Complex Network Theoretical Analysis on Information Dissemination over Vehicular Networks
Jingjing Wang, Chunxiao Jiang, Longxiang Gao, Shui Yu, Zhu Han, Yong Ren
aa r X i v : . [ c s . N I] J a n Complex Network Theoretical Analysis onInformation Dissemination over VehicularNetworks
Jingjing Wang ∗ , Chunxiao Jiang ∗ , Longxiang Gao † , Shui Yu † , Zhu Han ‡ , andYong Ren ∗ ∗ Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China † School of Information Technology, Deakin University, Burwood, VIC 3125, Australia ‡ Electrical and Computer Engineering Department, University of Houston, Houston, TX, USAE-mail: [email protected], { jchx, reny } @tsinghua.edu.cn, { longxiang.gao, syu } @deakin.edu.au,[email protected] Abstract
How to enhance the communication efficiency and quality on vehicular networks is one criticalimportant issue. While with the larger and larger scale of vehicular networks in dense cities, the real-world datasets show that the vehicular networks essentially belong to the complex network model.Meanwhile, the extensive research on complex networks has shown that the complex network theorycan both provide an accurate network illustration model and further make great contributions to thenetwork design, optimization and management. In this paper, we start with analyzing characteristicsof a taxi GPS dataset and then establishing the vehicular-to-infrastructure, vehicle-to-vehicle and thehybrid communication model, respectively. Moreover, we propose a clustering algorithm for stationselection, a traffic allocation optimization model and an information source selection model based onthe communication performances and complex network theory.
I. I
NTRODUCTION
Due to the emerging of intelligent transport system, vehicular networks have received lots ofattentions. Although cellular networks enable convenient voice communication and simple en-tertainment services to drivers and passengers, they are not well-suited for certain direct vehicle-to-vehicle (V2V) or vehicle-to-infrastructure (V2I) communications [1]. In particular, how toimprove the performances of the communication system has already been under development [2], where some key technologies [3], e.g., small cells, device-to-device (D2D) communication,mobile clouds, flexible spectrum management etc., can be considered to be employed in vehicularnetworks.In the literature of vehicular networks, many researches focused on improvement of the vehiclemobility models [4], communication channel models and the routing strategies [5] [6], while thenetwork properties as well as the complex characteristics of the vehicular networks have notbeen fully investigated. The vehicular networks are associated with a tremendous network size.Moreover, diverse hierarchical structures and node types give rise to more complex interactions.Furthermore, vehicular networks have a complex time-space relationship. The mobility of thevehicles on the road lead to the dynamic evolutionary topology. In terms of some hot communi-cation technologies, the ultra dense cellular deployment would lead to more than ever interactionsamong vehicle units (vehicles to infrastructures and infrastructure to infrastructure) and the D2Dbased vehicular-to-vehicular communication also lead to a more complex hybrid communicationnetwork. Therefore, it is necessary to view the vehicular networks from the other dimension,i.e., using complex network theory to discover the complex characteristics of vehicular networks,based on which the network performance can be improved.With the development of random graph model, the complex network theory emerged based onthe [7] and [8], which discovered the small-word property and the power-law distribution of thenode degree of the realistic complex networks. Based on the advantages of complex networkstheory, this paper proposes a complex network theoretic view on the vehicular networks withfollowing original contributions. For one thing, this is the first work to establish the vehicularnetwork V2V and V2I models with complex network theory. Moreover, We use the node degree,average path length, clustering coefficient and betweenness centrality to analyze the topology ofa vehicular network based on the taxis GPS database of Beijing [9] and study the relationshipbetween the network topological properties and communication parameters. For another thing,we propose a clustering algorithm, a traffic allocation model and an information source selectionmodel depending on the communication impedance.The rest of this paper is organized as follows. Section II establishes a vehicular network systemmodel based on the complex network theory, and gives some key parameters and their characters.Section III describes three typical vehicular communication models and three optimizationalgorithm models. Section IV gives the simulation results for the proposed models. Concluding T he l a t i t ude o f B e iji ng Taxi GPS
Fig. 1. The taxis GPS distribution in Beijing (longitude from 116.25 to 116.55 and latitude from 39.8 to 40.05). remarks and future work are given in Section V.II. D
ATA -D RIVEN C OMPLEX N ETWORK M ODEL
A. Dataset Analysis
In vehicular networks, vehicles can communicate with each other (V2V), and can also establishcommunication with the roadside infrastructures (V2I). In this subsection, we construct thecomplex network model for the vehicular networks based on a real-world dataset, which containsthe taxi GPS data of Beijing (longitude from 116.25 to 116.55, and latitude from 39.8 to 40.05)obtained from the Microsoft Research Asia [9].Based on the aforementioned GPS dataset, we plot the vehicles position distribution in theFig. 1 at one moment. The vehicles position distribution clearly reflects the shape planning struc-ture of Beijing and distinguishes its downtown and suburban areas. In the following subsection,we will construct a weighted and undirected graph model based on some key communicationparameters for vehicular networks.
