lifetime than VMaSC, and this is due to the use of the following vehicle with the highest priority. However, the average CH movement speed is an issue for VMaSC, so when a cluster’s speed increases, joining to other clusters is simple, and therefore, the lifetime of CM decreases.Although the duration of CMs decreases when the speed is increased, our proposed EtHgSC still maintains consistency. The EtHgSC achieves the highest lifetime for CMs of about 10%, 35%, and 86% compared with the JCV, PMC, and VMaSC, respectively, all vehicle speeds.


#### 4.2.3. CH Change Rate

Figure11 shows the relationship between the change rate of CH and the vehicle speed. The results show that the CH change rate increases with an increase in the maximum permitted speed. The CH change rate for our EtHgSC is lower than JCV, VMaSC, and PMC under all conditions. The change rate of CH in our EtHgSC is reduced by 6%, 14%, and 23% compared with JCV, VMaSC, and PMC, respectively, all vehicle speeds.


#### 4.2.4. Average Delay

The average delay is also calculated for our proposed model; it is the time taken for a packet to transfer from source to destination. The delay is a distance-dependent parameter, and the optimal CH’s location helps to reduce the delay, so the EtHgSC achieves less delay, approximately 24%, 66%, and 80% in a comparison with the JCV, PMC, and VMaSC algorithms, respectively, as seen in Figure12.


## 4.1. Effect of Predicted Next Vehicle’s Position on the Stability at Different Vehicle Speeds

Figures6–8 demonstrate the effect of using Tleave′ with predicted next vehicle’s position and without predicted next vehicle’s position (based on current position) on the CH lifetime, CM lifetime, and the change rate of CH, respectively. The proposed scheme is tested at different vehicle speeds for the worst-case scenario of 1000 vehicles. In a comparison with our proposed model using Tleave without predicated, we see that our proposed model using Tleave′ with the predicted next position achieves the highest CH lifetime, CM lifetime, and lowest CH change rate, especially at high speed (25 m/s). The Tleave′ with predicated next vehicle’s position increases the CH lifetime by 22% and the CM lifetime by 12% and decreases the CH change rate by 31% at all vehicle speeds in a comparison with our EtHgSC using the current vehicle’s position Tleave. The predicted next vehicle position increases the cluster stability because it solves the problem of changing the CH at junctions. So, it helps to select the vehicle as a CH, which has a high ability to stay in the cluster as long as possible.Figure 6
CH lifetimes for EtHgSC with and without predicted next vehicle’s position.Figure 7
CM lifetimes for EtHgSC with and without predicted next vehicle’s position.Figure 8
CH change rate for EtHgSC with and without predicted next vehicle’s position.


## 4.2. State-of-the-Art Comparison

Table3 compares the number of clusters and average CH lifetime for EtHgSC with other techniques from the literature. We compare with the CVoEG [6], Arkian et al. [15], JCV [24], VMaSC [14], and PMC [16]. The VMaSC and PMC are the most-cited clustering algorithms in the literature. This comparison occurs at low traffic density (100 vehicles), and the maximum vehicles speed is set to 25 m/s, as well as Tv is set to 200 m for all algorithms.Table 3
Comparative analysis of different algorithms at low traffic density.
AlgorithmAv. CH lifetime (%)Cluster numberEtHgSC815CVoEG65.520JCV7616Arkian et al. [15]5855VMaSC6617PMC41—The number of clusters that formed over time was another indicator of the algorithm’s success. The quality of the created clusters can be assessed using these numbers. Few clusters of limited mobility vehicles are able to connect effectively and maintain stable clustering. On the other hand, increased clustering eventually results in increased overhead and mergers. At low and high traffic levels, the EtHgSC constructs 5 and 13 clusters, respectively.Khan et al. [6] presented the CVoEG algorithm. They evaluated a graph spectral clustering technique on a network of highways. According to the I-5 highway analysis of the California environment, CVoEG [6] produces 20 clusters with low traffic density along a road length of 12 km. It is anticipated to reach a stability level of 65.5%. Vehicle speed is employed in this study to simulate graph edges. As a result, with low variance, low cluster formation results from the eigenvalues being nearly identical due to the almost identical speeds of the cars.Our proposed model has raised the concern of the CH stability at the junction, as the vehicle’s direction is unpredictable at the road junction. A similar problem is highlighted by Mukhtaruzzaman and Atiquzzaman [24] in their work. In the junction-based clustering in VANET (JCV) [24], the vehicle direction and transmission range are considered for the clustering, and the relative position, direction at the junction, time spent on the road, and node’s degree are considered as the CH selection parameters. With 16 clusters constructed, the JCV achieves 76% stability.In the method proposed by Arkian et al. [15], for a highway length of 3000 m, a large number of dynamic clusters are projected with a low variance of only 90 vehicles using two-lane analysis. This method uses neighborhood analysis, so when there are just 90 vehicles, there must be a lot of clusters to cover all the vehicles in a sparse region. With low traffic flow and a large number of clusters, the CH stability is 58%.The average CH lifetime of the VMaSC algorithm is 66%, with 17 clusters generated for 100 vehicles. This algorithm considers the relevant mobility metric as the CH selection factor. Because of the rapid movement of the vehicle nodes in VANET, the duration of each CH is extremely short.The PMC algorithm achieves 41% stability. This ratio is the lowest compared with all other algorithms although the CH selection is based on the neighbor following strategy. The authors did not mention the number of constructed clusters, such that Table3 lacks that.We can see that there is a significant effect of the number of constructed clusters on stability; a few clusters’ numbers improve the clustering stability. The highest stability is achieved using our EtHgSC. So, we can conclude that the formation of VANET as a hypergraph improves the clustering efficiency compared with other algorithms. Also, the use of the Eigen-trick method to improve the hypergraph algorithm as well as the CH selection scheme of our EtHgSC algorithm helps to achieve high stability.The results in Figures9–12 evaluate the clustering stability using average CH lifetime, average CM lifetime, CH change rate, and delay. Our EtHgSC is evaluated at different speeds (10, 15, 20, 25, 30 m/s) and under low traffic level (100 vehicles) and compared with the JCV, VMaSC, and PMC algorithms to show its supremacy.Figure 9
Average CH lifetime at different speeds.Figure 10
Average CM lifetime at different speeds.Figure 11
CH change rate at different speeds.Figure 12
Average delay at different speeds.


