The Impact of Vehicular Networks on Urban Networks

The objective of this paper is to study the impact of a vehicular network on a physical (road) network consisting of several intersections controlled by traffic lights. The vehicular network is considered to be a random graph superimposed on a regular Hamiltonian graph. The two graphs are connected by hyperlinks. The evolution of traffic at intersections given the existence of vehicular networks is measured by the method of reflective triangles.


Introduction
The objective of this paper is to study the impact of a vehicular network on a physical (road) network consisting of several intersections controlled by traffic lights.The communication network is represented by what is called a Vehicular Network (VN) that is considered to be a random graph [1] [2], superimposed on a regular Hamiltonian graph (road network).The two graphs are connected by hyperlinks.Each hyperlink signifies the interaction between the random graph (vehicular network) and the Hamiltonian graph (road network).Interaction is defined as a driver being at an intersection and talking to someone on a cell phone or connecting to Internet while idly waiting for the green light.Each hyperlink is characterized by (0, 1), (0) signifying waiting at an intersection without using a cell phone and (1) signifying waiting at an intersection using a cell phone.The variables that link the two graphs are queue length, delay, and average traffic density at intersections.Queue length, delay, and average density are functions of the physical position of the vehicle (intersection characteristics, traffic light duration, and any random event) and the state of the connectivity to the communication network or the Internet.It is assumed that connection to the communication network is random.The behavior of the two networks together can be studied using the method of reflection triangles.Each side of the reflection triangle represents one of the (3) variables.One side would be the queue length, one side delay and the third side the average density.The angles of the reflection triangle represent the degree of the dependency of the variables to each other.For example, an acute angle between queue length and delay indicates that delay is much greater than queue length.On the other hand, if the angle between queue length and delay is large, then it indicates that the two variables correlate.An extreme case is when the angle between the two variables is (180 degrees).This implies that the two variables have the same magnitude.To test the methodology of reflection triangles, the evolution of an urban intersection is analyzed given the existence of a communication network that is connected to the intersection.By this, it is meant that all vehicles at the intersection are communicating via cell phones or Internet.The simulation method used for the analysis is the method of (Reflective Triangles (RF)) [3]- [5].The complex Vehicular-Road network with reflective triangles is called a Reflective Network (RN).
Vehicular Networks (VN) is a focus of study for many researchers.This is due to the evolution of wireless communication that allows for vehicle to vehicle, and vehicle to Internet communication.The advantages of the VN network are the use of VN in accident avoidance, traffic jam, and traffic delay regulation, routing modification, almost instant emergency aid access, and immediate collection and distribution of safety information.The bulk of the research on the VN is on designing ways to render it more efficiently [6] [7].There is an abundant literature on rendering road networks efficiently and risk free effectively by reducing the impact on road users of network failure due to incident-related congestion and bottlenecks in particular, congestion from accidents, vehicle breakdowns, road works, lane blockages and road closures, [8] [9].Up to now, there have been very few studies done on the impact of vehicular networks on road networks [10] [11].The current main areas of research in this area are: in routing algorithms [12] [13] (e.g.shortest path algorithms), and adaptive traffic lights [14] [15].There is no overlap between the research on the (VNs), and the research on road networks.This paper adds to the existing literature by introducing a new algorithm that is aimed at improving and facilitating road traffic by introducing a VN and thus constructing a complex multilayer network (Vehicle-Road network) and providing a means of analyzing such a complex system using the reflective triangles.

