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8/12/2019 Between Presentation

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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
http://find/http://goback/


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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.



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Summary


Transportation

applications:e.g. routing

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Summary


Graph drawing



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Summary


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 )| .

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Summary


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 ) .

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Summary


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


M i i


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Summary


Deciency of previous approach

Overestimation of betweenness values of nodes near a pivot.

pivot

large overestimation small overestim a tion "fa lse ze ro"


Motivation Generalized Framework
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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"



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Summary


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 )}


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Summary


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




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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


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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


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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]


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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

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Motivationb


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Summary

The bisection scaling algorithm achieves the best results.

Future workefcient exact bisection scaling algorithm for st 2local searches to eliminate "false zeros"


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


E u c

l i d e a n

d i s t a n c e

Brandesbisection (unit)bisection (sh.path)linear


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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


% o

f p o s s

i b l e i n v e r s

i o n s


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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


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

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