B. Weighted and Undirected Graph Models for Vehicular Networks
In accordance with the analyses above, we build the vehicular network model as a weighedand undirected complex network in which the nodes represent the vehicles in the road segmentsand the undirected edges represent the interaction between the nodes. The interaction in thispaper means the communication between each two vehicles. The edges weights measure the communication performances on the vehicular networks which depend on the distance betweenthe communication pairs, communication channel fading, the environment disturbance and thecellular radius.To simplify modeling and calculation, we assume that the communication ability of each vehi-cle is identical and communication channel meets the COST 231-Bertoni-Ikegami model [10]. Inaddition, we neglect the cellular gaps and the cellular shapes, which are not affected by terrain.Accordingly, the weighted and undirected vehicular network is noted as a graph G = ( V, E, R ) ,where V is the set of vertices representing vehicles and E is the set of edges representingthe interaction among the vertices. Weights R reflect the communication performance on thevehicular network. R reflects the communication performance in the vehicular ad hoc network,where the specific definition of the communication impedance R is based on the following keycommunication technologies: Channel Model : Because of the city dotted with tall buildings and luxuriant trees, signalsfrom sources may be attenuated severely to destinations. This paper use the COST 231-Bertoni-Ikegami Model to analyze the transmission path loss. We assume that there exists a line-of-sighttransmission path between each two communication-capable vehicles. Therefore, the relativelyaccurate path loss in the urban area, L u can be calculated as: L u = 42 . d + 20 log f c dB , (1)where d is the transmission distance and f c represents the signal carrier frequency. Ultra Dense Cellular Handover : Communication system tends to construct a multi-layerheterogeneous network covering base stations and low power micro-stations. In order to improvespectrum efficiency and the transmission capacity, we have made unremitting endeavor on theenhancement of the modulation and encoding methods, while the decrease of cell radius canalso result in a sharp increase of system capacity. Therefore, an appropriate communication cellradius improves spatial multiplex ratio and reduces the system power consumption. Nonetheless,an ultra dense cellular handover means a frequency conversion, more shared-spectrum inter-ferences and more difficulties in multi-point coordination. Spontaneously, the time-delay andhandoff dropping probability are both increased due to the ultra dense cellular handover, whichincreases the impedance of communication of each communication link. We make a statistical calculation of the number of cellular switching on each communication link, noted as n s . Basedon the communication channel model and ultra dense cellular handover mentioned above andconsidering the node degrees and betweenness centralities in the complex network theory, wedefine the weight of the edge connecting node i and node j , marked as R ij , which is named aslink communication impedance: R ij = α ( k i B i + k j B j ) υ + βL uψ − µ ( ϑ/d ij ) ξ + ζ n s , d ij ≤ r ∞ , d ij ≤ r (2)where k i represents the degree of the node i and B i notes the betweenness centrality of vehicle i . ϑ shows the energy noise ratio, α, β, µ are characterized parameters varying with diffidentnetwork topology, and υ, ψ, ξ and ζ are nonlinear control parameters. Based on the abovedefinition, the communication impedance depends on the node degree, link distance, frequencyof communication, average signal energy noise ratio and the cellular switching times. First, avehicle with a large degree or high betweenness centrality means it participating in quantitiesof communication missions, which leads to a relatively long store-and-forward delay and highprobability of blocking. Second, long communication distance conduces high path loss andconsumes much more signal power. What is more, a small cellular radius leads to more cellhandovers n s , which also increases the time delay and deteriorates the communication per-formance. In these two aspects, the communication impedance should be positively correlatedwith k , B and n s . Third, a high average signal energy noise ratio per unit distance contributesa robust communication, naturally being negatively correlated to the impedance. In this way,we have completely established a complex network graph model for the vehicular networkcommunication. C. Complex Network Verification
In this section, we quantitatively analyze and verify the small-world property and scaling-freeproperty of the vehicular networks. In the first place, we propose some key parameters dependingon the complex network theory.