The Building Blocks of the Vehicle-Road Network
Each Reflective Network, RN is made of two or more networks.These networks are not necessarily compatible but can be compared through transformation into graphs.The graphs are related to each other by hyper-links.The representations of graphs are of outmost importance.For example, the urban road network should be represented as a stochastic Hamiltonian graph, and the Vehicle Network should be represented as a random graph.The stochastic Hamiltonian graph is defined as a graph ( ) ( ) H X E consisting of an ordered pair of disjoint sets ( ) where each ( ) T is the study period, and the set ( ) T X is the set of all vertices during the study period ( ) , , , , , , in the urban network.Each  ( ) , where ( ) ρ is the traffic density during time point ( ) i t on links with intersection vertices.Link density ( ) ρ is defined as the number of vehicles per unit area of a link.
Normally, the probability space contains graphs that are structured on a fixed set of distinct vertices, for ex-ample, . Contrary to this, the set of vertices ( ) ( ) is a variable set.The elements of each set vary with the number of vehicles at an intersection.For the initial period ( ) and the probability of occurrence ( ) 1 1 p = .This could be considered as an initial boundary condition for the space ( ) ( ) The justification for the assumption of ( ) , and therefore it has to be taken as a sure event otherwise the space is nonexistent.Theorem 1.In the initial probability space, the number of edges ( ) and the number of vertices , 2 n t ρ ≥ have to be at least (2).
Proof: Let ( ) V be a set with ( ) be an empty set.The probability of the elementary event that the subset ( ) , where ( ) m is the number of edges and is equal to zero ( ) The probability of ( ) . This is a contradiction since based on the Ramsey theorem [16], for every integer ( ) , there exists an ( ) every connected set of order at least ( ) n contains ( ) K ′ induced connected subsets.In this case, ( ) and   ( ) V should have at least one connected sub graph, which excludes the elementary subset ( ) 0 G .The implication of this theorem is that only those time points are selected that contain VN links.Thus there may be gaps between intervals, and so there is no continuity of time points.The most basic VN contains at least one link.
The number of edges for the graph , and the number of sub graphs is ( ) if the number of edges of sub graphs is ( ) , the probability space is The number of edges of sub graphs is ( ) ( ) The probability of occurrence of ( ) . The probability of the edge belonging to a sub graph is .
The probability of the number of edges of sub graphs ( ) , where the number of edges exceed ( ) the initial graph edge number is calculated as follows: let ( ) N ′ is the maximum allowable edges in the probability space δ is a fraction between [ ] 0,1 .This could occur if and only if ( ) max ρ ρ = and all vehicles in the network were communicating using wireless phones or internet connections, then , here ( ) t stands for the time interval corresponding to the event of ( ) , and the probability of occurrence of a sub graph with ( ) . ( ) M P ′ is the probability of occurrence of ( ) M ′ edges is defined as: , then ( ) , where ( ) , and let ( ) , where , , , , , where (l) is the number of intersections in the road network.Each node ( ) ( ) h t , where ( ) , where ( )

