Abstract and Figures

In this paper, we put two concepts together: Shortest paths and starant graphs. We calculate the costs involved in putting two randomly selected individuals in contact in a controlled network. That would be the costs in terms of public health. Disease spread became our main concern in what comes to the starant graphs in the year of 2002 because that is one of the directions the work of Comellas et al. and Watts et al. pointed at, and our work is inspired in theirs. Other factors, such as random, and unexpected, contact between individuals, are disregarded, so that if the individual visits the clinic that belongs to Mister X, his mate, but his usual doctor, Mister Y, is not there, and he is then served by Miss R, we will need new calculations, what means that we go from predictive power to disgrace power, and that frontally opposes our initial intentions with this work.
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Modern International Journal of Pure and Applied Mathematics 2017, 1(2): 24-29
DOI: 10.5923/j.mijpam.20170102.02
Some Weight on Our Edges
Marcia R. Pinheiro
IICSE University, USA
Abstract In this paper, we put two concepts together: Shortest paths and starant graphs. We calculate the costs involved in
putting two randomly selected individuals in contact in a controlled network. That would be the costs in terms of public health.
Disease spread became our main concern in what comes to the starant graphs in the year of 2002 because that is one of the
directions the work of Comellas et al. and Watts et al. pointed at, and our work is inspired in theirs. Other factors, such as
random, and unexpected, contact between individuals, are disregarded, so that if the individual visits the clinic that belongs to
Mister X, his mate, but his usual doctor, Mister Y, is not there, and he is then served by Miss R, we will need new calculations,
what means that we go from predictive power to disgrace power, and that frontally opposes our initial intentions with this
work.
Keywords Starant graph, Disease spread, Network, Shortest path, Cost
1. Introduction
We designed the Starant Graph in 2002 to address the problem human network for disease spread. Since then,
unfortunately working independently, and with little or no support, our progresses were slow, but perhaps reasonably steady.
Our first paper on the topic (produced in 2002) brought a bit of criticism to Comellas et al.’s theories. We found what we
believed to be a small inaccuracy with a lemma, and counter-examples made it all evident (Pinheiro, 2007).
We recently published a summary of our developments in what regards the Starant Graph. That was (Pinheiro, 2016).
In-between the first and the last paper on the topic, we produced Starants II or (Pinheiro, 2012). There we talked about
Colloquium Rectangles and Double Colloquium Rectangles.
We now want to put weights on our edges and apply the concepts associated with the name Dijkstra in the academic
products. We will talk about costs and we then will mean cost of putting a particular individual in physical contact with
another. That cost is directly connected to disease spread in our studies, so that we worry about impact of the spread each, and
every, time.
2. Development
We will quote Sedgewick (2017) here:
* Corresponding author:
drmarciapinheiro@gmail.com (Marcia R. Pinheiro)
Published online at http://journal.sapub.org/mijpam
Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved
Modern International Journal of Pure and Applied Mathematics 2017, 1(2): 24-29 25
Figure 1
Figure 2
Figure 3
26 Marcia R. Pinheiro: Some Weight on Our Edges
Figure 4
Figure 5
Modern International Journal of Pure and Applied Mathematics 2017, 1(2): 24-29 27
Figure 6
Our studies have to do with disease spread and impact of the contact between two randomly selected individuals in a
controlled network, impact in terms of population health.
The impact is going to be a nonnegative number, this as a rule, so that we can easily apply Dijkstra’s algorithm or way of
thinking to the problem.
From (Pinheiro, 2016), we got the following graph (Figure 7):
SC5,4.
Figure 7
Let’s now say that Alex has face-to-face contact with 100 people during week X. Peter has face-to-face contact with 500
instead. That is due to differences in employment arrangements, lifestyle, and so on. Jack has face-to-face contact with ten
people because he works at home and lives with his wife, only her and him. Paul has face-to-face contact with 50 people
because he works in IT, hardware, and everyone else who works with him has contact with people, sees others face-to-face,
but he doesn’t. He spends the entire day inside of an office full of equipment. Marcel has face-to-face contact with 200 people
instead. Asha, who always speaks to everyone else, students, fellows, superiors, relatives, and all else, has face-to-face
contact with 600 people. Every edge that leaves the centre of our Starant Graph will then have a weight to represent the
impact of Asha having contact with one of those people whose names we here mentioned, that is, Alex, Peter, Paul, Marcel,
and Jack. These weights, which we will not write on the graph for clarity purposes, are, respectively 700, 1100, 650, 800, and
610.
