GRAPH ALGORITHM TO FIND CORE PERIPHERY STRUCTURES USING MUTUAL K-NEAREST NEIG...
Using Networks to Measure Influence and Impact
1. Tongji University
Meritorious Winner
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With support from
Certificate of Achievement®
Lingjun Zhou
Yunhao Zhang
Zhihao Li
Long Meng
Interdisciplinary Contest In Modeling
Be It Known That The Team Of
With Faculty Advisor
Of
Was Designated As
D. Chris Arney, Contest Director
2014
Tina R. Hartley, Head Judge
2. For office use only
T1
T2
T3
T4
Team Control Number
24266
Problem Chosen
C
For office use only
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2014 Mathematical Contest in Modeling (ICM) Summary Sheet
Summary
With the trend of globalization, interactions among people become increasingly
frequent, leading to a fact that people nowadays are more eager than ever to know
their network, enhance their impact and eventually gain dominance in the network.
Network science is such an emerging subject to analyze the social and research
network.
In this paper, we first build the co-authorship network of researchers who has
collaborated with mathematician Paul Erd¨os. After analyzing the properties of this
network, we build an Entropy-Weight-Based Gray Relational Analysis (EWGRA)
model, a quantitative method for analyzing the correlation between two subjects
and could, to a large extent, avoid human interference in the weighting process. We
put classic centralities, including degree, betweenness and closeness, to the model,
and innovatively combine the EWGRA with PageRank to take interactions between
adjacent nodes into consideration. Applying this new model, we could obtain the
result that the most influential researcher in the co-authorship network is ALON,
NOGA M. To evaluate the significance of research papers, we propose a Food Chain
Model (FCM) measure, which simulates the nutrition-deriving procedure of food-
chain in ecosystem. Then we feed the significance calculated by FCM together with
other centrality measures to EWGRA model and get the result that the paper enti-
tled Collective dynamics of ’small-world’ networks is the most influential research
paper in the citation network.
We then extend our model to totally different areas. Building a network among
men’s single tennis players, we develop a metric for player’s influence and use the
model to get the predicted influence rank of those players at the end of 2013, which
nicely conforms to the real rank. Furthermore, we make an analysis on how to
make use of the influence model to be more successful in career.
Although bearing some slight weaknesses, our model is merited in many as-
pects. It comprehensively considered all the factors that could affect the network
influence and could be transplanted to other fields with minor modifications.
5. Team # 24266 Page 4 of 17
1 Problem Clarification
Network science is an emerging interdisciplinary subject that can be used to analyze the
hidden influence factor of a network such as co-authorship network and other social networks.
A list of researchers with co-author relationship is given and we are supposed to build the co-
authorship network of them for further analysis. Given the network, an influence measure is
needed to determine the rank of influence factor of the researchers.
Whether a research paper is cited by important works is a typical metric to determine the
influence of a paper, so that an algorithm is required to be used to seek the most influential
research paper in one scientific field. Such algorithm could also be transplanted to other social
and scientific area to conclude methodology for organizations and individuals to make wise
decisions.
2 Assumptions
• One researcher could be more influential if he collaborates with other researchers.
When two or more researchers cooperate, they are more likely to propose new findings
and get acquaintance with more researchers in their research field. These are facts that
may make them more influential.
• Restricting the network in the small circle of the members we concern does not influence
the relative influence of the members.
In our discussion, the circle of members consist most of the influential members in the
field. So that the influences from outside world have tiny impact the members of the
network.
• If one research paper is cited by another influential paper, we could confer that the paper
been cited also has great impact factor.
As we know that professional researchers hardly cite useless papers, if a paper is cited by
an influential paper, it must have some contribution to the paper that cited it.
