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.

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 consisting of an ordered pair of disjoint sets such that for where each is a time point, is the study period, and the set is the set of all vertices during the study period such that. Each is a set of vertices at any time point, and is the total number of vertices during each time point. Each vehicle traverses the set of vertices once and only once during each time point. The set is the set of all edges during time, where in the urban network. Each

is a set of edges at any time point, and is the total number of edges during each time point. The stochastic Hamiltonian graph includes cycles.

The Vehicle Network (VN) should be represented as a random graph. The reason for this choice is that it gives the possibility of attaching probabilities to the subsets of the set of random graph space of the VN. The VN is represented by a random graph, where is the probability space, and

is the probability that edge exists. The set of vertices is not fixed; it changes during each time point. Since the number of vertices in the VN depends on the number of vehicles at intersections in the road network, thus the vertices are functions of densities, generically represented as, where is the traffic density during time point on links with intersection vertices. Link density 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 example,. 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, the space is defined by letting and the probability of occurrence. This could be considered as an initial boundary condition for the space. The justification for the assumption of is that period is the birth of the probability space , 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 have to be at least (2).

.

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 is chosen to study the network, and if this period is divided into equal intervals, such as, then parent triangles are constructed during period. 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 -changes. - 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, but any sudden changes due to unexpected events during a time point cause spikes, and thus allow for either multiple parents or multiple off springs. The details are given later on in this section.

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, , and. Angles, and 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, delay, and traffic density. 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 at intersection, the length of the second side represents traffic density, , and the length of the third side represents delay. The magnitude of the angles, and and indicate the degree of the dependability of the three traffic characteristics at any intersection. The parent triangle is constructed during the initial time point. 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 2. 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

length and delay compared to density. 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.

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

Figure 4. Method of bisectors.

Figure 5. The parent triangle.

Figure 6. The off spring or reflection triangle.

Figure 7. The parent and the off spring triangles.