Graph Theory in Communication Network Analysis
Maghy S J
1*and Robinson Chelladurai
21*
Research Scholar, Department of Mathematics,
Scott Christian College (Autonomous), Nagercoil-629 003, Tamil Nadu, INDIA.
2
Associate Professor, Department of Mathematics,
Scott Christian College (Autonomous), Nagercoil-629 003,Tamil Nadu, INDIA.
Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627 012, Tamil Nadu, INDIA.
email: [email protected], [email protected]
(Received on: December 8, 2018)
ABSTRACT
Graph theory plays vital role in various networks and the graph theoretical ideas can be used in different networks such as communication network, electrical network, transportation network etc. Network analysis is an essential tool used in various sectors such as transportation, electrical, communication and also for the flow of matter and energy. Moreover choosing efficient route is essential for businesses and industries which aid distribution of goods and services optimally. This paper addresses how the concept of graph labeling, graph coloring and Dikstra’s algorithm can be applied to communication networks.
AMS Subject Classification: 68R10.
Keywords: Communication network, Dijkstra’s algorithm, Graph labeling, Graph coloring.
1. INTRODUCTION
Graph theory can be used to represent communication networks and it is a collection of links, nodes and terminals, in order to connect a communication between different users.
Each terminal has a suitable address so that messages can easily reach the recipients. A
communication network is composed of nodes, each of which has computing power and can
transmit and receive messages over communication links, wireless or cabled .Address space is nothing but the collection of addresses in the network and communication network is to transmit packet of data between computers, telephones, processors or other devices. Graphs play an important role in communication networks.
Based on mathematics the study of graphs is called graph theory and it is a fundamental structures used to generate a model pair relation between objects. Normally a graph is made up of vertices, nodes or points and these are connected by edges, lines or arc. If a graph is undirected then it is shown that there is no distinction between two vertices linked within the each edge or edges may exist from one vertex to another.
Definition 1.1.A graph G is defined as 𝐺 = (𝑉, 𝐸) V is the set of all vertices and E is set of all edges in the graph.
(a) Example 1 (b) Example 2 Figure 2. Examples of graph
Definition 1.2. G is called a labeled graph if each edge e=UV is given the value f (UV) = f (u)*f (v), where * is a binary operation such as addition, multiplication, modulo addition or absolute difference, modulo subtraction or symmetric difference
Definition 1.3. Graph coloring is assigning colors to the vertices or edges of graph so that no two adjacent vertices or edges will have same color. Graph coloring should be done using minimum number of colors called the chromatic number. Graph colored using minimum number of colors is called properly colored graph. Graph coloring technique is a special case for graph labeling and it is assigned for labels, traditionally called colors in order to subject to the certain constraints. While graph coloring, the constraints that are set on the graph are colors, order of coloring, the way of assigning color, etc.
Definition 1.4 A radio labeling or multilevel distance labeling for a graph G is an injective function f: V (G) →N union {0} such that for any vertices u and v, |f (u) - f(v)| >= diam (G) - d (u, v) +1.
2. GRAPH COLORING IN GSM MOBILE PHONE NETWORK
In Mobile phone network like Groups Special Mobile (GSM) the geographical area of
the network is divided into hexagonal regions called cells. Mobile phones within the cell will
be connected to communication tower that exist in that cell. Mobile phones connect to the GSM network by searching for cells in the neighbors. Four color theory is used in GSM Since it operates only in four different frequency ranges. The cellular regions are properly colored using four colors. The vertex coloring algorithm can also be used to assign at most four different frequencies for any GSM mobile phone network.
Map drawn on the plane uses a four color theorem to color the regions of a map, properly using at most four distinct colors such that no two adjacent regions are assigned the same color. For the given map we construct the dual graph. The dual graph is constructed by placing vertex inside each region of a map. The vertex inside two regions will be connected by an edge if those two regions have a common edge forming the boundary. So the four color theory can be used in GSM network where four colors are the four frequency ranges in which the network operates. It is always possible that if a given map can be drawn on the plane it can be color properly using atmost four distinct colors such that no two adjacent regions are assigned the same color.
3. GRAPHS IN MOBILE AD HOC NETWORK (MANET)
A Mobile Ad hoc Network (MANET) is a dynamically changing infrastructure less and resource-constrained network of wireless nodes that may move arbitrarily, independent of each other. The transmission range of the wireless nodes is often limited, necessitating multi- hop routing to be a common phenomenon for communication between any two nodes in a MANET. Various routing protocols for unicast, multicast, multi-path and broadcast communication have been proposed for MANETs. The communication structures that are often determined include: a path for unicast – single-path and multi-path routing, a tree for multicast routing and a connected dominating set – CDS for broadcast routing. Within a particular class, it is almost impossible to find a single routing protocol that yields an optimal communication structure with respect to different route selection metrics and operating conditions.
Figure 3. Mobile Ad Hoc Network
4. GRAPH LABELING IN COMMUNICATION NETWORKS
In a communication network, it might be useful to assign a node number to each user terminal subject to the constraint that all connecting edges that means the communication links
Figure 3. Mobile Ad Hoc Ne
receive distinct numbers. In this way, the numbers of any two communicating terminals automatically specify the link number of the connecting path and the path number uniquely specifies the pair of user terminals which it interconnects. For the fast communication in sensor networks radio labeling can be used. Given a set of transmitters, each station is assigned a channel by a positive integer such that interference can be avoided. The smaller the distance between stations then stronger the interference becomes, and hence the difference in channel assignment has to be greater. Here each vertex represents a transmitter and any pair of vertices connected through an edge corresponds to neighboring transmitters. Here the kind of labeling used is Radio labeling. The maximum distance between any pair of vertices is called the diameter of G denoted by diam (G).