Alex
Marcel
Peter
Paul
Asha
Jack
28 Marcia R. Pinheiro: Some Weight on Our Edges
SC5,4.
Figure 8
If Peter visits Paul at home and Paul then decides to see Marcel with Peter at his place, we get an impact of Peter + Paul +
Marcel = 500 + 50 + 200 = 750.
That means that, during week X, 750 people are potentially infected with the viral disease that Peter got.
That is then the impact of Peter’s existence in the public health arena.
If Peter visits Paul at his place, and that is all he does, the impact is reduced to 550, which is the value of the edge that goes
from the node we called Peter to the node we called Paul in the figure we here have named Figure 8.
Of course, we are assuming that the people each person we mention here meets face to face is not someone the other person
we have mentioned here would be meeting, that is, we are assuming that there is no intersection between the groups of people
attached to each person we here mention.
3. Conclusions
In conclusion, we have now created a way to tell the impact of the face-to-face contact between any two randomly chosen
individuals in a controlled network in terms of public health or population contamination.
We make use of Dijkstra’s algorithm to measure such an impact and our calculations are based on how many people the
individual who is randomly chosen meets face-to-face during a certain period of time.
Each edge of our Starant Graph starts having a weight, and that weight represents the sum of the weights of the networks of
both nodes or individuals involved.
We could think of a social network environment such as Linkedin here to make it easier: Assume all the people we can
meet in face-to-face mode appear in our Linkedin network. In this way, the number of members of our Linkedin is the weight
of our network or the weight of our personal network in the problem we here consider; that of disease spread.
We make very naïve assumptions, such as that there are no common members to these networks and those networks do not
include any member of the main group we study, just for the purpose of presenting the tool, which we will be using later, in
more advanced studies.
We cannot really add the weight of the edges if we talk about more than two members and face-to-face contact amongst
them. In this case, we would have to diminish the weight of the individual’s network attached to the common nodes when
considering the edges instead. For instance, in the example we here give, of Peter, Paul, and Marcel meeting face-to-face, if
we added the weights of the edges between them, if we simply did that, we would get 550 + 250 = 800, but the impact of these
contacts is actually 750 instead because Paul is represented by a node that connects to more than one node involved in this
calculation and his weight is 50, precisely what has to be subtracted from the straight sum. In conclusion, we have to add all
edges and subtract the weight of the common nodes multiplied by the number of participating nodes that it connects to minus
one. If that node only connects to one participating node, we would subtract zero then.
REFERENCES
Pinheiro, M. R. (2007). Medium World Theories I. Applied Mathematics and Computation, (188), 1061–1070. Retrieved from
http://dx.doi.org/10.1016/j.amc.2006.05.213
Pinheiro, M. R. (2016). Starants: A New Model for Human Networks. Applied Mathematics, (7), 267–271. Retrieved from
http://file.scirp.org/pdf/AM_2016022916113966.pdf
Jack
210
Alex
Marcel
Peter
Paul
Ashasha
600
110
Modern International Journal of Pure and Applied Mathematics 2017, 1(2): 24-29 29
Pinheiro, M. R. (2012). Starants II. Asian Journal of Current Engineering, 1, 259–265.
Sedgewick, R. (2017). Shortest Paths. Retrieved June 5, 2017, from https://www.cs.princeton.edu/~rs/AlgsDS07/15ShortestPaths.pdf
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Article
Full-text available
Refining the results from the work of Comellas et al. regarding deterministic small-world networks, and intending to apply results deduced from theirs into traffic networks, we introduce new constraints, extend their work to networks of circulants, criticize the choice of the name ‘small worlds’ for large circulants, with a number greater than 64 for their vertices and, as a side result, we introduce a new form of graph: starants as a replacement of the circulants for the case of disease spread and social networks. In order to reach our goals we make use of standard combinatorial tools, graph analysis, and general algebraic procedures.
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