3 Notations
Notations Descriptions
⇠i(k) the correlation coefficient of the kth measure of the ith evaluation object
x0
0(k) the best value of the kth measure
⇢ recognition differential
Hk the entropy of the kthmeasure
w(k) the entropy weight of the kth measure
CD(ni) Degree centrality
CC(ni) Closeness centrality
CB(ni) Betweenness centrality
d(ni, nj) the distance of ni between nj
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4 Model design
4.1 Network analysis
4.1.1 Network Building
Figure 1: Co-authorship Network
As there are over 9,000 researchers in the Erdos1 table, we have to limit the size of the
network for more accurate analysis. We only retain the 511 researchers who have the co-author
relationship with Erd¨os (Assumption 1). By regarding the researchers as nodes and the co-
author relationships as links, we build the network in the right:
In Fig 1, The size of the nodes represent it’s degree and the nodes with the same degree is
aligned in the same ray from the center of the circle.
4.1.2 Properties Analysis
Global Graph metrics:
Table 1: Matrix of Network
Properties Diameter Mean node number of clustering
distance components coefficient
Value 10 3.844 511 0.263
Small world: Our network has a clustering coefficient of 0.263, and a characteristic path
length of 3.844. With a similarly sized connected random graph, the clustering coefficient is
7. Team # 24266 Page 6 of 17
0.31 and the characteristic path length is 3.66. This means that the co-authorship network of
Erdos1 is a small world graph as can be expected.
Individual actor properties:
Table 2: Properties of Individual Nodes
Properties Closeness Betweenness PageRank Degree
centrality centrality
Name HENRIKSEN, HARARY, HARARY, ALON,
MELVIN FRANK FRANK NOGA M.
4.2 Co-authorship Network
In network analysis, it is a common practice to use centrality, including degree centrality,
betweenness centrality, closeness centrality and eigenvector centrality, as the impact measure of
a node. Existing models usually consider the centrality measures respectively, leading to a not
thorough and comprehensive enough analysis of an author’s influence. Meanwhile, existing
models usually ignored the significance of an author’s publications on his or her influence in
the scientific social network. Therefore, in this section, we propose a novel Entropy-Weight-
Based Gray Relational Analysis model. In this model, we comprehensively considered the
centrality measures, and we also took the author’s publications into consideration.
4.2.1 Entropy-Weight-Based Gray Relational Analysis Model (EWGRA)
Gray Relational Analysis:
GRA is a quantitative method for analyzing the correlation between two subjects[9][10].
Higher correlation coefficient means higher correlation between the two subjects. If one of the
subjects is an ideal subject, we could get the correlation between a real subject and the ideal
subject. Then we could sort the real subjects by their correlation coefficients, the higher, the
better.
step 1. Since different measures of a subject vary in scale and dimension, we first apply nor-
malization to the measures. Xi = {xi(1), xi(2), · · · , xi(k), · · · , xi(m)} ,i = 1, 2, · · · , n;n
is the number of subjects which need evaluating. k = 1, 2, · · · , m;m is the number of
measures.Then xi is the ith row of the matrix Xi , and xi(k) is the kth measure of xi.
Therefore, we normalize the data using formula:
x0
i(k) =
xi(k) min xi(k)
max xi(k) min xi(k)
(1)
where x0
i is normalized xi.
step 2. Determine the measures of the ideal subject. Since we normalized the data in Step 1,
the ideal subject here is simply a subject with all the measures of 1. Therefore, the ideal
subject X0 = {1, 1, · · · , 1}.
step 3. Calculating correlation coefficients.
⇠i(k) =
min
j
min
k
(x0
0(k) x0
j(k)) + ⇢ max
j
max
k
(x0
0(k) x0
j(k))
(x0
0(k) x0
j(k)) + ⇢ max
i
max
k
(x0
0(k) x0
j(k))
(2)
8. Team # 24266 Page 7 of 17
in this equation, i = 1, 2, · · · , n; k = 1, 2, · · · , m. Besides, ⇠i(k) represents the correlation
coefficient of the kth measure of the ith evaluation object; x0
0(k) represents the best value
of the kth measure; ⇢ is the recognition differential, using to improve the significance of
the difference between the coefficients of correlation. We usually assign 0.5 to it.
step 4. Calculating weighted correlation coefficient.
i =
mX
k=1
w(k)⇠i(k) (3)
the w(k) means the weight of the kth measure.
Entropy Weight
In information theory, entropy is a measure of uncertainty in the information content[10].
The lower the entropy of the information is, the higher the usefulness of the information. By
applying entropy into our model, we could avoid taking artifact factors into the consideration.
