IN
DEGREE PROJECT TEKNIK, FIRST CYCLE, 15 CREDITS STOCKHOLM SWEDEN 2018 ,
Modellering av
kommunikationsnätverk för trafikledning
JENS FAGERBERG LUCAS BERGSTRAND
KTH ROYAL INSTITUTE OF TECHNOLOGY
SKOLAN FÖR ELEKTROTEKNIK OCH DATAVETENSKAP
www.kth.se
IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
STOCKHOLM SWEDEN 2018 ,
Modelling Communication Networks for Active Traffic Management
JENS FAGERBERG LUCAS BERGSTRAND
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
www.kth.se
Sammanfattning
Restider, trafikstockningar och utsl¨ app av v¨ axthusgaser i stadsomr˚ aden ¨ ar ett st¨ andigt v¨ axande problem med ¨ okad urbanisering. ¨ Overvakning av trafikfl¨ odet vid trafikljus m¨ ojligg¨ or aktiv trafikledning och stora f¨ orb¨ attringar till dessa problem. F¨ or att trafikljus skall kunna kommunicera beh¨ ovs en grundl¨ aggande n¨ atverksstruktur. Detta kommunikationsn¨ atverk kan centraliserat eller decen- traliserad, d¨ ar b˚ ada har f¨ or- och nackdelar. Om befintliga n¨ atverksmodeller, s˚ asom Bar´ abasi-Albert och Erd˝ os-R´ enyi modellerna, studeras och modifieras kan en algoritm som genererar ett robust och s¨ akert n¨ atverk utvecklas, vilket har varit projektets huvudsakliga m˚ al. F¨ or att s¨ akerst¨ alla att en optimal modell anv¨ andes analyserades denna ur ett robusthetsperspektiv d¨ ar parametrar s˚ asom genomsnittlig nodgrad, n¨ atverksdiameter, klustring och effektivitet j¨ amf¨ ordes.
N¨ atverkets robusthet prioriterades d˚ a det starkt bidrar till n¨ atverkets drifts¨ akerhet vilket ocks˚ a ledde till att en decentraliserad modell anv¨ andes.
1
D3A: SMART INFRASTRUCTURES NETWORKS
Modelling Communication Networks for Active Traffic Management
Jens Fagerberg and Lucas Bergstrand
Abstract—Emissions, travel times and traffic congestions in urban areas have become a central issue with increasing urban- isation. By actively monitoring and managing traffic flow via traffic lights there can be extensive improvements in these areas.
To enable communication between infrastructure components an underlying network is needed to route information. These networks can be both centralised and decentralised which both have advantages and disadvantages. By studying and modifying existing network models, such as the Bar´abasi-Albert and Erd˝os- R´enyi, an algorithm that generate a robust and safe communi- cation network can be constructed, which has been the main goal of the project. To ensure that an optimal model is being used, robustness was analysed by comparing parameters such as average node degree, network diameter, clustering and effi- ciency. Network robustness was prioritised as it ensures network reliability and resilience and this led to the conclusion that a decentralised network model was suitable for the application.
I. I NTRODUCTION
With increasing urbanisation and population growth, society face extensive environmental changes and new problems arise that require innovative solutions. The concept of applying Internet of Things-technology to infrastructure is a rapidly growing field of study and market with an estimated annual revenue of $88.7 billion by year 2025 [1]. This technology may be used for autonomous vehicles, smart energy man- agement, home automation, traffic management and structural health monitoring to mention a few.
As the number of vehicles has been increasing since year 2000 [2] and by 2030, 60% of the world’s population are estimated to live in urban areas [3], it’s clear that congestion problems have to be resolved in order to decrease emissions, travelling times and economic loss. As of today, economic loss caused by traffic congestion, in the U.S alone, are ap- proximately $180 billion annually [4].
Changes to the road network are either extremely expensive or not possible. It has grown over time and countless road segments are independent projects being connected to each other. Therefore, solutions has to be developed with this in mind and adapt to the current infrastructure.
One solution to congestion is active traffic management. By enabling traffic lights to connect and exchange information with one another, traffic flows at junctions can be monitored and actively managed. Similar studies have initially shown that it’s possible to achieve a 26% reduction of travel times, 40%
idle time reduction and 21% reduction in emissions [5].
