INTRODUCTION:
Network analysis is one of the applications of graph-theoretic principles to study the relationship between components in a complex dynamic system8. It gives a procedure for further investigating the structure of the system. The devices or components in a network can do similar assignments simultaneously and the other components on its path have to pay observation to it. This may cause network efficiency. In order to discuss this phenomenon, we usually use the term collision domains2. It is a network system connected to different components through different paths in the network for which a simultaneous data transmission collides with one another. This may allow increasing the latency of the network. So, the network execution can be remarkably intensified by detecting the component collision domain.
A network can be exhibited as a graph in which nodes represent the devices and edges represents the connection between them. Information gives the relative importance of nodes and edges in a graph can be done through graphical measures such as centrality, average path length, closeness etc.
The objective of this paper generally determines a class of graphical measures to be used so as to determine the relationship between the components in the network. This paper also discusses a computational perspective to concede the likelihood significant difference between fast and slow progressors of diabetic patients3.
Basic Concepts:
A graph G = (V(G), E(G) consists of two finite
sets V(G), the vertex set of the graph, often denoted by just V,
which is a non -empty set of elements called vertices, and E(G) the edge set of
the graph, often denoted by 
, which is possibly empty set of elements
called edges, such that each edge in 
in is assigned an unordered pair of vertices(u,v)
called the end vertices of e.
Adjacent vertices are two vertices that are joined by an edge are said to be adjacent or neighbors.
The degree d(v) of a vertex v is the
number of edges of graph G incident with v counting loop twice,
that is the number of times 
is an end vertex of an edge4.
Graphical Measures:
Average path length: Let G be a connected graph
with vertex set 
, and let d(v) denote the
averagelength of the shortest path from vertex u to any other vertex v
in G and is defined as
The average path length 
is defined as
Closeness5: Consider a (strongly) connected
graph 
. The closeness 
of a vertex 
is defined as
.
Example of the above definition:
Figure 1: A simple connected graph
From the figure 1 we can calculate the eccentricity, average path length and closeness
Eccentricity of some vertex: e (1) = 3, e (2) =3, e (4) =2
Average path length of the vertex: d (1) =1.66, d (2) =1.833, d (3) = 1.833, d(4)=1.5, d(5)=1.66, d(6)=1.66
d (7) =1.833.
Average path length of graph d(G) = 1.711
Closeness of vertex: Cc(1)=0.60, Cc(2)=0.545, Cc(3)=0.545, Cc(4)=0.666, Cc(5)=0.602, Cc (5)=0.602
Cc(6)=0.602, Cc(7)=0.545.
Collision Domain in a Network:
A collision domain is a network segment connected by a shared medium or through repeaters where simultaneous data transmissions collide with one another. Using graphical measures, we can analyse the most important device in the network6,7. Most of all the routes may pass through this particular device and thus one or more devices may collide with each other in its path. The following table represents the average path length and closeness of all the devices.
Table 1: Table Plotted by Taking Network Diagram to Calculate Their Average Length of Shortest Path
|
Device |
Average Length of Shortest Path |
Closeness |
|
Switch 1 |
3.66 |
0.273 |
|
Switch 2 |
2.6 |
0.384 |
|
Switch 3 |
3.266 |
0.306 |
|
Bridge |
2.6 |
0.384 |
|
Router |
2.2 |
0.454 |
|
Hub 1 |
3.4 |
0.294 |
|
Hub 2 |
3.866 |
0.258 |
|
Hub 3 |
2.6 |
0.384 |
|
Hub 4 |
3.4 |
0.294 |
|
Hub 5 |
3.33 |
0.300 |
|
PC-1 |
4.33 |
0.230 |
|
PC-2 |
4.33 |
0.230 |
|
PC-3 |
4.33 |
0.230 |
|
PC-4 |
4.33 |
0.230 |
|
PC-5 |
4.33 |
0.230 |
|
PC-6 |
4.33 |
0.230 |
OBSERVATION:
The devices can be taken as nodes in the network. As per Figure 1, the network has no cycles and is connected, each of the PCsis at the same length from the router and has the same average shortest path length. Also, from the table, it is clear that whenever the average length increases, the closeness of each device decreases.