In a given problem where there are n evaluation objects and m measures. The kth entropy
is defined by:
Hk = p
nX
j=1
fkj ln fkj (4)
fkj =
x0
ikj
Pn
j=1 x0
ikj
(5)
p = 1/ ln n (6)
Besides, k = 1, 2, · · · , m; x0
ikj represents the normalization of the kth measure of the jth evalua-
tion objects. And then the entropy weight of the kth measure can be calculated by :
w(k) =
1
Hk
Pm
k=1
1
Hk
(7)
mX
k=1
w(k) = 1 (8)
w(k) is the weight of the kth measure; Hk is the entropy of the kth measure.As mentioned above,
the entropy and weight are of antidependence relationship.
4.2.2 EWGRA Based Influence Model for Coauthor Network
Centrality Measures
It is proved that a person in central position can influence the group by withholding or dis-
torting information in transmission[8]. Therefore, we should consider centrality measures[4]
in our model. Meanwhile, as mentioned above, an author’s influence is also determined by an
author’s publications, and hence, we also take the total number of publications of an author,
the maximum citation of an author, and the total citations of an author into consideration.
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• Degree centrality This equals the number of ties that a node has with other nodes. Nodes
with higher degree usually have more impact in the network. In this case, the higher de-
gree an author has indicates that he or she actively cooperates with others, and, therefore,
it is more probable for him or her to exert impact on others.
CD(ni) = d(ni) (9)
where d(ni) is the degree of the node ni.
• Closeness centrality This measures the geodesic distance from each node to others and
focuses on the extent of influence over the entire network. In this case, it is a metric of
“how long it will take information to spread from a given author to others in the networ”.
CC(ni) =
NX
i=1
1
d(ni, nj)
(10)
• Betweenness centrality This is based on the number of shortest paths passing through
a node. Nodes with higher betweenness are“pivot points of knowledge flow in the net-
wor”, and it represents the interdisciplinarity of scholars in the co-author network.
CB(ni) =
X
j,k6=i
gjik
gjk
(11)
In this equation, gjk is the geodesic distance. between the vertices of j and k.
• Eigenvector centrality and PageRank[7] Eigenvector centrality is based on the idea that
the importance of a node depends on the importance of its neighbors. However, it ne-
glects the fact that nodes with more adjacent nodes should exert less influence on each
of its adjacent nodes. Therefore, we take advantage of PageRank, which is the principal
eigenvector of the transition matrix M:
Mij =
1 d
N
+ d
1
C(pi)
Aij,
and therefore can be seen as a variation of eigenvector centrality.
• Number of publications This is the total number of publications published by the author.
• Maximum Citation This is the number of citation of the paper that is cited by most other
papers.
• Total Citations This is the total number of times that the papers of the author are cited
for.
Experiment
In this model, we’ve got 7 parameters(measures).We collected 511 data records of number
of publications, maximum citation, and total citations from MathSciNet[2]. Put the network
data from the question 1 into the model, we then could get the respective measures of each
node(see Table 3). Then we calculate the correlation coefficients, and sort the data (see Table 4).
10. Team # 24266 Page 9 of 17
Table 3: Rank by Principle Measures
Closeness Centrality Betweenness Centrality PageRank Degree
HENRIKSEN, MELVIN HARARY, FRANK* ALON, NOGA M. ALON, NOGA M.
GILLMAN, LEONARD SOS, VERA TURAN GRAHAM, RONALD LEWIS GRAHAM, RONALD LEWIS
BOES, DUANE CHARLES STRAUS, ERNST GABOR* HARARY, FRANK* HARARY, FRANK*
GAAL, STEVEN A. RUBEL, LEE ALBERT* RODL, VOJTECH RODL, VOJTECH
SCHERK, PETER* POMERANCE, CARL BERNARD TUZA, ZSOLT TUZA, ZSOLT
HERZOG, FRITZ* ALON, NOGA M. SOS, VERA TURAN SOS, VERA TURAN
BONAR, DANIEL DONALD GRAHAM, RONALD LEWIS BOLLOBAS, BELA BOLLOBAS, BELA
CARROLL, FRANCIS WIILLIAM FUREDI, ZOLTAN FUREDI, ZOLTAN FUREDI, ZOLTAN
DARLING, DONALD A. PACH, JANOS SPENCER, JOEL H. SPENCER, JOEL H.