Communicating traffic lights provides a solution to enable increased efficiency for existing infrastructure. Providing an alternative solution, instead of large scale changes in infras-
tructure which requires large monetary investments and takes time.
Another use for this technology is for public transportation routing and urban planning. Data collected can not only be used to reroute bus lines in case of accidents or construction sites but also be used to evaluate future investments in infras- tructure.
This project has focused on the communication infrastruc- ture needed to connect traffic lights to each other. The network needed can be created and analysed with network science and models existing in this field of research. It has also provided a method of robustness and performance analysis.
II. N ETWORK S CIENCE
Network science is the study of networks. This involves communication networks, computer networks, biological net- works, social networks etc. Network Science provides a large number of tools for modelling, analysis and optimisation of a network.
Networks consist of nodes, connected by links. There are a number of properties to consider in a network. These properties and how they correspond to parameters of a com- munication network will be introduced in this section.
• Node Degree: The number of links associated with the node.
• Weight: A link can be assigned a weight which can represent i.e. transmission distance or cost. In our network model, all links are unweighed as distances are small enough to consider transmission times as instant.
• Adjacency Matrix: Matrix representation of links and nodes in the network. Elements of the square matrix indicate if two nodes are connected. If they are, the element is equal to the weight of the link otherwise the element is equal to zero. The adjacency matrix for the undirected network in 1 is
A =
0 1 0 1
1 0 1 1
1 1 0 1
0 1 1 0
(1)
• Degree Distribution: Degree distribution of a network is
defined as the distribution of probability that a random
node has the degree k. This is an important aspect, as the
degree distribution describes the network in an efficient
manner and also enables simple comparison between
networks. This aspect will be discussed further in section
VI.
D3A: SMART INFRASTRUCTURES NETWORKS
• Undirected and Directed Networks: As shown in figure 1, networks can be both directed and undirected. In com- parison to the directed network, its undirected counterpart enables two-way communication. A communication net- work is therefore in most cases undirected, as two-way communication is desirable. Using undirected graphs will reduce computing time as the adjacency matrix becomes symmetrical.
Fig. 1. Undirected (left) and directed network (right).
• Shortest Path: As there are not only one path between two given nodes. We are interested in the shortest path between the two nodes, shown in figure 2.
Fig. 2. Shortest path between node 0 and 5 in yellow.
• Network Diameter: The longest shortest path between any two nodes in a network is called the diameter, d max . This parameter determines how closely connected the network is, increasing L will generally decrease the diameter of the network.
• Connectedness: An important parameter of a network is connectedness. A network is connected if and only if all paths between two nodes are finite. If a network is not connected there are parts of the network which is inaccessible, meaning you cannot reach all nodes in a network from any given node. An unconnected network is not suitable for the purpose of communication.
• Clustering Coefficient: Clustering coefficient, C i , mea- sures the local link density around a node, i, in other words, how many of the nodes connected to node i are connected to each other.
C i = 2L i
k i (k i − 1) (2)
where L i denotes the number of links between the k i
neighbours of node i. A clustering coefficient of 1 corre- sponds to all local nodes being connected and a clustering
coefficient of 0 corresponds to all local nodes being disconnected, illustrated in figure 3.
Fig. 3. Two networks with C
0= 0 and C
6= 1
III. C ENTRALISED AND D ECENTRALISED N ETWORKS
An important aspect to consider in the design of a com- munication network is whether the network should be cen- tralised or decentralised. Two different approaches with dif- ferent strengths and weaknesses. It’s important to analyse and compare these two approaches to be able to choose the approach most suitable for the application.
A. Decentralised Network
In comparison to its centralised counterpart, which can be seen in figure 4, decentralised networks have no hubs. The degree distribution is more uniform, and messages are relayed from node to node until it reaches its final destination. A decentralised network is harder to maintain as there are no hubs and all nodes has to handle multiple connections and in- formation routing. The increased complexity and connectivity of these nodes result in a more robust system as information has multiple routes to the final destination.
Fig. 4. Decentralised network.
1) Erd˝os-R´enyi Model: The Erd˝os-R´enyi (ER) model was first introduced in 1959 [6]. This model is used to generate decentralised, random networks from a given number of nodes.
The most important aspect of the ER models is that the probability of connecting two nodes is uniform and hence, the degree distribution of the network is Poisson-distributed.
This is an important aspect to consider and will be discussed
further later in the article.
