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Re: Katz' Rank Prestige, PageRank or Bonacich Power?

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Wed, 14 Nov 2007 06:25:11 -0800

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 ```***** To join INSNA, visit http://www.insna.org ***** Dear Jack, I'm unclear whether a hospital increases in centrality as it receives from central hospitals or as it sends to central hospitals. I'll assume the former - status comes from receiving patients from important hospitals. Aij is the number of patients i sends to j.     One big difference between Katz and Bonacich is that Katz standardizes his adjacency matrices so that column sums or row sums are all 1.00. This has the effect of making everyone equal in out-degree or in-degree. In measuring centrality this might mean that all hospitals send the same number (100%) of its patients (rows sums of 1.00), and differences are allowed in the number received. You would solve the equation Transpose(A)x=x The Bonacich-Lloyd measure is also designed for non-symmetric relations but it equalizes vertices in another way. It is assumed that every actor has an initial centrality independent of network effects and that this centrality can be magnified by network effects. In other words, all hospitals, even if they don't receive any patients, have some centrality. PageRank seems to use both strategies: the adjacency matrix is normalized so that the column sums are 1.00 and there is an exogenous factor as well. Since A is not symmetric it is probably not a good idea to use the eigenvector Ax = Lx, where L is the largest eigenvalue.     The measures are not functions of one another although I suspect they are correlated. Phillip Bonacich Department of Sociology University of California Los Angeles, CA 90095-1551 (310) 825-3017 -----Original Message----- From: Social Networks Discussion Forum [mailto:[log in to unmask]] On Behalf Of Jack Iwashyna Sent: Friday, November 09, 2007 6:13 AM To: [log in to unmask] Subject: Katz' Rank Prestige, PageRank or Bonacich Power? ***** To join INSNA, visit http://www.insna.org ***** Colleagues, I have a directed graph with continuous values on each edge -- the number of transferred patients that flow between any pair of hospitals in the United States. I need a continuous measure of centrality that increases with the degree of the receiving hospital, the number of patients transferred between any pair of hospitals, and also increases in the centrality of the hospitals that themselves send the patients. Katz' Rank Prestige, PageRank and Bonacich Power all have these properties, and all are more-or-less readily implemented. Is there a simple mathematical relationship between the three? Is there a compelling reason to choose among them? (And for those who prefer Bonacich, do you have an a prioiri way to choose the beta?) I would welcome advice. thanks,    Jack The relevant references are: Bonacich, P. 1987. "Power and Centrality: A Family of Measures." American Journal of Sociology 92:1170-1182. Brin, Sergey, and Lawrence Page. 1998. "The Anatomy of a Large-Scale Hypertextual Web Search Engine." WWW7 / Computer Networks 30:110-117. Page, Lawrence, Sergey Brin, Rajeev Motwani, and Terry Winograd. 1999. "The PageRank Citation Ranking: Bringing Order to the Web." Katz, L. 1953. "A New Status Index Derived from Sociometric Analysis." Psychometrika 18:39-43. -- -=-=-=-=-=-=-=-=-=-=-=-=-=- Theodore J. Iwashyna, M.D., Ph.D. Division of Pulmonary and Critical Care Department of Medicine Hospital of the University of Pennsylvania email: [log in to unmask] -=-=-=-=-=-=-=-=-=-=-=-=-=- _____________________________________________________________________ SOCNET is a service of INSNA, the professional association for social network researchers (http://www.insna.org). To unsubscribe, send an email message to [log in to unmask] containing the line UNSUBSCRIBE SOCNET in the body of the message. _____________________________________________________________________ SOCNET is a service of INSNA, the professional association for social network researchers (http://www.insna.org). To unsubscribe, send an email message to [log in to unmask] containing the line UNSUBSCRIBE SOCNET in the body of the message.```