Team Control Number. Problem Chosen Mathematical Contest in Modeling (ICM) Summary Sheet. Summary

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1 For office use only T1 T2 T3 T4 Team Control Number Problem Chosen C For office use only F1 F2 F3 F 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 Erdös. 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 foodchain 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 entitled 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 aspects. It comprehensively considered all the factors that could affect the network influence and could be transplanted to other fields with minor modifications.

2 !! Using Networks to Measure Influence and Impact!! Team # 24266!

3 Team # Page 3 of 17 Contents 1 Problem Clarification 4 2 Assumptions 4 3 Notations 4 4 Model design Network analysis Network Building Properties Analysis Co-authorship Network Entropy-Weight-Based Gray Relational Analysis Model (EWGRA) EWGRA Based Influence Model for Coauthor Network Determine Model for Networks Food Chain Model (FCM) Experiment Tennis Player Network Analysis Tennis Player Influence Metric Experiment the Science, Understanding and Utility of Networks Sensitive Analysis 16 6 Strength and Weakness Strength Weakness

4 Team # 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 coauthorship 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 i (k) x 0 0 (k) H k w(k) C D (n i ) C C (n i ) C B (n i ) d(n i,n j ) Descriptions the correlation coefficient of the k th measure of the i th evaluation object the best value of the kth measure recognition differential the entropy of the k th measure the entropy weight of the k th measure Degree centrality Closeness centrality Betweenness centrality the distance of n i between n j

5 Team # Page 5 of 17 4 Model design 4.1 Network analysis 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 Erdös (Assumption 1). By regarding the researchers as nodes and the coauthor 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 Properties Analysis Global Graph metrics: Table 1: Matrix of Network Properties Diameter Mean node number of clustering distance components coefficient Value Small world: Our network has a clustering coefficient of 0.263, and a characteristic path length of With a similarly sized connected random graph, the clustering coefficient is

6 Team # Page 6 of and the characteristic path length is 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 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 normalization to the measures. X i = {x i (1),x i (2),,x i (k),,x i (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 x i is the i th row of the matrix X i, and x i (k) is the k th measure of x i. Therefore, we normalize the data using formula: where x 0 i is normalized x i. x 0 i(k) = x i(k) min x i (k) max x i (k) min x i (k) 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 X 0 = {1, 1,, 1}. step 3. Calculating correlation coefficients. i (k) = min min(x 0 0 (k) x0 j j k (x 0 0 (k) x0 j (k)) + max max j (k)) + max i (1) k (x0 0 (k) x0 j (k)) max k (x0 0 (k) x0 j (k)) (2)

7 Team # Page 7 of 17 in this equation, i =1, 2,,n; k =1, 2,,m. Besides, i (k) represents the correlation coefficient of the k th measure of the i th evaluation object; x 0 0 (k) represents the best value of the k th 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 w(k) i (k) (3) k=1 the w(k) means the weight of the k th 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 k th entropy is defined by: H k = p nx f kj ln f kj (4) j=1 f kj = x 0 ikj P n j=1 x0 ikj (5) p = 1/ ln n (6) Besides, k =1, 2,,m; x 0 ikj represents the normalization of the kth measure of the j th evaluation objects. And then the entropy weight of the k th measure can be calculated by : w(k) = 1 H k Pm k=1 1 H k (7) mx w(k) =1 (8) k=1 w(k) is the weight of the k th measure; H k is the entropy of the k th measure.as mentioned above, the entropy and weight are of antidependence relationship 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 distorting 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.

8 Team # Page 8 of 17 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 degree 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. where d(n i ) is the degree of the node n i. C D (n i )=d(n i ) (9) 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. C C (n i )= NX i=1 1 d(n i,n j ) (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 networ, and it represents the interdisciplinarity of scholars in the co-author network. C B (n i )= X j,k6=i g jik g jk (11) In this equation, g jk 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 neglects 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: M ij = 1 d N + d 1 C(p i ) A ij, 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).

