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SOCNET  November 2007

SOCNET November 2007

Subject:

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

From:

Phillip Bonacich <[log in to unmask]>

Reply-To:

Phillip Bonacich <[log in to unmask]>

Date:

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

Content-Type:

text/plain

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text/plain (97 lines)

*****  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]
-=-=-=-=-=-=-=-=-=-=-=-=-=-

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