D3A: SMART INFRASTRUCTURES NETWORKS
The network G is defined with, G(N,p), where N is the number of nodes and p is the probability threshold. The network is generated by choosing two nodes and generating a random number between 0 and 1. If the random number is greater than p, a link is placed between the two nodes. This procedure is repeated for every possible link in the network.
This model inherits some important properties [7]:
• If p > (1+) ln N N the network will surely be connected, and vice versa.
• If N p < 1 the network will surely not have connected components larger than O(log(N ))
• If N p = 1 the network will surely have a component with maximum size of N
23.
• If N p > 1 the network will surely have a giant compo- nent.
These characteristics are of importance to the ER model as it allows the user, to some extent, to control network structure.
Decentralisation is the most important factor for our com- munication network. Downtime will increase congestions and eventually expose drivers to unnecessary risks, leading to potential economic loss and larger environmental pollution which is the direct opposite of what’s desired.
With these definitions in mind, it was decided that for this application, a decentralised approach is most suitable. This approach would provide a network with higher robustness and connectivity but also an infrastructure where an expensive data centre that needs maintenance and personnel is avoided, effectively reducing cost.
Choosing a decentralised network also simplifies the analy- sis of the network as they can be modelled with the ER model.
B. Centralised Network
In the centralised approach there’s a clear hierarchy. Mes- sages are communicated to a central node and then relayed to its final location. All communication passes this central node.
In the mathematical models these central nodes are hubs, with a lot of links to the smaller children. This approach with a central server will result an easier system to maintain, the hubs needs to be very sophisticated whilst the children are simpler and only need to gather data and relay to the hub, as seen in figure 5.
The hubs are the greatest strength and the biggest weakness of the centralised network. It provides a natural node for anal- ysis of gathered data and can be used to easily communicate with other nodes. Although if the hub is experiencing problems and seize to function the whole network can be crippled as there is no way for the nodes to communicate with each other.
A centralised approach would be suitable if a data centre is needed to perform large, real-time calculations. Something that could become necessary if big data analysis is implemented to prevent congestions before they occur or if public transport routes can be effectively changed whenever problems arise.
Although, these calculations can be done remotely, and control signals could be sent to the network of traffic lights.
1) Barab´asi-Albert Model: The Barab´asi-Albert (BA) model was developed by R´eka Albert and Albert-L´aszl´o Barab´asi during the late 1990s. Several real networks, such as
the World Wide Web, possess scale-free properties and growth by preferential attachment, which led to the development of this network model [8]. Most real network have power law degree distributions and not Poisson-distributions. The degree distribution of these networks can be described mathematically as
p k ∼ k −γ (3)
where γ denotes at which rate degree probability decrease.
Networks whose degree distribution follow a power law have scale-free properties. In these networks hubs exist, which are nodes with higher than average degree. The number of hubs are small, and the number of low degree nodes are high.
The BA model will produce networks with a power law degree distribution, where a small number of nodes has a large number of links and many nodes have very few links. This property is the main difference between the ER model and the BA model and will have an effect on how the communication network will be constructed, investigated further in section VI-A.
Fig. 5. Centralised network.
The method of generating a network using the BA model is more complex compared to generating a network with the ER model. As you start off with two nodes and connect the nodes.
Then you add a third node. This node will connect to either node 1 or node 2 or both. This is where the random aspect is present. For every new node, the probability for the node to connect to another node is proportional to the number of links of the other node. A node with twice the number of links will have twice the probability to be connected. The process is repeated until the desired number of nodes is reached.
With this method of generating the network, hubs will be formed as the probability is weighed by the degree of the node.
Highest degree nodes have higher probability of connecting.
This method is called preferential attachment.
D3A: SMART INFRASTRUCTURES NETWORKS
IV. R EQUIREMENTS
As traffic flows and the probability for congestion heavily varies during a 24-hour period, it is clear that situations like morning and afternoon traffic define what requirements the network has to satisfy. If information is transmitted at a too low frequency, congestions will still occur, and desired func- tionality wouldn’t be reached. Consequently, if information is transmitted at a too high frequency, network capacity will be affected resulting in data loss or unnecessary hardware costs.
A. Technical Specification
Every traffic light, seen as a node in the network, has to communicate certain information at a certain frequency.
During i.e. midday and at night when very little congestion occurs, there is small or no need for active control and therefore the system could be in standby. When the system is actively managing traffic, we’ve estimated that each node should transmit the following information every 15 seconds.