Node Degree Distribution : The node degree of a vehicle i in the vehicular network, markedas k i , is defined as the number of the vehicles it can communicate with. Moreover, p ( k ) is theprobability that a randomized node’s degree is k . And the distribution of p ( k ) is defined as the node degree distribution. Clustering Coefficients : The characteristic that neighbors can also communicated with eachother is called the clustering characteristic, which measures the tightness of the network. Thevehicle i ’s clustering coefficient is defined as the following: C i = E i k i ( k i − / , (3)where k i represents the node degree of vehicle i and E i is the number of communication linksamong neighbors. Further more, the general clustering coefficient of the entire network is theaverage of C i . Betweenness Centrality : The normalized betweenness centrality B , and therefore, is definedto measure the importance of the node from another dimension, i.e., B i = 2( N − N − X s = i = t n ist g st , (4)where g st is the number of the shortest path from s to t , and n ist notes the number of the shortestpath via i from s to t .A data-driven numerical simulation is conducted for the vehicular network and we verify thecomplex network properties based on the Taxi GPS dataset. Fig. 2 demonstrates the parametersmentioned above of the proposed network with communication distance r = 500 . Moreover, wecalculated the average network clustering coefficient C = 0 . and the average path length l = 6 . .The simulation results conform to the small world property (a high degree of clustering anda short average path length) and a scaling free distribution in node degree and betweennesscentrality. In consequence, we can quantitatively treat the vehicular network as a complex networkand the complex network theory bring us a new perspective in network design, optimization andmanagement for the communication on vehicular networks. Next section, we will propose threeoptimization models under different communication models.III. C OMMUNICATION ON THE V EHICULAR N ETWORKS
In Section II, we have discussed the network topology of vehicular networks. Based on theanalysis above, we establish the V2I (Section III-A), V2V (Section III-B) and the hybrid com- N ode deg r ee k Node degree (a) Node Degree C l u s t e r i ng c oe ff i c i en t Clustering coefficient (b) Clustering Coefficient − ne i ghbo r C l u s t e r i ng c oe ff i c i en t (c) 2-neighbor Clustering Coefficient B e t w eene ss c en t r a li t y Betweeness centrality (d) Betweenness CentralityFig. 2. The Complex Network Parameters Verification. munication model (Section III-C), respectively, with the communication impedance. Moreover,we propose a clustering algorithm for station selection, a traffic allocation optimization modeland an information source selection model.