Figure 2: Collision domains
The average path length differentiates an easily accessible network from one, which is complex and unproductive, with a shorter average path length being more desirable. However, the average path length is simply what the path length will mostly likely be. The network itself might have some very remotely linked nodes and many nodes, which are neighbours of each other.
The graph given below explains the relation between average shortest path lengths and closeness about the devices in the network. The measures closeness gives the communication efficiency of a definite vertex within a network while the average path length measures that of the whole network.
Figure 3: Average path length inversely proportional to closeness
Diabetic Slow and Fast Progressors:
Diabetes is a substantial health fear with more than 30 million living in each country with diabetes. The beginning of diabetes increases the risk for various complications, including kidney disease, myocardial infractions, heart failure, stroke, retinopathy and liver disease. We study and predict these complications using a network-based approach by identifying fast and slow progressors.
Table 2: Fast Progressor Diabetes Five Major Symptoms of People in the Age Group 30 to 70
|
Age |
Polyuria |
Polydipsia |
Weakness |
Polyphagia |
Delayed healing |
|
30 |
Yes |
No |
Yes |
Yes |
Yes |
|
31 |
Yes |
Yes |
Yes |
Yes |
No |
|
32 |
No |
No |
No |
No |
Yes |
|
33 |
No |
Yes |
No |
No |
No |
|
34 |
Yes |
Yes |
Yes |
Yes |
No |
|
35 |
Yes |
No |
No |
Yes |
Yes |
|
36 |
Yes |
Yes |
No |
Yes |
Yes |
|
37 |
No |
No |
Yes |
No |
Yes |
|
38 |
Yes |
Yes |
No |
Yes |
Yes |
|
39 |
Yes |
No |
No |
No |
No |
|
40 |
No |
Yes |
Yes |
No |
Yes |
|
41 |
Yes |
No |
Yes |
Yes |
Yes |
|
42 |
No |
No |
Yes |
Yes |
No |
|
43 |
Yes |
Yes |
Yes |
Yes |
No |
|
44 |
No |
Yes |
No |
No |
Yes |
|
45 |
Yes |
Yes |
Yes |
No |
Yes |
|
46 |
Yes |
No |
Yes |
No |
Yes |
|
47 |
No |
No |
Yes |
Yes |
No |
|
48 |
Yes |
Yes |
Yes |
Yes |
Yes |
|
49 |
Yes |
Yes |
Yes |
No |
No |
|
50 |
Yes |
Yes |
No |
Yes |
Yes |
|
51 |
Yes |
Yes |
Yes |
No |
Yes |
|
52 |
Yes |
Yes |
Yes |
Yes |
Yes |
|
53 |
Yes |
No |
No |
No |
Yes |
|
54 |
Yes |
Yes |
Yes |
No |
Yes |
|
55 |
Yes |
Yes |
No |
Yes |
Yes |
|
56 |
Yes |
Yes |
Yes |
No |
No |
|
57 |
Yes |
No |
Yes |
Yes |
No |
|
58 |
No |
Yes |
Yes |
Yes |
No |
|
59 |
Yes |
No |
No |
No |
No |
|
60 |
Yes |
Yes |
Yes |
Yes |
Yes |
|
61 |
Yes |
Yes |
Yes |
Yes |
No |
|
62 |
Yes |
Yes |
Yes |
Yes |
No |
|
63 |
Yes |
Yes |
Yes |
Yes |
No |
|
64 |
No |
Yes |
No |
No |
Yes |
|
65 |
Yes |
Yes |
Yes |
Yes |
Yes |
|
66 |
Yes |
Yes |
Yes |
Yes |
No |
|
67 |
No |
Yes |
Yes |
Yes |
Yes |
|
68 |
Yes |
Yes |
Yes |
Yes |
Yes |
|
69 |
Yes |
No |
Yes |
Yes |
Yes |
|
70 |
No |
Yes |
Yes |
Yes |
No |
We collected raw data about diabetic patients of different age group and analyze their symptoms with respect to a dominating set D = {Polyuria, Polydipsia, Weakness, Polyphagia, Delayed healing} which consists of five important symptoms that are commonly seen. In table III we analyze the average shortest path length and closeness of these age groups having minimum of four symptoms out of the selected major five symptoms.