### 4.2.1. Average CH Lifetime

We can see that the average CH lifetime indicates a decreasing trend as vehicle speed increases. The network’s architecture significantly changes as a result of the speed increase, breaking the connection. From Figure9, the average CH lifetime for JCV is higher than that of VMaSC and PMC because the JCV algorithm takes into consideration the change of vehicle route at the junctions when selecting the CH, so it achieves a high CH duration. We can see that our EtHgSC has the highest duration time of CH in a comparison with JCV, VMaSC, and PMC. This achievement is due to our intelligent cluster formation algorithm and the CH selection scheme, which helps to select the vehicle that stays in the cluster as long as possible based on different measures. Also, our proposed model solved the problem of unpredicted changes for CH at the junctions, which led to an increase in the stability of the CH.


### 4.2.2. Average CM Lifetime

Figure10 shows the lifetime of CM for different speeds. The PMC algorithm has a higher CM lifetime than VMaSC, and this is due to the use of the following vehicle with the highest priority. However, the average CH movement speed is an issue for VMaSC, so when a cluster’s speed increases, joining to other clusters is simple, and therefore, the lifetime of CM decreases.Although the duration of CMs decreases when the speed is increased, our proposed EtHgSC still maintains consistency. The EtHgSC achieves the highest lifetime for CMs of about 10%, 35%, and 86% compared with the JCV, PMC, and VMaSC, respectively, all vehicle speeds.


### 4.2.3. CH Change Rate

Figure11 shows the relationship between the change rate of CH and the vehicle speed. The results show that the CH change rate increases with an increase in the maximum permitted speed. The CH change rate for our EtHgSC is lower than JCV, VMaSC, and PMC under all conditions. The change rate of CH in our EtHgSC is reduced by 6%, 14%, and 23% compared with JCV, VMaSC, and PMC, respectively, all vehicle speeds.


### 4.2.4. Average Delay

The average delay is also calculated for our proposed model; it is the time taken for a packet to transfer from source to destination. The delay is a distance-dependent parameter, and the optimal CH’s location helps to reduce the delay, so the EtHgSC achieves less delay, approximately 24%, 66%, and 80% in a comparison with the JCV, PMC, and VMaSC algorithms, respectively, as seen in Figure12.


## 4.2.1. Average CH Lifetime

We can see that the average CH lifetime indicates a decreasing trend as vehicle speed increases. The network’s architecture significantly changes as a result of the speed increase, breaking the connection. From Figure9, the average CH lifetime for JCV is higher than that of VMaSC and PMC because the JCV algorithm takes into consideration the change of vehicle route at the junctions when selecting the CH, so it achieves a high CH duration. We can see that our EtHgSC has the highest duration time of CH in a comparison with JCV, VMaSC, and PMC. This achievement is due to our intelligent cluster formation algorithm and the CH selection scheme, which helps to select the vehicle that stays in the cluster as long as possible based on different measures. Also, our proposed model solved the problem of unpredicted changes for CH at the junctions, which led to an increase in the stability of the CH.