The Method of Reflective Networks
In this section a new algorithm is introduced that analyses and calculates the impact of a Vehicle Network on an urban traffic network.The methodology is based on the application of reflective triangles.Reflective triangles demonstrate the state of a network at intersections, and they reflect the state of the network in the near future.For example, if a period ( ) T is chosen to study the network, and if this period is divided into ( ) , , , t t t T = , then parent triangles are constructed during period ( )  The objective behind building reflective triangles is to define the dynamic nature of each intersection in a concrete manner, where a definite structure can be formed in a manner that is called "deterministic chaos".The evolution of initial or parent triangles to their off springs show the way to deciphering the ordered or chaotic nature of each dynamic system, which in this case is a Road-Vehicle Network represented by the characteristics and the communication activities around intersections.The second advantage of applying the method of reflective triangles is that it allows for an in depth analysis of a system without having to depend on historical data.Each time a system analysis is required, a whole new base data (parent triangles) could be constructed that reflect more realistically the nature of the system given that the environmental (external) and the internal vectors of the system are entirely different from the their historical counterparts.
The method of reflective triangles consists of building the parent triangle during the initial or starting period, and then constructing off-spring triangles for the consecutive intervals.The reflective or off spring triangle is obtained by reflecting the vertices of the parent triangle.The parent triangle is either generative or degenerative.A degenerative parent triangle is most likely to produce degenerative off springs which is an indicator of a chaotic evolution.Generative parent triangles are more likely to produce generative off springs.Obviously, each triangle is made up of three angles and three sides.The sum of the three angles must add up to 180 degrees; these angles are represented by ( ) α , ( )  length ( ) ϑ and delay ( ) Λ compared to density ( ) ρ .The acute angle between the queue length and delay confirms the correlation between the two variables.This situation is due to the existence of an active Vehicle Network.The off spring (reflective) triangle is generative, the queue length, delay, and density are similar in magnitude to the parent triangle and thus predict that if the Vehicle Network has a fixed number of edges, the traffic around the intersection will exhibit stable behavior and this will continue for many consecutive intervals.M edges, as in Equation ( 1): It is possible to have many parent or initial triangles during interval ( ) if there are micro-stochastic changes.Multiple parent triangles are created by generating several queue lengths as is given in Equation ( 2): In Equation ( 2), ( ) a is a constant, and ( ) ε is a small perturbation that causes a spike in the parent queue length ( ) ( ) The angles ( ) α , ( ) β , and ( ) γ are calculated as: The coefficients in Equation (11), are the ones suggested by IJzeren [5], but in general any coefficient would work as long as ( ) , and ( ) ≠ .This condition is demonstrated by [16].The same condition also assures the non-degeneracy of the off spring triangles.The first step towards obtaining an off spring or a reflective triangle, is to reflect the vertices of the parent or (initial) triangle.To find the reflective vertices, a reflection circle of radius ( ) ( ) M P ′ is the probability that there are ( )      14): , i R t ρ is the average of the densities of all the links of an intersection during ( ) i t .The angles corresponding to the (3) sides of the off spring triangle are calculated as follow: It is possible to have more than one off spring triangle; in other words an off spring can have siblings.This would be the result of the occurrence of micro-stochastic events during ( ) i t .In that case, the off shoots of the three sides of the off spring triangle are expressed as follow: Calculation of the corresponding angles for the sibling triangles is similar to the off spring triangle.It suffices to replace the variables by their micro-stochastic versions.The off spring triangle is constructed following the procedure described here.As is shown in Figure 3, to construct the reflection or off spring triangle, one must draw the angle bisectors of the parent triangle, details of which are given earlier.The point of the intersection of the three angles bisectors becomes the center of a new reflection circle, subset of the disk ( )

Conclusion
Reflective network is an urban network connected to a vehicle network through hyperlinks.It is called reflective network because it contains reflective triangles.In this type of an urban network, each intersection is represented by a multitude of triangles, each one representing the state of traffic at an intersection during a time interval.Each triangle is constructed based on the previous triangle given a particular procedure.The process is repeated and each triangle is analyzed after each iteration.The set of triangles forms a dynamic system which could demonstrate either a stable (generative triangles) or a chaotic (degenerative) evolution depending on the situation of traffic and the state of communication at intersections.The advantage of such method is primarily that it allows for a complex multilayer network that consists both of physical links (roads) and communication links that can be related to each other.The dependency on historical data is significantly reduced since for each situation, it is now possible to create its own particular history starting with the parent triangle to the off spring triangle and the next generations.The ancestor triangle constitutes the historical data.The method of reflection triangles also allows for a rapid macroscopic analysis of the system through the parent and the reflection triangles of all intersections included in the reflective network.The method of reflective networks opens a whole new way of analyzing traffic by focusing at intersections rather than tracking traffic along each link of a network.
Since the two events ( ) G , and ( ) G′ are independent, and occur with probabilities ( ) M P and ( ) M P ′ , then let ( ) P be the product measure of the two measures ( ) the nodes corresponding to hyper links, where the set