• Current state: Green or red signal.
• Consumed time in current state.
• Any errors detected. If the traffic light is out of order, then other rules at that intersection (signs etc.) should be applied.
• Current traffic flow by evaluating sensor data.
This will be a data packet. The amount of information might be trivial, but doing appropriate optimisation from the beginning ensures that the network is allowed to expand, and hardware costs can be reduced.
B. Economic Specification
As current infrastructure has to be replaced or modified, large investments are needed and small reductions in cost for each unit lead to large decreases in the overall project cost.
Therefore, when constructing the network, the economical aspect should be considered. Cost can be reduced by min- imising hardware requirements. This can be done by reducing the amount of data transmitted, minimise connections to other units and for wireless systems, reduce transmission range.
V. M ETHOD
A. Software
Algorithms have been programmed in Python in which the library NetworkX has been utilized for network construction and analysis. NetworkX provides a framework to construct graphs. It also provides methods for calculating parameters, degree distributions and creating graphical representations of networks.
NetworkX, and the authors personal preferences were the main reasons why Python was chosen as programming lan- guage over MATLAB. The documentation for NetworkX [9]
has been thoroughly studied.
B. Network Models
As networks appear in technological, biological and social applications there is, as a result of this, various models that quantifies different characteristics of these networks. Intu- itively, we do not expect any similarities between a social network and a communication network, but this is not the case.
Networks in different applications exhibits similar behaviour.
We’ll investigate if some of these models may be suitable for generating the network that connects traffic lights.
1) Barab´asi-Albert Model: Introduced in section III-B1, this model generates a random network with scale-free proper- ties, where a preferential attachment-mechanism is used, and the node connectivity is chosen beforehand.
Preferential attachment is a process which propose that nodes with high degree has an increased probability to connect to new nodes in the network.
The probability for a newly added node to connect to another node i is calculated by dividing the degree of node i by the sum of degrees of pre-existing nodes, as shown in equation 4.
p i = k i
P
j k j (4)
which states that nodes with higher degrees achieve a higher probability.
As the network possess the scale-free property the degree distribution follows a power law. Average path length, l, increases proportionally to the network size
l ≈ ln N
N ln N (5)
2) Unit Disc Model (UD): Proposed in 1995 [10], this model aims to better model the behaviour of wireless transceivers. It is a popular model in which the transmission range of each node is taken into consideration. One flaw of this model is that it assumes that every transceiver has a perfectly circular broadcast range, which in reality isn’t true and affected by antenna construction, antenna placement and physical obstructions.
The probability for two nodes to connect in this model are based on the Euclidean distance
p i,j = |x i − x j |< d t (6) between the two nodes. This distance has to be lower than a threshold distance, which is defined by the transmission ranges of the nodes.
3) Adjacent Nodes Model (AN): The project group at- tempted to develop their own network model in order to gain understanding about the complexity of networks and their generating algorithms. The base idea is to connect one node to a number of nodes in its proximity. This value can be changed together with a maximum distance between two nodes. Com- pared to the Unit Disc and BA model, this takes both node connectivity and maximum distance in consideration.
When the algorithm generates the network, it will connect to
the adjacent nodes until it reaches either the maximum distance
or maximum connections thresholds.
D3A: SMART INFRASTRUCTURES NETWORKS
C. Geographical Data
Fig. 6. Traffic lights in Stockholm.
In order to generate a communication network for traffic lights in Stockholm, information regarding their positioning throughout the town is needed. Data was acquired from Stockholm Municipality. GPS-position provide not only valu- able information regarding network approach but also the geographical constraints needed if wireless communication is being considered. In Appendix A an extraction of the geographical data is shown, where the coordinates are in RT90 and SWEREF format.
D. Network Generation
To compare and analyse networks it is needed to program algorithms that generate these. This has been done with Python and NetworkX. The procedure to generate ER and AB networks has been discussed in section II.
As it has been decided to have decentralised approach, the model needs to reflect this approach, while using the geographical data gathered. It’s theorised a traffic light will affect nearby traffic lights more than distant ones. Therefore, the model has to be complemented to take into account the geographical location of the traffic lights.
In order to satisfy the need for larger connectivity in closer proximity and having a decentralised approach to the network, the authors propose a modification of the ER model.