## 4.2.2. Average CM Lifetime

Figure10 shows the lifetime of CM for different speeds. The PMC algorithm has a higher CM lifetime than VMaSC, and this is due to the use of the following vehicle with the highest priority. However, the average CH movement speed is an issue for VMaSC, so when a cluster’s speed increases, joining to other clusters is simple, and therefore, the lifetime of CM decreases.Although the duration of CMs decreases when the speed is increased, our proposed EtHgSC still maintains consistency. The EtHgSC achieves the highest lifetime for CMs of about 10%, 35%, and 86% compared with the JCV, PMC, and VMaSC, respectively, all vehicle speeds.


## 4.2.3. CH Change Rate

Figure11 shows the relationship between the change rate of CH and the vehicle speed. The results show that the CH change rate increases with an increase in the maximum permitted speed. The CH change rate for our EtHgSC is lower than JCV, VMaSC, and PMC under all conditions. The change rate of CH in our EtHgSC is reduced by 6%, 14%, and 23% compared with JCV, VMaSC, and PMC, respectively, all vehicle speeds.


## 4.2.4. Average Delay

The average delay is also calculated for our proposed model; it is the time taken for a packet to transfer from source to destination. The delay is a distance-dependent parameter, and the optimal CH’s location helps to reduce the delay, so the EtHgSC achieves less delay, approximately 24%, 66%, and 80% in a comparison with the JCV, PMC, and VMaSC algorithms, respectively, as seen in Figure12.


## 5. Discussion

This work has three novel contributions: the first one is the improved hypergraph algorithm using the Eigen-trick method to optimally cluster vehicles. The time to leave estimationTleave′ is another novel addition to our work. Also, based on the grey relational analysis model (GRA), a novel metric for selecting the CH is proposed that meets the requirements of maximum relative speed ψvehi, neighborhood degree nd, eccentricity Ecc, and maximum time to leave estimation Tleave′.The change in the vehicle’s direction at the road junction hammers the stability. Finding the estimated next vehicle position with the aid of PGRP prediction aids in choosing the CH that has a high ability to stay in the cluster as long as possible, thus improving the overall stability. It is important to find the estimated next vehicle position when calculating the time to leave parameter. So, from the results, we can see that the CH is more stable when using the estimated time to leaveTleave′ instead of the time to leave without estimation Tleave. As a result, the proposed approach achieves more stability and avoids frequent cluster breakage at the junctions.Also, compared to the most common clustering algorithms in the literature, the JCV method follows our proposed EtHgSC method in terms of stability, because the two methods solve the problem of CH stability at junctions by preventing the frequent cluster breakage. The two methods take into consideration the change of vehicle route at the junctions when selecting the CH.Finally, although the JCV method achieves good stability, our proposed EtHgSC method is the best, achieving 81% stability at low traffic levels while the JCV method achieves 76% stability. This is because of the intelligent cluster formation method using the Eigen-trick method, which is used to improve the hypergraph algorithm and make clustering more efficient. The scheme of CH selection using the four measures and the GRA model is also a reason why stability is getting better. In addition, we mentioned earlier that a few clusters number is able to connect effectively and maintain stable clustering. So, from the results section, we see that our efficient hypergraph algorithm constructs the fewest number of clusters (5 clusters) compared with other algorithms.


## 6. Conclusion and Future Work

In this article, an Eigen-trick-based stable clustering algorithm (EtHgSC) in VANET has been introduced with the goal of increasing clustering stability by employing efficient cluster formation and cluster head selection methods. The proposed scheme includes two main parts: the cluster formation phase and the cluster head selection phase.The cluster formation has been handled using an improved hypergraph-based spectral clustering algorithm. The hypergraph algorithm has been improved using the Eigen-trick method; this method is used to partition both vertices and hyperedges, which provides an approach for reducing the computational complexity of the clustering. By utilizing higher-order information in eigenvalues, the Eigen-trick improves clustering efficiency. Four parameters have been used for selecting the cluster head: in addition to relative speed, neighboring degree, and eccentricity, the vehicle’s time to leave parameter has been presented. In this parameter, due to the possibility of a vehicle changing its direction, the estimated next vehicle’s position is used to get good and more stable results. So, the proposed approach achieves more stability and avoids frequent cluster breakage at the junctions. Our proposed approach has been applied to an urban scenario at low and high traffic levels and at different vehicle speeds. In comparison to other state-of-the-art algorithms, the results show that our proposed scheme is superior in terms of cluster stability and packet delay.In future work, to achieve more effective clustering, we will try to employ a modularity matrix rather than an adjacency matrix to generate more efficient clustering along with a vehicle’s lane index corresponding to other vehicles to improve the CH stability.



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*Source: 6327247-2023-03-15.xml*