VAN KAMPEN, EGBERTUS RUDOLF* HAJNAL, ANDRAS POMERANCE, CARL BERNARD PACH, JANOS
WINTNER, AUREL FRIEDRICH* BOLLOBAS, BELA PACH, JANOS CHUNG, FAN RONG KING
HUNT, GILBERT AGNEW TUZA, ZSOLT HAJNAL, ANDRAS HAJNAL, ANDRAS
SIRAO, TUNEKITI RUZSA, IMRE Z. CHUNG, FAN RONG KING LOVASZ, LASZLO
BAGEMIHL, FREDERICK ODLYZKO, ANDREW MICHAEL STRAUS, ERNST GABOR* FAUDREE, RALPH JASPER, JR.
KHARE, SATGUR PRASAD SARKOZY, ANDRAS SARKOZY, ANDRAS POMERANCE, CARL BERNARD
SMITH, BRENT PENDLETON KLEITMAN, DANIEL J. LOVASZ, LASZLO NESETRIL, JAROSLAV
DARST, RICHARD BRIAN SPENCER, JOEL H. NESETRIL, JAROSLAV KLEITMAN, DANIEL J.
FELLER, WILLI K. (WILLIAM)* RODL, VOJTECH FAUDREE, RALPH JASPER, JR. SZEMEREDI, ENDRE
JACKSON, STEPHEN CRAIG SCHINZEL, ANDRZEJ KLEITMAN, DANIEL J. GYARFAS, ANDRAS
VIJAYAN, KAIPILLIL SHIELDS, ALLEN LOWELL* SZEMEREDI, ENDRE STRAUS, ERNST GABOR*
Table 4: Rank by Correlation Coefficients
Rank GRA Name
1 0.730005579 ALON, NOGA M.
2 0.602836556 HARARY, FRANK*
3 0.588431962 GRAHAM, RONALD LEWIS
4 0.565383561 BOLLOBAS, BELA
5 0.546964138 RODL, VOJTECH
6 0.545439769 SHELAH, SAHARON
7 0.51811051 LOVASZ, LASZLO
8 0.511504097 TUZA, ZSOLT
9 0.505746662 SPENCER, JOEL H.
10 0.503490531 FUREDI, ZOLTAN
11 0.499874352 SOS, VERA TURAN
12 0.487420155 HENRIKSEN, MELVIN
13 0.48300931 PACH, JANOS
14 0.475723158 CHUNG, FAN RONG KING
15 0.46562863 SZEMEREDI, ENDRE
16 0.461775187 NESETRIL, JAROSLAV
17 0.460549514 POMERANCE, CARL BERNARD
18 0.458779327 GILLMAN, LEONARD
19 0.457257127 FAUDREE, RALPH JASPER, JR.
20 0.457006357 HAJNAL, ANDRAS
Result Analysis
We use SPSS to do Spearman analysis to analyze the sort result of the correlation coeffi-
cients and the sort result of other common measures. The result (see Table 5) presents that our
measure is plausible.
Table 5: Correlations Sig.
Name Spearman’s rho Sig.
Betweenness Centrality 0.062
Closeness Centrality 0.000
Degree 0.452
PageRank 0.205
Total Publication 0.020
Max Reference 0.095
Total Reference 0.044
4.3 Determine Model for Networks
When analyzing the significance of a research paper, some important works that follow
from it could be utilized as a measure. In this section, we collect 327 important papers that
follow the fundamental set of publications and construct a citation network among them. Then,
11. Team # 24266 Page 10 of 17
we propose a Food Chain Model to measure the significance of a paper basing on its followers
and evaluate the relative influence of the fundamental papers.