Instead of having a completely random network with a uniform probability, the probability is weight with respect to distance between the two nodes. The probability should be proportional to the inverse of the distance, r ij . Thus, the probability can be described as
p ij = C i
r ij (7)
Where C i is a scale coefficient. The scale coefficient is different for every node i and it defined as follows,
C i = 1
P N j=1
1 r
ij(8)
Another version of this model has also been considered, when the probability is proportional to the inverse of the distance squared. These models will be referred to as the R and R2 model respectively.
VI. A NALYSIS AND OPTIMISATION
In this section we analyse network parameters in order to determine the properties of the networks and thus determine which models are suitable for connecting traffic lights.
A. Degree Distribution
The degree distribution is a statistical presentation of all node degrees in a network. It’s an intuitive representation and contains information about the network structure. This measure has been mentioned in section II and in this section, histograms will be presented for each network type.
1) Erd˝os-R´enyi Network: In the ER model, the event that pairs of nodes connect are independent from each other, therefore, this model yields a network, that for large N, are Poisson-distributed. The structure of this network can be modified by changing the probability for two nodes to connect, which has been explained in section II.
Fig. 7. Degree distribution for Erd˝os-R´enyi model.
In figure 7 we can observe the typical appearance of a
Poisson-distribution with a mean of approximately 30 con-
nection, lowest node degree around 14 and the highest degree
around 44.
D3A: SMART INFRASTRUCTURES NETWORKS
Fig. 8. Degree distribution for Bar´abasi-Albert model.
2) Bar´abasi-Albert Network: In comparison to the ER model, this model captures phenomenons observed in real- world network. The degree distribution follows a power- law, indicating the existence of hubs caused by preferential attachment.
Figure 8 represents the degree distribution of BA network with edge connectivity of five. We can see that the The exponential reduction in node degrees indicate that we have a few hubs of different sizes, and that our network probably is split into several components.
3) Unit Disc Model: The unit disc models yields a degree distribution that share similarities with the degree distribution for the Erd˝os-R´enyi network.
Fig. 9. Degree distribution for Unit Disc model.
It can be observed in figure 9 that the mean node degree is around 8. Not taking preferential attachment into consideration in our model might be one of the reasons why we don’t observe a degree distribution following a power law.
B. Network resilience
Data transmitted across the network has to be protected in order to ensure the integrity of the network as well as safety of drivers, pedestrians and vehicles. Unauthorized users getting access to control signals could have catastrophic consequences such as traffic accidents and road blocks which obstruct emergency vehicles.
The information can be protected by using data encryption and firewalls, but also by isolating the network to communicate on a separate frequency or with a private protocol rather than public options like 802.11 WLAN and 4G.
Ensuring that a network can withstand malicious attacks are of great importance but is beyond the scope of this project.
Although, robustness analysis do affect network resilience, which is discussed in the next section.
C. Robustness analysis
Robustness states how well a network can ensure function- ality when nodes, link or other parts of the network break down which obstructs data transmission. Networks with high robustness will avoid cascading failures when equipment fail at one location by having alternative paths that data can be transmitted through.
There are several important parameters that’ll be considered when analysing robustness of a network [11].
1) Average shortest path: There can be multiple paths between a pair of nodes. Although, the shortest path denotes how many nodes the information has to pass to reach its destination. The average shortest path distance across the network indicates how dependent nodes are of each other.
If the average distance is 1, then it can be concluded that all nodes directly connected to each other. Therefore, a lower value indicates higher robustness.
˜
a = 1
N (N − 1) X
i,j∈G
d i,j (9)
Where d i,j is the shortest distance between node i & j and n is the total number of nodes in the network G.
2) Edge connectivity: The number of edges that can be removed prior to the networking becoming disconnected. A network is robust if it stays connected even though edges are being removed, hence, a large edge connectivity indicates the network is robust.
3) Average clustering coefficient: The local clustering coef- ficient of a node is calculated as the proportion between links in its neighbourhood divided by the number of possible links.
hCi = 1 N
N
X
i=1
C i (10)
C i is the local clustering coefficient, defined in section II.
A higher average clustering coefficient means more links
between neighbouring nodes and therefore the network is more
robust.
D3A: SMART INFRASTRUCTURES NETWORKS
4) Network efficiency: Network efficiency is based on the idea that information between two nodes travels along the shortest path. The efficiency of this communication is defined as the inverse of the shortest distance. The average efficiency across the network measures the mean flow-rate of information [12].