A. Clustering Algorithm of the V2I Model
In the following, we will focus on the V2I communication model. Similarly, the vehicleimpedance in the V2I model is defined based on the Massive MIMO in vehicular communicationsystem, which is a technology to enhance the overall networks performance. With a large excessof service antennas over terminals and time-division duplex operation, the extra antennas focusesenergy into ever smaller regions of space and bring huge improvements in communicationthroughput and energy efficiency. In [11], the authors proposed the throughput f R k (achievable rate of the uplink transmission from user k to measure the behavior of massive MIMO systems): f R k , (1 − τ − ς ) E[ log (1 + γ k )] , (5)where γ k shows the the signal-to-interference-plus-noise-ratio (SINR) which is a function con-taining channel model parameters and antennas parameters. τ is the channel estimation (CE)time, and ς is the wireless energy transfer (WET) time. In our model, we only consider thevalue of f R k instead of its impact factors. We assume that the base stations directly communicatewith vehicles within its control range, which means that the distance from a vehicle to a basestation is less than the cellular radius in the V2I Model. In this way, we define the communicationimpendence of vehicle i as follows: R i = α ( k i B i ) υ + β f R kψ , i = 1 , , ..., N. (6)Similarly, k i represents the degree of the node i and B i notes the betweenness centrality of thevehicle i . f R k shows the throughput of a certain vehicle-to-station communication link. Besides, α and β are characterized parameters varying with diffident network topologies, while υ and ψ are nonlinear control parameters. A clustering algorithm based on the generalized distance D ispresented. D ij = ǫ ( R i + R j ) + (1 − ǫ ) d ij , (7)where R i represents the vehicle impendence, d ij represents the realistic distance of two vehiclesand ǫ denotes the weighting coefficient. Clustering algorithm based on generalized distance.
Step1: Select one sample point as the clustering center c .Step2: Calculate the generalized distances to the center, and select the i with max i D ic as center c .Step3: Calculate all the generalized distances to the two centers, and select the j with max { min { D jc , D jc }} as center c , the rest can be done in the same manner.Step4: Based on the nearest neighbouring rule classifying other samples.Fig.3 shows a clustering example based on the generalized distance, which provides a con- C l u s t e r i ng v e r ti ca l a x i s y ( m ) Fig. 3. A clustering example based on the generalized distances (different color dots distinguishing the categories). structive suggestion on the base station selection and cellular division.
B. Traffic Allocation on the V2V Model
In terms of the complex communication missions in vehicular networks, a variety of serviceslike real-time voice services, high definition video services and Internet access services shouldbe supported whenever and wherever. However, how to allocate the communication traffic inan optimal fashion is worth discussing in details. For simplification, we assume that there arecertain quantities of communication tasks transmitting from n vehicles to a destination vehicle.The total communication demand quantity is marked as Q . Let v be the vehicle node set and thestarting vehicle set is denoted by S = s , s , ..., s n and X = x , x , ..., x n represents the allocatedcommunication traffic allocation set, where x i is the actual communication task quantity on the i th communication link. We define the cost function C ( x ) as: C ( x ) = n X i =1 X u,v x i R iuv , (8)where R iuv is the communication impedance from vehicle u to vehicle v on the Dijkstra pathunder the condition of transferring the communication traffic x i . Let c be the communicationcapacity of each communication link, which denotes the maximum number of communicationtasks and let m uv represents the total communication tasks on the communication link between vehicle u and v , m uv ≤ c . We have the following optimization problem: min C ( x ) = n X i =1 X u,v x i R iuv