Table 3: Table Plotted with Patients Having Minimum Four Symptoms Out of the Major Five and Calculating their Average Length, Closeness
|
Age |
Average Length of Shortest Path |
Closeness |
|
30 |
1.666 |
0.600 |
|
31 |
1.571 |
0.636 |
|
36 |
1.619 |
0.617 |
|
38 |
1.619 |
0.617 |
|
41 |
1.666 |
0.600 |
|
43 |
1.571 |
0.636 |
|
45 |
1.666 |
0.600 |
|
48 |
1 |
1 |
|
50 |
1.619 |
0.617 |
|
51 |
1.666 |
0.600 |
|
52 |
1 |
1 |
|
54 |
1.666 |
0.600 |
|
55 |
1.619 |
0.617 |
|
60 |
1 |
1 |
|
61 |
1.571 |
0.636 |
|
62 |
1.571 |
0.636 |
|
63 |
1.571 |
0.636 |
|
65 |
1 |
1 |
|
66 |
1.571 |
0.636 |
|
67 |
1.857 |
0.538 |
|
68 |
1 |
1 |
|
69 |
1.666 |
0.600 |
It is observed that the age group 48, 52, 60, 65, 68 has the same features and their average shortest path length and closeness equals one.
For comparison, we analyze the relationship between these measures with the patients with a minimum of three symptoms with respect to the dominating set D.
Table 4: Table Plotted with Patients Having Minimum Four Symptoms Out of the Major Five and Calculating their Average Length, Closeness
|
Age |
Average Length of Shortest Path |
Closeness |
|
30 |
1.2 |
0.833 |
|
31 |
1.1 |
0.909 |
|
34 |
1.466 |
0.681 |
|
35 |
1.6 |
0.625 |
|
36 |
1.266 |
0.789 |
|
38 |
1.266 |
0.789 |
|
40 |
1.7 |
0.588 |
|
41 |
1.366 |
0.731 |
|
43 |
1.1 |
0.909 |
|
45 |
1.133 |
0.882 |
|
46 |
1.666 |
0.600 |
|
48 |
1 |
1 |
|
49 |
1.466 |
0.681 |
|
50 |
1.266 |
0.789 |
|
51 |
1.166 |
0.857 |
|
52 |
1 |
1 |
|
54 |
1.133 |
0.882 |
|
55 |
1.3 |
0.769 |
|
56 |
1.466 |
0.681 |
|
57 |
1.533 |
0.652 |
|
58 |
1.566 |
0.638 |
|
60 |
1 |
1 |
|
61 |
1.2 |
0.833 |
|
62 |
1.1 |
0.909 |
|
63 |
1.133 |
0.882 |
|
65 |
1 |
1 |
|
66 |
1.2 |
0.833 |
|
67 |
1.266 |
0.789 |
|
68 |
1 |
1 |
|
69 |
1.233 |
0.811 |
|
70 |
1.6 |
0.625 |
Figure 4: Closeness coincides with average path length for different age group
People are not conscious that due to bad diet habits the signs of diabetes become visible before several years. The classic symptoms of diabetes such as polyuria, polydipsia and polyphagia occur commonly in most of all diabetic patients, which has a prompt expansion of severe hyperglycemia.
The graph shows that the closeness and average path length of each group coincide. Therefore, these age groups show similar features expressing that they are at high risk or belong to the fast progressors of diabetes.
Figure 5. Closeness coincides with average path length with three symptoms
DISCUSSION:
It is detected that patient having four symptoms out of the five symptoms in the dominating set has the same features as that of the patients with three symptoms with respect to D. As per the data, the age group 48, 52, 60, 65, 68 is more likely to become diabetic. Since the measures are closely the same as the two categories, it is concluded that both are fast progressors of diabetes.
CONCLUSION:
In this paper, we examined how close a node is to all other nodes in the network. We also show which symptoms are most appropriate to lead to specific diabetic complications. Moreover, a disease diagnosis graph can be a useful tool for physicians to understand the disease management proposal.
ACKNOWLEDGMENT:
We thank anonymous referees for their valuable suggestions.
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Received on 22.05.2021 Modified on 14.03.2022
Accepted on 21.09.2022 © RJPT All right reserved
Research J. Pharm. and Tech 2023; 16(4):1749-1753.
DOI: 10.52711/0974-360X.2023.00288