E = 1
N (N − 1) X
i6=j∈G
1
d i,j (11)
As the efficiency is proportional to the sum of inverse of the distance, a high efficiency indicates higher robustness.
5) Algebraic connectivity: It has been proven [13] that al- gebraic connectivity is defined as the second lowest eigenvalue of the Laplacian, which has been described in previous section.
The magnitude of this is a measure of how well the network is connected and is zero only if and only if the graph is disconnected. Considering this, a higher value result in higher robustness.
6) Number of spanning trees: A spanning tree is defined as a graph which includes all available nodes without any cycles.
In order for a spanning tree to exist in a network, it has to be connected and therefore a higher number of spanning trees indicate a more robust network. Kirchoff’s Theorem states that the number of spanning trees in graph G can be calculated as:
t(G) = 1 N
N
Y
i=2
λ i (12)
Where λ i are the non-zero laplacian eigenvalues.
7) Effective graph resistance: By looking at the network as an electrical circuit, all links can be considered (for an unweighed network) having resistances of 1 Ohm. The ef- fective resistance corresponds to the total resistance over the circuit when applying a current over these series- and parallel connections. Once again, the non-zero laplacian eigenvalues can be used to calculate this measure [14]:
R e = N
N
X
i=2
1 λ i
(13) As in an electrical circuit, a lower resistance is desired and indicates higher robustness.
8) Percolation limit: Assuming that node removal occurs randomly, the percolation limit is calculated as [15]:
P c = 1 − 1
hx
2i
hxi − 1 (14)
where hxi and hx 2 i is the average node degree and the variance respectively. The percolation limit yields the critical threshold of how many nodes that can be removed before disconnecting the network. As of this, a higher value indicates higher robustness.
All parameters explained in this section has been collected for following networks:
• Barab´asi-Alb´ert Network with different connectivity.
• Network generated with unit-disc model.
• Network generated with regard to 5 closest nodes.
These parameters for networks generated with the Barab´asi- Alb´ert model with different connectivity are presented in table I.
TABLE I
R
OBUSTNESS PARAMETERSBA
MODELNetwork generated with BA model
Edge Connectivity 2 3 4 5 6
Average Distance 3.86 3.32 3.00 2.83 2.71
Efficiency 0.29 0.32 0.35 0.37 0.39
Average Clustering 0.040 0.040 0.044 0.054 0.061 Algebraic Connectivity 0.53 1.18 2.11 2.93 3.79 No. of Spanning Trees 3.99 5.97 7.95 9.92 11.88
Eff. Graph Resistance 170e3 98e3 69e3 53e3 43e3 Percolation limit 0.89 0.91 0.93 0.94 0.95
It can be observed from the data in table I, that the Barab´asi-Alb´ert network model, in which desired connectivity is selectable, robustness increases as we increase the connec- tivity. All parameters increase in magnitude except for average distance and graph resistance which decrease with increasing robustness.
TABLE II
R
OBUSTNESS PARAMETERS RANDOM AND UNIT-
DISC MODELR model R2 model AN model
Edge Connectivity 1 1 1
Average Distance 3.10 3.95 11.55
Efficiency 0.346 0.277 0.126
Average Clustering 0.03 0.11 0.13
Algebraic Connectivity 0.551 0.301 0.004 No. of Spanning Trees 9.61 6.70 4.40
Eff. Graph Resistance 49e3 84e3 375e3
Percolation limit 0.90 0.85 0.74
In table II robustness parameters are presented for the R, R2 and unit-disc model. All three networks have an edge connectivity of 1, which is caused by that two clusters are connected through 1 link. This can be altered with to increase robustness. But as mentioned earlier if the distance is large between nodes the effect is decreased. Thus this might not be an issue, but analysing this is beyond the scope of this project.
Comparing the AN model to the R and R2 models, it can be concluded that the 5 closest model is inferior. Seen in table II the effective graph resistance is a factor 10 larger than the R and R2 models. The average distance is four times larger for the 5 closest model and algebraic connectivity is a factor 100 smaller. With respect to these aspects we discard the AN model in favour of the R and R2 models.
The R and R2 models, provides a good alternative to the the Barab´asi-Alb´ert model with the exception for edge connectivity. When the probability of a node establishing a link is reduced in the R2 model, average clustering increase, increasing robustness, but remaining parameters indicate that the network generated with the R2 model is less robust compared to the R model.