4.3.1 Food Chain Model (FCM)
Food Chain Model (FCM) is inspired by the ecosystem. The significance of first-comers
(fundamental papers) feeds on that of the latter-comers (following papers).
step 1. Normalization. In order to make the FCM significance standardized, which means to
make the FCM significance of different papers possible to compare with each other, we
need to first normalize the latter-comers’ significance before calculating. The normal-
ization formula is:
x0
i =
xi min xi
max xi min xi
(12)
where x0
i is the normalized xi.
step 2. Calculation. The first-comer significance is calculated simply by add up the normalized
FCM significance.
Since citation network could be abstracted as a tree structure, the model could be applied in
a level-to-level bottom-up manner. We assume that the leaf nodes, which are almost the latest-
comers, have the FCM significance of 0 and nodes of which all children are leaf nodes have the
FCM significance of normalized citations per year.
4.3.2 Experiment
Data Source
We collected 811 data records made up by 327 research papers following the fundamental
paper set from Google Scholar[3]. Our criteria for choosing these papers are based on the rule
that important works should be cited more than 500 times. Our fundamental paper set consists
of 18 papers, 16 of them are from the paper set provided by the problem description, and the
other 2 are papers of high citation (more than 10,000 times) in the Network Science field and
follow the provided fundamental papers .
Table 6: Fundamental Papers in Network Science Field
reference name
A family of measures
Collective dynamics of ’small-world’ networks
Community structure in social and biological networks
Emergence of scaling in random
Exploring complex networks
Identifying sets of key players in a network
Identity and search in social networks
Models of core/periphery structures
Navigation in a small world
Networks, influence, and public opinion formation
On properties of a well-known graph or what is your ramsey number?
On random graphs
Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality
Social network thresholds in the diffusion of innovations
Statistical mechanics of complex networks
Statistical models for social networks
The structure and function of complex networks
The structure of scientific collaboration networks.
12. Team # 24266 Page 11 of 17
Figure 2: Citation Network of Research Papers
Model calculation
• FCM Since our major concern is the relative influence of papers in the fundamental set,
we treat every paper outside the fundamental set as a node of which the children are all
leaf nodes for simplicity. Then, we could get the FCM significance for each node in the
network.
• Centrality Measures Then, we calculate the centrality measures including degree cen-
trality and closeness centrality. Betweenness is ignored here since the citation network is
unidirectional, and the betweenness is meaningless in the network.
• EWGAR Finally, we put the centrality measures and FCM significance into EWGAR mod-
el and get the relative ranking as follow(see Table7)
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Table 7: Rank of Influence Measures of Fundamental Papers
Rank Label GRA
1 Collective dynamics of ‘small-world’ networks 0.828191566
2 Emergence of scaling in random 0.638534799
3 Identity and search in social networks 0.565388583
4 A family of measures 0.525524382
5 Community structure in social and biological networks 0.49109065
6 The structure of scientific collaboration networks 0.473075466
7 Navigation in a small world 0.469285373
8 The structure and function of complex networks 0.465774706
9 On Random Graphs 0.455678303
10 Exploring complex networks 0.453830022
11 Statistical mechanics of complex networks 0.45169438
12 Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality 0.44187918
13 Models of core/periphery structures 0.370988004
14 Social network thresholds in the diffusion of innovations 0.370706228
15 Networks, influence, and public opinion formation 0.369317764
16 Identifying sets of key players in a network 0.333333333
17 On properties of a well-known graph or what is your ramsey number? 0.333333333
18 Statistical models for social networks 0.333333333
Result
In the EWGAR model, several measures was considered and their final weight is listed
below:
Table 8: Weight of Measures
Degree Closeness Centrality Cite Weight
0.264753 0.33200714 0.40323985
Methodology
The EWGAR model could be used to measure networks other than citation network as long
as members of the network have analogous relationship with the citation network. We would
need the following data: the relationship of the members including the nodes and weight of
the links, the initial values of their influence or measures that could be regard as their influ-
ence. For example, if we want to measure the impact of a university, we would need to build a
network of universities by gathering such information like the number of academic activities a
university held and participate, the number and quality of research papers that several univer-
sities collaborated and the rank of comprehensive strength of universities. With these data, we
could analyze the network of universities and get the final rank of influence of universities.