5. SHORTEST PATH ALGORITHM IN NETWORK ANALYSIS
In this part, we considered the problem of routing in communication networks. The most used approach for solving this problem is shortest paths. If the network is represented by a weighted graph, where vertices model nodes of the network, edges model links connecting such nodes and the weight of an edge models the time required by packets for traversing the corresponding link, the problem can be solved by the distributed computation of all-pairs of shortest paths.
Several algorithms are implemented to solve the problem in the network. The shortest path algorithm is classified into three types. They are
To find the direct path from source node to all nodes around the graph, find the shortest path from single source.
Problem occur in finding the direct path to a single source and it aim is to find the shortest path between each connected pair present in the graph.
Problem occurs by finding the shortest path between two nodes and also finds the shortest path between connected pair in the graph.
In network area the shortest path problem is mainly concentrate to calculate the path from source to destination with minimum weight. The edge length of the network represents the real life quantities such as time, cost, etc and in classical shortest problem different parameters such as time, distance between nodes are to be assumed. For the particular reason several algorithm is used to find the shortest path between the nodes and here Dijkstra’s algorithm is the best method to find the shortest path between source nodes to another node in the graph.
6. DIJKSTRA’S ALGORITHM
Dijkstra’s algorithm is the shortest path algorithm and it is similar to prim’s algorithm.
It finds the shortest path from one vertex to the remaining vertices and solves the shortest path
problem in directed graph. The main characteristics of the algorithm is starting point extends
to the outer layer at the starting point to the end. Assume 𝐺 = (𝑉, 𝐸) is the weighted directed
graph and the vertex V is divided into two set of groups. In first group the set of vertices for
shortest path is obtained and it is denoted by S. For this group only one source point is present at the beginning of S and for each vertices the shortest path is obtained, it vertices is added to S until all the remaining vertices are added to S. The second set contains the set of vertices and it is denoted by U for the remaining shortest path. The vertices present in the second group in order of the shortest path length are added to S and in this procedure the shortest path length of the vertices from source point v to S is less compared to shortest path length of the vertices from source point v to U. The vertex v to the source point and the distance from the vertex in U, the vertex in S acts as the middle vertex. Figure 4 shows the flow chart of Dijiksra’s algorithm.
If 𝐺 = (𝑉, 𝐸, 𝑊) is the weighted directed graph and the vertex
= (𝑣1, 𝑣2, … . . 𝑣𝑛), (𝑖, 𝑗) ≤ 𝑛. E is the edge from 𝑣
𝑖to 𝑣
𝑗and 𝑊(𝐸) is the weight of E, D is the shortest path from 𝑣
𝑖to 𝑣
𝑗.Vertex set V is divided into two set of groups ,one set of group S is the shortest path it is calculated and other set of group is opposite of S.S has only one source at the beginning(16).
Calculate the shortest path 𝐷(𝑣
𝑖𝑛) from𝑣
1to 𝑣
𝑛.The following are the steps for Dijkstra’s algorithm.
Step 1: Generate distance matrix
Figure 4. Flow chart of Dijkstra’s algorithm
Let,𝑆 = {𝑣
𝑖}, i=1,𝑆 = {𝑉
2, 𝑉
3, … . . 𝑉
𝑁} and let { 𝑊(𝑣
1) = 0
𝑇(𝑣
𝑗) = ∞ 𝑣
𝑗𝜖𝑠 (1) From equation 1 calculate the distance matrix
Step 2: Calculate𝑇(𝑣
𝑗), 𝑣
𝑗𝜖𝑠
𝑇(𝑣
𝑗) = 𝑚𝑖𝑛{𝑇(𝑣
𝑗), 𝑊(𝑣
𝑖) + 𝑤
𝑖𝑗} (2) 𝑇(𝑣
𝑗) is the distance from 𝑣
𝑖to 𝑣
𝑗,𝑗 = (2,3, … 𝑛)
Step 3: Determine the shortest path from 𝑣
𝑖to 𝑣
𝑗, 𝑣
𝑗𝜖𝑠.
𝑊(𝑣
𝑘) = 𝑚𝑖𝑛{𝑇(𝑣
𝑗)} (3) 𝑊(𝑣
𝑘) is the shortest from 𝑣
1to𝑣
𝑗.
Step 4: Determine the shortest path 𝐷(𝑣
𝑖𝑛)
If𝑣
𝑘= 𝑣
𝑛, then the shortest path𝐷(𝑣
𝑖𝑛) = 𝑊(𝑣
𝑘).
If 𝑣
𝑘≠ 𝑣
𝑛, then let 𝑘 = 𝑖 and delete 𝑘 from𝑠. Put 𝑣
𝑘in step 2 for the iteration until 𝑣
𝑘= 𝑣
𝑛. By using this method finds the shortest path𝐷(𝑣
𝑖𝑛). Figure 5 shows the set V network.
Figure 5. Set V network
For example vertex set 𝑉 = (𝑣
1, 𝑣
2, 𝑣
3, 𝑣
4, 𝑣
5, 𝑣
6),from 𝑣
1to 𝑣
6calculate the shortest path 𝐷(𝑣
16). In figure 1 the weight of the path is marked.
Table 2. Path distance matrix
Vertex V1 V2 V3 V4 V5 V6
V1 0 4 1 ∞ ∞ ∞
V2 4 0 1 4 8 ∞
V3 1 1 0 7 9 ∞
V4 ∞ 4 7 0 1 4
V5 ∞ 8 9 1 0 1
V6 ∞ ∞ ∞ 4 1 0
Calculate the first time iteration, 𝑗 = 2,3,4,5,6.
Figure 5.Set V network