As a modern communication network can handle large
information of data with low hardware requirements and
D3A: SMART INFRASTRUCTURES NETWORKS
transmission speeds being very high. We conclude that a modified ER model with a probability proportional to the inverse of the distance between the nodes is suitable for a decentralised network approach. The R model is from a robustness perspective better compared to the R2 model, although both models can be used.
VII. R ESULTS
The network study performed and presented in this re- port verifies that random models, even after modification, is suitable for the application of connecting traffic lights.
By developing a new model and comparing to known ones, such as Erd˝os-R´enyi and Bar´abasi-Albert, characteristics and performance was able to be estimated.
Three models were developed and tested, being mentioned as ”Adjacent Nodes (AN)”, ”R” and ”R2”, where R and R2 are the same model but with probability of connecting reducing with linearly and quadratically. The AN-model were discarded as it didn’t provide satisfactory results in either performance or robustness. The other two were based on the Erd˝os-R´enyi and Unit Disc models.
This proved to be a well functioning model in regard to robustness with e.g. low resistance and high percolation limit.
As a decentralised approach were chosen, it was decided to modify the ER model as the random model doesn’t create any hubs. Consequently, the network do not follow a power- law degree distribution which is common for real networks and provides an almost constant average path length when the network is growing.
The model needs improvement as edge connectivity, as seen in table II, is only 1. This is related to nodes having a maximum number of connections and the order in which nodes are connected to the network. If all adjacent nodes within the transmission range of a node i has reached this limit, it will not be able to connect to something. Also, if network growth is of importance, it might be a good idea to do modifications to the BA model instead of the ER, which was done here.
Fig. 10. Network generated with ”R”-model.
VIII. F UTURE W ORK
As network structure is the foundation to enable communi- cation between traffic lights, further work is required before implementation. In terms of network analysis, the algorithm can be further developed in regard to network growth and by modifying the Bar´basi-albert model instead of Erd˝os-R´enyi model. One important improvement to the chosen algorithm is to increase edge connectivity.
A communication protocol suitable for this application has to be chosen and evaluated through simulation, which is also needed to analyse network performance. The project group researched and planned to perform simulation in OmNet++, but this test was discarded as it is beyond the scope of this project. When simulation has been conducted one might look into hardware solutions and investigate the economic cost of this project.
IX. C ONCLUSION
The goal of this project was to model and analyse a network model for traffic lights in an urban environment with the long- term goal of active traffic management. This has been done by researching network models and implementing appropriate models with gathered geographical data of traffic lights in Stockholm, Sweden.
The authors decided, after having evaluated both a cen- tralised and decentralised approach, that a decentralised ap- proach is better with the main argument of higher network robustness even though economical costs will be larger. In order to compare results a centralised model was analysed.
The models used was the Barab´asi-Alb´ert model, Erd˝os- R´enyi model and models based on the Erd˝os-R´enyi model , R and R2 models and a model connecting the 5 closest traffic lights. The R, R2 and AN models was introduced to take into consideration the geographical location of the traffic lights. As the authors theorised that nodes in close proximity will communicate more frequently and the effect of traffic management diminish as distance between nodes increases.
Analysis of the different models yielded results resulting in the AN being discarded as the performance and robustness was inferior to the R and R2 models. Comparing the R and R2 models, with data from II, the performance and robustness are similar but as average distance and effective graph resistance is larger and algebraic connectivity and efficiency is lower.
The R model is superior.
The Barab´asi-Alb´ert model produced similar results to the R model. Acting as a confirmation that the R model is applicable for the application. As the Barab´asi-Alb´ert model models a centralised network, the R model provides an alternative for a decentralised approach.
It is concluded that using a random model which takes geographical data into consideration is suitable for use in traffic light communication with a decentralised approach.
Further research is required to implement such a network.
A CKNOWLEDGMENT
The authors would like to thank their supervisor Lamia
Varawala for her guidance, support and swift answers to
questions during the project.