4.4 Tennis Player Network Analysis
Tennis is a highly professionalized sports game. Tennis players need to pay for their train-
ing team, including salaries, travelling allowance, and etc. of the team members. Therefore, it
is crucial for the professional tennis players to fully mine their commercial potentials. As their
commercial value is based on their influence, it is necessary for them to know their influences.
In this section, we extend our model to the field to men’s professional tennis and try to analyze
the relative influence among the top 15 players.
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4.4.1 Tennis Player Influence Metric
Relative Challenge Influence Index (RCII)
It is commonsensical that when a low-ranked player competes with a high-ranked player,
the low-ranked player would have a greater chance to enhance his reputation and influence,
especially if he wins. Even if he loses, he would also gain some influence since he could be
exposed to the mass media more. On the contrary, a high-ranked player would not benefit
much in a competition with a low-ranked player when he wins. What’s worse, he would
suffer a tremendous loss of influence if he loses the game. Bearing this knowledge in mind, we
present the Relative Challenge Influence Index (RCII) to measure this relative win-lose impact
on players.
Let player A be the low-ranked player and Play B be the high-ranked player. The RCII from
A to B is
W |2/(PA PB)| L |(PA PB)/5|
W + L
and that from B to A is
W |(PA PB)/2| + L |(PA PB)/10|
W + L
.
W is the number chances that A beat B. L is the number chances that B beat A. PA and PB
is the strength of A and B. The equations above is derived from an empirical formula which is
widely used in predicted the possibility of winning.
World Tour Points Index (WTPI)
There are many tennis matches around the world over the year, each with different ranking
points, indicating different significance of a match. There are five different tier matches, includ-
ing Grand Slam, which worth 2000 points, ATP World Tour Masters 1000, which worth 1000
points, ATP World Tour 500, which worth 500 points, and ATP World Tour 250, which worth
250 points.
It is obvious that attending a more important championship could considerably promote
one’s influence, especially if one could get a good rank in the championship. Here, we present
a World Tour Points Index (WTPI) to measure the influence accumulated by a player in world
tours. It is important to know that instead of using the event points a player gained in the cham-
pionship, we use (ranking points / one’s ranking in the game) to represent the performance of
a player for simplicity.
We define that if a player gets the number n rank in a tennis match worth Q points, then the
WTPI he gets from the match is
p
Q/n.
4.4.2 Experiment
Data Source
We collected the world ranking[1] right after the 2012 Davis Cup, which is the last cham-
pionship in the year of 2012, representing a players’ strength at the beginning of the year of
2013. Then we collected the all the playing activities of top 15 players from ATP Official. There
15. Team # 24266 Page 14 of 17
are 977 records, involving 161 different players. We construct a network(Figure 3.) among the
players. The weights of edges RCII.
Figure 3: Network of Tennis Players
Model Calculation
As mentioned above, we use the RCII as the weight of edges in the network. According
the weighed network, we could get the weighted degree centrality, closeness centrality and
betweenness centrality. As we’ve considered the influence of adjacent nodes in the weight of
edges, we do not include eigenvector centrality this time. We put the data together with WTPI
into EWGAR model and get the weighted coefficient ranking.
16. Team # 24266 Page 15 of 17
Table 9: Ranks of Tennis Players
name GRA GRA rank real rank
Rafael Nadal 0.840016376 1 1
David Ferrer 0.673792761 2 3
Novak Djokovic 0.601445885 3 2
Roger Federer 0.560249532 4 6
Andy Murray 0.52617534 5 4
Juan Martin Del Potro 0.519789108 6 5
Nicolas Almagro 0.513121459 7 11
Tomas Berdych 0.510810309 8 7
Richard Gasquet 0.508854339 9 8
Juan Monaco 0.497876022 10 15
Janko Tipsarevic 0.493357647 11 13
John Isner 0.489311272 12 12
Jo-Wilfried Tsonga 0.487057834 13 9
Milos Raonic 0.482946792 14 10
Marin Cilic 0.481200603 15 14
Result
We use SPSS to make a Spearman analysis between the ranking after 2013 Davis Cup and
the ranking predicated by our model, the Spearman’s rho Sig. is 0, which means our model
suited the real-world condition accurately.