x
* are the set of intersection nodes in the Hamiltonian network, intersection nodes.Since a hyper link represents the existence of wireless communication at intersections, it does not matter which link in the set of VN network is used as long as they are correlated with the right intersection nodes in the road network.Hyper links flag out the existence of wireless communication and their role does not extend beyond this.Each hyperlink is characterized by (0, 1), (0) signifying waiting at an intersection without using a cell phone, (1) signifying waiting at an intersection using a cell phone.The nodes ( ) j h are used to locate the parent triangles in reflective networks.In Figure1, the communication links(3,2), and (1, 2) represent internet connection or conversation with someone outside of the zone of the intersection by road users (1), and (3).Communication link(1,3) represents that the two road users on links (1), and (3) are in communication with each other.

1 t.
During each subsequent interval offspring triangles are constructed based on the parent triangles.Each parent triangle can have an off spring or many off springs (siblings) during each period.The occurrence of multiple parents or multiple siblings depends on whether changes during an interval are static or micro-stochastic.Micro-stochastic refers to changes in variables that occur during an instance of time which from here on will be designated as ( ) ε ∧ -changes.( ) ε ∧ - changes can be regarded as spikes in the values of variables used in constructing either parent or offspring triangles.Normally, the values of the variables used are continuous during each time point ( ) i t , but any sudden changes due to unexpected events during a time point ( ) i t cause spikes, and thus allow for either multiple parents or multiple off springs.The details are given later on in this section.

Figure 1 .
Figure 1.The Road-Vehicle Network with hyper links.

β 2 .
must be determined in a systematic manner each time a triangle is constructed.Each side of a triangle is representative of traffic and communication characteristics at an intersection.Intersection characteristics are designated as: queue length ( ) ϑ , delay ( )Λ ,and traffic density ( ) ρ .Queue length, delay, and density are normally defined as functions of traffic behavior only.In the new formulation, these variables are formulated as functions of the physical position of a vehicle (intersection characteristics, and any random event) and the state of the connectivity to the communication network.The length of one side of a triangle represents queue length ( ) ϑ at intersection, the length of the second side represents traffic density, ( ) ρ , and the length of the third side represents delay ( ) Λ .The magnitude of the angles ( ) α , and ( ) β and ( ) γ indicate the degree of the dependability of the three traffic characteristics at any intersection.The parent triangle is constructed during the initial time point ( ) 0 t .The off spring triangle is born from the parent triangle during the next time point.Each off spring gives birth to its own off spring during the consequent time point.This way the parent triangle becomes the ancestor of the future generation off spring triangles.The evolution from the parent triangle to the off spring (reflective) triangle and the next generations is depicted in Figure In Figure 2, the intersection of Figure 1 is revisited.The objective is to demonstrate the evolution of traffic through reflective triangles.Initially, the parent triangle is a generative triangle, which indicates long queue

Figure 2 .
Figure 2. Parent triangle at (t 0 ) and its off spring and the next generation at (t i ) s.
the number of cars that are likely to remain at the intersection by the end of the initial time point.The parent queue length ( ) number of cars that are left on intersection links from before the initial time point, ( ) λ η , and ( ) M P is the probability that VN has ( )

1 a 1 b 1 c′ and ( ) 2 c′ as is shown in Figure 4 . 1 c
. The second reflection circle of radius ( ) .Both reflection circles cross the parent average density at points ( ) The three points ( ) of the (3) vertices of the parent triangle.It is postulated that the three points ( ) are in fact the points of intersections of the bisectors of the parent triangle with its sides.The point of intersection of the bisectors of the parent triangle ( ) O is the starting point of the off spring or reflection triangle.The off spring or reflective queue length

Figure 3 .
Figure 3. Parent and reflection triangles drawn from subsets of disk (D).

µ
is fixed for all time points.
communication length during ( ) i t .The average off spring or reflective density is calculated as in Equation (

Figure 6 .
Figure 6.The off spring or reflection triangle.

Figure 7 .
Figure 7.The parent and the off spring triangles. , T , where , )