D3A: SMART INFRASTRUCTURES NETWORKS
A PPENDIX
E XTRACT OF T RAFFIC L IGHT G EOGRAPHICAL D ATA
NTS_ID Namn RT90_X RT90_Y Extra_In Sweref_N Sweref_E 160038 Lidingovagen - Fortifikationsforvaltningen6582968 1629888 RT90 6582815 675602 160039 Karlbergsvagen - Dalagatan6582374 1627409 RT90 6582191 673131 160040 Odengatan - Sveavagen6582525 1628031 RT90 6582349 673751 160041 Karlbergsvagen - Vastmannagatan6582386 1627522 RT90 6582204 673244 160042 Karlbergsvagen - Upplandsgatan6582400 1627619 RT90 6582219 673341 160043 Odengatan - Upplandsgatan6582320 1627660 RT90 6582140 673383 160044 Norrtullsgatan - Vanadisvagen6582827 1627541 RT90 6582645 673257 160045 Tegnergatan - Dalagatan6581732 1627766 RT90 6581553 673496 160046 Dalagatan - Radmansgatan6581841 1627704 RT90 6581661 673432 160047 Odengatan - Dobelnsgatan6582595 1628166 RT90 6582421 673885 160048 Odengatan - Roslagsgatan6582688 1628343 RT90 6582516 674061 160049 Roslagsgatan - Surbrunnsgatan6582771 1628293 RT90 6582598 674010 160050 Odengatan - Birger Jarlsgatan6582735 1628430 RT90 6582564 674147 160051 Birger Jarlsgatan - Surbrunnsgatan6582817 1628385 RT90 6582645 674101 160052 Odengatan - Tulegatan6582642 1628258 RT90 6582469 673977 160053 Sveavagen - Surbrunnsgatan6582605 1627996 RT90 6582429 673715 160054 Sveavagen - Frejgatan6582783 1627899 RT90 6582605 673616 160055 Frejgatan vid Tulegatan6582904 1628117 RT90 6582729 673832 160056 Valhallavagen - Roslagstull6583342 1628022 RT90 6583166 673732 160057 Valhallavag - Ingemarsgatan - Birger Jarlsgatan6583147 1628201 RT90 6582973 673913 160058 Sveaplan 6583187 1627685 RT90 6583007 673397 160059 Sveavagen - Vanadisvagen6582963 1627798 RT90 6582784 673513 160060 Norrtull 6583186 1627383 RT90 6583002 673095 160061 Norra Lanken - Uppsalavagen6583318 1627302 RT90 6583133 673013 160062 Norra Stationsgatan - S:t Eriksgatan6583045 1627213 RT90 6582859 672927 160063 Valhallavagen - Lidingovagen6582537 1629374 RT90 6582377 675093 160064 Valhallavagen - Engelbrektsgatan6582674 1629049 RT90 6582510 674767 160065 Valhallavag - Danderydsg. - Dr Kristinas Vag6582745 1628951 RT90 6582580 674668 160066 Valhallavagen - Uggelviksgatan6582730 1628842 RT90 6582564 674559 160067 Valhallavagen - Odengatan6582868 1628676 RT90 6582700 674392 160068 Valhallavagen - Frejgatan6583057 1628406 RT90 6582886 674119 160069 Birger Jarlsgatan - Frejgatan6582993 1628281 RT90 6582820 673995 160070 Frejgatan - Roslagsgatan6582948 1628200 RT90 6582774 673915 160071 Valhallavagen vid Surbrunnsgatan6582940 1628585 RT90 6582771 674300 160072 Valhallavagen - Sibyllegatan6582343 1629705 RT90 6582187 675427 160073 Valhallavagen - Artillerigatan6582323 1629832 RT90 6582169 675554 160074 Valhallavagen - Vartavagen6582137 1630254 RT90 6581988 675978 160075 Valhallavagen - Banérgatan6582089 1630391 RT90 6581942 676116 160076 Valhallavagen vid Gyllenstiernsgatan N & S6581976 1630675 RT90 6581832 676401 160077 Karlavagen - Oxenstiernsgatan6581663 1630641 RT90 6581519 676371 160078 Valhallavagen - Erik Dahlbergsgatan6582240 1630034 RT90 6582089 675757 160079 Sveavagen - Kungsgatan - Apelbergsgatan6581652 1628490 RT90 6581482 674220 160080 Sveavagen - Olof Palmes Gata6581692 1628472 RT90 6581522 674202 160081 Sveavagen - Adolf Fredriks Kyrkogata6581821 1628404 RT90 6581650 674132 160082 Sveavagen - Tegnergatan6582025 1628297 RT90 6581853 674023
