Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
ELSEVIER
Social Networks
November 2004
A measure of betweenness
centrality based on random walks
M.E.J. Newman
1DCenter
for the Study of Complex Systems, University of Michigan
Presented by Oleg Kolgushev
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Contents
• Introduction to Centrality of a graph node
• Random walk betweenness
– A current flow analogy
– Calculation of betweenness in random walks
• Examples and applications
– Simple graph example
– Correlation with other measures
– Example applications
• Conclusion and future work
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Centrality measures
•
Importance of vertices in a graph
– Roles played by people on social network
– Communication stability of a network
– Epidemiological application, etc.
• Degree (number of edges on a node)
• Closeness
Vertices A and B have high (shortest-path) betweenness in
this configuration, while vertex C does not.
• Betweenness
– Shortest-path betweenness
– Flow betweenness
– Random walk betweenness
• Power centrality
• Random walk centrality
• Information centrality
•
In calculations of flow betweenness, vertices A and B in
this configuration get high scores while vertex C not.
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Random Walk Betweenness
• Current flow analogy
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Simple Graph Examples
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Network Examples
The network of intermarriage relations between the 15th century Florentine
families studied by Padgett and Ansell (1993)
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Applications
The largest network component of
a sexual contacts of high-risk actors
in Colorado Springs, CO (Potterat et
al. 2002). The size of the vertices
indicates their random walk
betweenness. The pointed shaded
vertices are those for which the
random-walk betweenness is much
greater than shortest-path
betweenness (twice or more).
Scatter plots of the random-walk
betweenness of vertices in the network
above against vertex degree (left) and
standard shortest-path betweenness
(right). The dotted lines indicate the best
linear fits in each case, which have the
correlation coefficients indicated.
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Applications
The largest component of
the co-authorship network
of scientists working on
networks from Newman
and Park (2003).
Size of vertices represents
their random-walk
betweenness measure.
Vertices on a single path
from one part of the
network to another
(labeled “A”) get a high
score.
So, do those labeled “B”
however, even though
they lie on one of several
paths between different
parts of the network.
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
Presentation: A measure of betweenness centrality based on random walks M.E.J. Newman
Conclusions
• The measure of betweenness based on random walks counts all
paths between vertices and makes no assumptions of optimality
• It can be calculated using matrix inversion in O(n3)
• It correlates to other measures of centrality and gives more realistic
scores to vertices in mentioned applications
• Conclusion and future work
Computational Epidemiology Research Lab (CERL) - Department of Computer Science and Engineering - University of North Texas - 2011/02/21
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Random Walk Centrality