s.t. x i ≥ , ∀ i = 1 , , ..., n, n X i =1 x i ≥ Q,m uv = n X i =1 x i a iuv ≤ c, ∀ u, v ∈ V, (9)where x = [ x , x , ..., x n ] T and a iuv = 1 , when the traffic x i goes through the link connectingthe vehicle u and v , otherwise a iuv = 0 . The network traffic allocation optimization problem canbe casted as a convex optimization problem in (11) by the definition of traffic-edge incidencematrix A ∈ R E × n , and A ij = , traffic j passing the edge i , otherwise (10)where E is the total number of probable links, x = [ x , x , ..., x n ] T , and = [1 , , ..., T . Then,we have min C ( x ) s.t. x ≥ , x T ≥ Q, Ax ≤ c . (11)Furthermore, we can add a eigenfunction to this linear programming problem and rewrite it asfollows: min x T R w + n + E +1 X i =1 − t log( − f i ( x )) s.t. f i ( x ) = − x i , i = 1 , , ..., n,f i ( x ) = Q − x T , i = n + 1 ,f i ( x ) = A i x − c, i = n + 2 , n + 3 , ..., n + E + 1 . (12) where A i represents the row vector of matrix A and auxiliary variable t > controls thecomputational accuracy. R w is the sum of the communication impendence of each the allocationrouting.The solution of the problem (12) is marked as x ∗ ( t ) , which satisfies the condition: t R w − x + 1 Q − x T · + A T c − Ax = , (13)where let x = [ x , x , ..., x n ] T , ∀ x ∈ R n . And we can prove that the deviation between x ∗ ( t ) and the optimal solution of primal problem is not more than ( n + E + 1) /t . Many computersimulation algorithms can solve the above optimization problem. C. Information Source Selection on the Hybrid Model
The criterion for selecting the information source location is to make the network capacitymaximize. In another word, the information broadcasting facilities should be located near thesource vehicles associated with information replicas. In this subsection, we focus on the hybridcommunication model, where we study the optimal source vehicles selection strategy. Let q ( i ) indicate the probability of any packet to pass node i , and n ist and g st are defined identically as(4): q ( i ) = X s ( s = i ) X t ( t = i ) p ( s, t ) n ist g st , (14)where p ( s, t ) is the probability of a packet to choose source vehicle s and vehicle t as itsdestination. Instead of uniform distribution, the source vehicles obey the probability p ( s ) , whilewe assume that the destination vehicles of packets are uniformly distributed and are independentlyselected. We have: p ( s, t ) = p ( s ) p ( t ) = p ( s ) N − . (15)Then, the probability of any packet to pass vehicle i can be calculated as follows: q ( i ) = 1 N − X s = i X t = i p ( s ) n ist g st . (16) Define the p ( i | s ) measuring the conditional probability of the situation where packet starts fromvehicle s to pass vehicle i , p ( i | s ) = 1 N − X t ( t = s,t = i ) n ist g st . (17)Then, R c can be estimated as: R c = C max i { R i P s p ( s ) p ( i | s ) } , (18)where R c indicates the upper bound packets generated per time step to maintain in a flow state,and serves as a measure of the overall capacity of the network system, which is a function ofbetweenness centrality and communication impendence R i .The base station selection model, therefore, reduces to a a min-max problem: min max i { R i X s p ( s ) p ( i | s ) } s.t. ≤ p ( s ) ≤ , X s p ( s ) = 1 . (19)After introducing an auxiliary variable Λ : Λ = max i { R i X s p ( s ) p ( i | s ) } ( i = 1 , , ..., N ) , (20)the optimization problem can be casted as a linear programming problem as follows: min Λ s.t. RAp − Λ ≤ , p T = 1 , p ≥ , (21)where A = [ p ( i | s )] , p = [ p ( s ) , s = 1 , , ..., N ] T and = [1 , , ..., . R is defined in (22) R = R · · · R ...... . .. · · · R N . (22) T he a v e r age c o mm un i c a t i on i m pedan c e R r=0.2kmr=0.5kmr=0.8kmr=1kmr=1.5km (a) The impact of carrier fre-quency on the impendencewith different communicationranges. c (km) C e ll u l a r s w i t c h i ng t i m e s n s r=0.2kmr=0.5kmr=0.8kmr=1kmr=1.5km (b) Cellular switching timeswith different communicationranges. V eh i c l e c o mm un i a t i on i m pedan c e Vehicle communiation impedance (c) Communicationimpedance of each vehicle inthe descend order.