9 Team # 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 ALON, NOGA M HARARY, FRANK* GRAHAM, RONALD LEWIS BOLLOBAS, BELA RODL, VOJTECH SHELAH, SAHARON LOVASZ, LASZLO TUZA, ZSOLT SPENCER, JOEL H FUREDI, ZOLTAN SOS, VERA TURAN HENRIKSEN, MELVIN PACH, JANOS CHUNG, FAN RONG KING SZEMEREDI, ENDRE NESETRIL, JAROSLAV POMERANCE, CARL BERNARD GILLMAN, LEONARD FAUDREE, RALPH JASPER, JR HAJNAL, ANDRAS Result Analysis We use SPSS to do Spearman analysis to analyze the sort result of the correlation coefficients 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 Closeness Centrality Degree PageRank Total Publication Max Reference Total Reference 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,

10 Team # 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 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 normalization formula is: x 0 i = x i min x i max x i min x i (12) where x 0 i is the normalized x i. 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 latestcomers, have the FCM significance of 0 and nodes of which all children are leaf nodes have the FCM significance of normalized citations per year 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.

11 Team # 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 centrality 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 model and get the relative ranking as follow(see Table7)

12 Team # Page 12 of 17 Table 7: Rank of Influence Measures of Fundamental Papers Rank Label GRA 1 Collective dynamics of small-world networks Emergence of scaling in random Identity and search in social networks A family of measures Community structure in social and biological networks The structure of scientific collaboration networks Navigation in a small world The structure and function of complex networks On Random Graphs Exploring complex networks Statistical mechanics of complex networks Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality Models of core/periphery structures Social network thresholds in the diffusion of innovations Networks, influence, and public opinion formation Identifying sets of key players in a network On properties of a well-known graph or what is your ramsey number? Statistical models for social networks 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 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 influence. 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 universities 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 training 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.

13 Team # Page 13 of 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/(P A P B ) L (P A P B )/5 W + L and that from B to A is. W (P A P B )/2 + L (P A P B )/10 W + L W is the number chances that A beat B. L is the number chances that B beat A. P A and P B 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, including 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 championship, 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 Experiment Data Source We collected the world ranking[1] right after the 2012 Davis Cup, which is the last championship in the year of 2012, representing a players strength at the beginning of the year of Then we collected the all the playing activities of top 15 players from ATP Official. There

14 Team # 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.

15 Team # Page 15 of 17 Table 9: Ranks of Tennis Players name GRA GRA rank real rank Rafael Nadal David Ferrer Novak Djokovic Roger Federer Andy Murray Juan Martin Del Potro Nicolas Almagro Tomas Berdych Richard Gasquet Juan Monaco Janko Tipsarevic John Isner Jo-Wilfried Tsonga Milos Raonic Marin Cilic 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 influence 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 enthusiasm 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 Centrality, 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 importance 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.

16 Team # 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 collaboration relationship, such like the media impact and the influence of big events. Although

17 Team # 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. References [1] Official Men s Tennis Rankings, Home.aspx/ [2] MathSciNet, [3] Google Scholar, [4] Yan, E., Ding, Y. (2009). Applying centrality measures to impact analysis: A coauthorship network analysis. Journal of the American Society for Information Science and Technology, 60(10), [5] BÃűrner, K., Dall Asta, L., Ke, W., Vespignani, A. (2005). Studying the emerging global brain: Analyzing and visualizing the impact of co-authorship teams. Complexity, 10(4), [6] Liu, X., Bollen, J., Nelson, M. L., Van de Sompel, H. (2005). Co-authorship networks in the digital library research community. Information processing management, 41(6), [7] Brin, S., Page, L. (1998). The anatomy of a large-scale hypertextual Web search engine. Computer networks and ISDN systems, 30(1), [8] Freeman, L. C. (1979). Centrality in social networks conceptual clarification. Social networks, 1(3), [9] Kuo, Y., Yang, T., Huang, G. W. (2008). The use of grey relational analysis in solving multiple attribute decision-making problems. Computers Industrial Engineering, 55(1), [10] Deng, J. L. (1989). Introduction to grey system theory. The Journal of grey system, 1(1), 1-24.