4.5 the Science, Understanding and Utility of Networks
With the information explosion of human society, the connections between people, either
in scientific field or social relationship, is becoming increasingly complex. By studying the
properties and characteristics of networks, some hidden information could be dug out from
the complex representations.
In our model, every member of any social groups, as long as they have the collaboration
relationship or citation relationship, has weighted influences to others. We could dig out the in-
fluence measures which represent the impact of an individual or organization by analyzing the
information of the network using the algorithm we proposed. After getting the measures, we
could easily find out the researchers or companies with marked impact and determine whom
or which company to cooperate with. For example, when looking for co-authors, we can use an
appropriate algorithm and measures to determine which researcher has the most remarkable
impact in our research field and he/she should be the one that we have the greatest enthu-
siasm to cooperate with. Just take our model as an example. We developed a new influence
measure-GRA, as stated before, which consider the Closeness Centrality, Betweenness Central-
ity, PageRank and Degree of a network. These indexes of the network represent the efficiency
of spreading information, the potential of a point for control of communication, the impor-
tance of its neighbors and the tendency to have more links to others respectively. We apply the
gray relational analysis model to mix these indexes to get our final measure-GRA which could
accurately determine the most remarkable researcher.
17. Team # 24266 Page 16 of 17
5 Sensitive Analysis
Considering that there might be some faults in our data and the list of members and links
in a network might be incomplete, our model should be robust enough to cope with such
problems. Now we provide a reasonable scenario for test.
Dr. Who was a coeval mathematician with Paul Erdos, but he was a mysterious person
and no one knows his real identity. He was idiosyncratic in cooperating with others. He only
cooperated with scientists whose Erdos Number is 1 and he only cooperated with 10 scientists
during his life. He published 75 papers. The maximum citation per paper is 76. And his papers
were cited 450 times totally. All the 4 numbers (10, 75, 76, 450) are the average number of those
of other scientists. Due to some reasons, we forget to take him into considerations. Now, we
include Dr. Who into our analysis. The result we got is listed below:
Table 10: the Rank Before and After Inserting New Data
Before After
ALON, NOGA M. ALON, NOGA M.
HARARY, FRANK* HARARY, FRANK*
GRAHAM, RONALD LEWIS GRAHAM, RONALD LEWIS
BOLLOBAS, BELA BOLLOBAS, BELA
SOS, VERA TURAN RODL, VOJTECH
RODL, VOJTECH SHELAH, SAHARON
SHELAH, SAHARON LOVASZ, LASZLO
TUZA, ZSOLT TUZA, ZSOLT
POMERANCE, CARL BERNARD SPENCER, JOEL H.
LOVASZ, LASZLO FUREDI, ZOLTAN
SPENCER, JOEL H. SOS, VERA TURAN
FUREDI, ZOLTAN HENRIKSEN, MELVIN
PACH, JANOS PACH, JANOS
HENRIKSEN, MELVIN CHUNG, FAN RONG KING (GRAHAM)
HAJNAL, ANDRAS SZEMEREDI, ENDRE
STRAUS, ERNST GABOR* NESETRIL, JAROSLAV
CHUNG, FAN RONG KING (GRAHAM) POMERANCE, CARL BERNARD
NESETRIL, JAROSLAV GILLMAN, LEONARD
KLEITMAN, DANIEL J. FAUDREE, RALPH JASPER, JR.
GILLMAN, LEONARD HAJNAL, ANDRAS
We use SPSS to do Spearman analysis to analyze the two result. The result-Correlation Sig.
is 0.01-presents that our model is stable.
6 Strength and Weakness
6.1 Strength
• Our model takes all sorts of factors that may influence the impact measure of network
members in to consideration comprehensively.
• Our model could determine the influence measure of members in any networks as far as
the data of relationship is available.
6.2 Weakness
The influence measure of in a network can be affect by some other factors other than col-
laboration relationship, such like the media impact and the influence of big events. Although
18. Team # 24266 Page 17 of 17
such factors could be represented as a parameter and then added to our model, we have not
done it yet. So our model cannot eliminate the influence of these impact factors.
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