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MotivationOur Contributions
Summary
Better Approximation of BetweennessCentrality
Robert Geisberger Peter Sanders Dominik Schultes
Workshop on Algorithm Engineering & Experiments, 2008
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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8/12/2019 Between Presentation
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Motivation
Automatic analysis of networks requires fast computation ofcentrality indices.The networks grow faster than the speed of our computers sofast approximation algorithms gain importance.
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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8/12/2019 Between Presentation
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Transportation
applications:e.g. routing
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Graph drawing
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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8/12/2019 Between Presentation
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Denition Betweenness Centrality
Let G = ( V , E ) be a weighted directed (multi)-graph,SP st = set of shortest paths between source s and target t SP st (v ) = set of shortest paths that have v in their interior.
Then the betweenness centrality for node v is
c (v ) :=s , t V
st (v )st
, where st := |SP st | and st (v ) := |SP st (v )| .
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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8/12/2019 Between Presentation
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Exact algorithm
Brandes [Brandes01] exact algorithm:solve single source shortest path problem (SSSP) fromeach nodebackward aggregation of counter values
s v
w
w
s v
w
w
+ =s v
w
w
Time requirements:(nm ) for unit distance, otherwise nm + n 2 log (n ) .
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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8/12/2019 Between Presentation
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Approximation approach
Brandes and Pich [BrandesPich06] approximation algorithm:choose subset k of starting nodes ( pivots )solve only k single source shortest path problem (SSSP)extrapolate betweenness values
This yields an unbiased estimator for betweenness.
pivot
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
M i i
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MotivationOur Contributions
Summary
Computation of centrality indicesPrevious Work
Deciency of previous approach
Overestimation of betweenness values of nodes near a pivot.
pivot
large overestimation small overestim a tion "fa lse ze ro"
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Motivation Generalized Framework
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MotivationOur Contributions
Summary
Generalized FrameworkEfcient ImplementationExperiments
Main idea
Consider the length to the pivot to scale contributions.
pivot
large overestimation small overestim a tion "false ze ro"
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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Motivation Generalized Framework
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MotivationOur Contributions
Summary
Generalized FrameworkEfcient ImplementationExperiments
Generalized Framework (continuation)
For each shortest path of the formQ
s , . . . , v , . . . , t
P
we dene a scaled contribution
P (v ) :=f ( (Q )/ (P ))
st
Overall, v gets a contribution from a pivot s
s (v ) :=t V
{ P (v ) : P SP st (v )}
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Motivation Generalized Framework
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8/12/2019 Between Presentation
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MotivationOur Contributions
Summary
Generalized FrameworkEfcient ImplementationExperiments
Proposed Parameters
Length function :edge weight function used for shortest-path calculationunit distance
Scaling function f :Brandes and Pich constant
linear scaling f (x ) = x bisection scaling
f (x ) =0 for x [0 , 1/ 2)1 for x [1/ 2, 1]
0 1
0
1
0 1 0 1
Brandes and Pich linear scaling bisection scaling
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Motivation Generalized Framework
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Our ContributionsSummary
Efcient ImplementationExperiments
Linear Time Computation
Brandes [Brandes01] :compute st on the y during the shortest path calculationsubsequent aggregation phase, like exact algorithm
linear scaling:
Let st denote the shortest path distance from s to t ,aggregate 1 / st instead of 1, multiply with sv at the end.
s
t
tv
w
1
st
1
st
sv 1
sw+ 1
st+ 1
st
sw 1 st
+ 1 st
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Motivation Generalized Framework
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Our ContributionsSummary
Efcient ImplementationExperiments
Linear Time Computation of bisection scaling
use unit distancedepth rst traversal of shortest path DAG, keep an arraystoring the current path from s increment counter of current node v and decrement
counter of middle node v
s v v
0 d2
1 d
f
+1 1
Comments:only efcient for st {0 , 1}for st 2 sampling of shortest paths required
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Motivation Generalized Framework
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Our ContributionsSummary
Efcient ImplementationExperiments
Overview of used graphs
graph nodes edges sourceBelgian road network 463 514 596 119 PTV AGBelgian road network (unit dist.) 463 514 596 119 PTV AGActor co-starring network 392 400 16 557 451 [NotreD]
US patent network 3 774 769 16 518 947 [NBER]World-Wide-Web graph 325 729 1 497 135 [NotreD]CNR 2000 Webgraph 325 557 3 216 152 [LabWA]CiteSeer undir. citation network 268 495 2 313 294 [Citeseer]CiteSeer co-authorship network 227 320 1 628 268 [Citeseer]CiteSeer co-paper network 434 102 32 073 440 [Citeseer]DBLP co-authorship network 299 067 1 955 352 [DBLP]DBLP co-paper network 540 486 30 491 458 [DBLP]
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Motivation Generalized Framework
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Our ContributionsSummary
Efcient ImplementationExperiments
Belgium road network
0 . 5
1
2
3
0 . 5
1
2
3
16 32 64 128 256 512 1024 2048 4096 8192
pivots for Brandes' algorithm
% o
f p o s s
i b l e i n v e r s
i o n s
Brandesbisection (unit)bisection (sh.path)linear
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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Motivationb
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Our ContributionsSummary
Summary
The bisection scaling algorithm achieves the best results.
Future workefcient exact bisection scaling algorithm for st 2local searches to eliminate "false zeros"
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
A di Additi l E i t
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Appendix Additional Experiments
Belgian road network
1 0
4
1 0
3
1 0
2
1 0
4
1 0
3
1 0
2
16 32 64 128 256 512 1024 2048 4096 8192
pivots for Brandes' algorithm
E u c
l i d e a n
d i s t a n c e
Brandesbisection (unit)bisection (sh.path)linear
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Appendix Additional Experiments
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Appendix Additional Experiments
Belgian road network (Brandes and Pich)
2
3
2
3
24 25 26 27 28 29 210 211 212 213 214 215 216 217 218
pivots for Brandes' algorithm
% o
f p o s s
i b l e i n v e r s
i o n s
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
Appendix Additional Experiments
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Appendix Additional Experiments
Additional networks
1
5
2
1 0
2 0
1
5
2
1 0
2 0
16 32 64 128 256 512 1024 2048 4096 8192
pivots for Brandes' algorithm
c o m p a r e
i n v e r s
i o n
B r a n
d e s
/ b
. s a m p
l . ( 2 )
Belgium unitBelgiumDBLP copaperCiteSeer citationCiteSeer copaperDBLP coauthorCiteSeer coauthor
Robert Geisberger, Peter Sanders, Dominik Schultes Better Approximation of Betweenness Centrality
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