Vehicle Index n0 50 100 150 200 250 300 350 400 450 O p t i m a l s ou r c e s e l e c t i on p r obab ili t y p ( s ) Selection probability (d) Optimal source selectionprobability p ( s ) distribution.Fig. 4. Communication impedances analysis and optimal information source selection on the hybrid model. Thus, we can easily find the minimal Λ by linear programming algorithms and get thenumerical solution with the help of calculating computer.IV. S IMULATION R ESULTS
In this section, we conduct simulation on the extensive studies about the network topology andthe communication performances based on our models. First of all, we analyze the influenceof the maximum communication distance r and other key communication parameters on thenetwork topology.Section III-B proposed a vehicular network V2V communication model based on the complexnetwork theory, relying on which we elaborated some complexity parameters to analyze theperformance of the network in the respect of topology structure. In the following, we analyzethe effect of communication parameters on the communication impedance. On this score, weonly concentrate on the topology properties of the vehicular network based on the Taxis GPSin Beijing for the time being and give constructive suggestions on the traffic management andcommunication design.The carrier frequency mainly determines the transmission path loss L u . We obtain five curveswith different maximum communication distances, as in Fig. 4 subgraph (a). The vertical coor-dinates represents the average communication impedance for each of links and in this situationwe neglect the effect of node importance by letting α = 0 in (2). With the increasing ofcarrier frequency under each scenario, the average communication impedance R ascends corre-spondingly. Obviously, the conclusion can be deduced from the definition of the communicationimpedance R . Likewise, a large maximum communication range r contributes the communi-cation impedance with more power loss. Specifically, with a small maximum communication range r (200m ∼ R maintains a relatively small value but ahigh growth rate with r . However, when the r reaches a specific distance (above 800m), thegrow is slowing and communication impedance is tending towards stability. To our knowledge,the carrier frequency in communication may apply a high carrier frequency, but it needs acomprehensive consideration on the path loss and the communication range. Fig. 4 subgraph(b) shows the relationship between cellular radius and switching times under the condition ofdifferent maximum communication ranges. Generally speaking, the switching times descendwith the increasing of cellular radius and the declining rate tends mildness. Even though we canimprove spectrum efficiency and the transmission capacity by narrow down the cellular radius,large switching times reduce the communication performance in the same way. When we extendthe maximum communication range, obviously there is a soaring incasement in the switchingtimes under the scenarios of relatively small radiuses. As cellular radius r c > m , the switchingtimes have no significant changes. The simulation results are consistent with the actual situation.The distribution of taxis in the city are concentrated in crowded areas, which is just the clusteringfeature of the small-world network. The average path length l is surprisingly to a limited extent.As a consequence, in terms of an appropriate cellular radius r c , the average switching timeshover in a narrow range. To summarize, the communication parameters to some extent affectthe impedance of communication. In the realistic engineering, we should synthetically considerthe carrier frequency, maximum communication distance, energy utilization efficiency, cellularradius etc, where a trade-off may contribute a communication effects. This paper provides aperformance analysis method rather than the specific parameters.As for the information selection model, Fig. 4 subgraphs (c) and (d) shows the related simula-tion results with the maximum communication distance r = 500m . Subgraph (c) demonstrates thecommunication impedance of each vehicle in the descend order. Subgraph (d) is the simulationresult about how to select the information sources. Obviously, we can conclude that the vehiclesplay highly symmetrical roles in the information spreading. As shown in subgraph (d), onlya few vehicles should act as sources in heterogeneous vehicular networks. That’s means thesource vehicle should be distributed within a small number of the nodes. Therefore, we candirect or manage fewer vehicles to control the entire vehicle network. More than that, anappropriate communication distance means a small range communication defined above dueto the dispersed degree distribution and the low path loss. It makes great contribution to the green communication with a low power dissipation. As the vehicular network is a large-scaleheterogeneous network, our work suggests that to improve the network capacity, informationlike traffic accident, congested roads or the traffic control should be broadcasted deriving fromcertain source vehicles. V. C ONCLUSION
In this paper, we analyzed the V2V and V2I communication performances on vehicularnetworks based on complex network theory. Furthermore, we proposed a clustering algorithmfor station selection, a traffic allocation optimization model and an information source selectionmodel, respectively which were viewed as examples for illustration of the concrete applicationof the defined communication impedance.A
CKNOWLEDGMENT
This research was supported by the NSFC China under projects 61371079, 61271267 and91338203. R
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