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Network Science (Part II)
Daniel Martin KatzMichigan State University - College of Law
Quantitative Methods for Lawyers Class #24
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A Brief Word About Tables in Stata
Estimates Table (esttab is best as it easily has the R^2 values)
http://repec.org/bocode/e/estout/esttab.html
You have to Install this as it is not here by defaulthttp://repec.org/bocode/e/estout/installation.html
There are Several Ways To Go Here:
Might Have to Say “Estimates Table”
A Brief Word About Tables in Stata
Another Option is “Outreg”ssc install outreg
Remember you need to store the models first - so it knows what return to you
It is going to write the output to a file
http://www.kellogg.northwestern.edu/rc/stata-outreg.htmhttp://www.ats.ucla.edu/stat/stata/faq/outreg.htm
Some Tutorials (aka #hiveknowledge )
Random Graphs
Power Law networks
Generating Power Law Distributed Networks
Pseudocode for the growing power law networks:
Start with small number of nodes
add new vertices one by one
each new edge connects to an existing vertex in proportion to the number of edges that vertex already displays (i.e. preferentially attach)
Growing Power Law Distributed Networks
The previous pseudocode is not a unique solution
A variety of other growth dynamics are possible
In the simple case this is a system that extremely “sensitive to initial conditions”
upstarts who garner early advantage are able to extend their relative advantage in later periods
for example, imagine you receive a higher interest rate the more money you have “rich get richer”
Just To Preview The Application to Positive
Legal Theory ....
Power Laws Appear to be a Common Feature of Legal Systems
Katz, et al (2011)American Legal Academy
Katz & Stafford (2010)American Federal Judges
Geist (2009)Austrian Supreme Court
Smith (2007)U.S. Supreme Court
Smith (2007)U.S. Law Reviews
Post & Eisen (2000) NY Ct of Appeals
Smith (2007)U.S. Law Reviews
Some Additional Thoughts on the Question...
Back to Network Measures
Node Level Measures
Sociologists have long been interested in roles / positions that various nodes occupy with in networks
For example various centrality measures have been developed
Degree
Closeness
Here is a non-exhaustive List:
Betweenness
Hubs/Authorities
DegreeDegree is simply a count of the number of arcs (or edges) incident to a node
Here the nodes are sized by degree:
Degree as a measure of centrality
Please Calculate the “degree” of each of the nodes
Degree as a measure of centrality
ask yourself, in which case does “degree” appear to capture the most important actors?
Degree as a measure of centrality
what about here, does it capture the “center”?
Closeness Centrality
Closeness is based on the inverse of the distance of each actor to every other actor in the network
Closeness Formula:
Normalized Closeness Formula:
Closeness Centrality
Closeness Centrality
Betweenness Centrality
Idea is related to bridges, weak ties
This individual may serve an important function
Betweenness centrality counts the number of geodesic paths between i & k that actor j resides on
Betweenness Centrality
Betweenness centrality counts the number of geodesic paths between i & k that actor j resides on
Betweenness Centrality
Check these yourself:
gjk = the number of geodesics connecting j & k, and
gjk = the number that actor i is on
Note: there is also a normalized version of the formula
Betweenness Centrality
Betweenness is a very powerful concept
We will return when we discuss community detection in networks ... If you want to preview check out this paper:
Michelle Girvan & Mark Newman, Community structure in social and biological networks, Proc. Natl. Acad. Sci. USA 99, 7821–7826 (2002)
High Betweenness actors need not be actors that score high on other centrality measures (such as degree, etc.)
[see picture to the right]
Hubs and Authorities
The Hubs and Authorities Algorithm (HITS) was developed by Computer Scientist Jon Kleinberg
Similar to the Google “PageRank” Algorithm developed by Larry Page
Kleinberg is a MacArthur Fellow and has offered a number of major contributions
Hubs and Authorities
We are interested in BOTH:
to whom a webpage links
and
From whom it has received links
In Ranking a Webpage ...
Hubs and Authorities
Intuition --
If we are trying to rank a webpage having a link from the New York Times is more of than one from a random person’s blog
HITS offers a significant improvement over measuring degree as degree treats all connections as equally valuable
Hubs and Authorities
Relies upon ideas such as recursion
Measure who is important?
Measure who is important to who is important?
Measure who is important to who is important to who is important ?
Etc.
Hubs and Authorities
Hubs: Hubs are highly-valued lists for a given query
for example, a directory page from a major encyclopedia or paper that links to many different highly-linked pages would typically have a higher hub score than a page that links to relatively few other sources.
Authority: Authorities are highly endorsed answers to a query
A page that is particularly popular and linked by many different directories will typically have a higher authority score than a page that is unpopular.
Note: A Given WebPage could be both a hub and an authority
Hubs and Authorities
Hubs and Authorities has been used in a wide number of social science articles
There exists some variants of the Original HITS Algorithm
Here is the Original Article : Jon Kleinberg, Authoritative sources in a hyperlinked environment, Journal of the Association of Computing Machinery, 46 (5): 604–632 (1999).
Note: there is a 1998 edition as well
Calculating Centrality Measures
Thankfully, centrality measures, etc. need not be calculated by hand
Lots of software packages ... in increasing levels of difficulty ... left to right
Difference in functions, etc. across the packages
easy: acceptsmicrosoft excel files
Medium: requires the .net / .paj
file setup
Hard: has lots of features
(R or Python)
Daniel Martin Katz Eric Provins!
Introduction to Computing for Complex Systems (Session XVII)!
Access A Full Step By Step
Tutorial for Pajek
The Slides From My Intro to Computing
for Complex Systems@ UMich ICPSR
Access Using this Tab
http://computationallegalstudies.com/icpsr-class/
Network Analysis Software
Just Download Pajek and Use the Tutorial
You should download it to your personal machine
MAC Users Note: It is a PC only Program so you will need something like crossover or you will have to multiboot
http://pajek.imfm.si/doku.php?id=download
Advanced Network Science Topics
Community Detection
ERGM Models
Diffusion / Social Epidemiology
http://computationallegalstudies.com/2009/10/11/programming-dynamic-models-in-python/
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!"#$%&'$()*%+$,-$.%/012*$'3%%% Mason A. Porter, Jukka-Pekka Onnela and Peter J. Mucha. 2009. “Communities in Networks.” Notices of the American Mathematical Society 56: 1082-1166. ))Santo Forunato. 2010. “Community detection in graphs.” Physics Reports. 486: 75-174.)
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We Will Discuss This Later ...
In Both Citation and Social Networks -- Algorithm Choice Matters
Due to Time Constraints
Those Who Are Interested in
Applications to Law
Network Analysis and the Law
Daniel Martin KatzMichigan State University
College of Law
Michael J. Bommarito IICenter for Study of
Complex Systems
Jurix 2011 Tutorial @ Universität Wien
!
http://computationallegalstudies.com/network-analysis-and-law-
tutorial/
