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Feb 01, 2023 178 likes •756 views

The Manipulability of Centrality Measures An Axiomatic Approach Tomek Ws, Marcin Waniek, Talal Rahwan, Tomasz Michalak University of Warsaw, NYU Abu Dhabi Motivation Investigation of criminal networks Investigation of criminal networks

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SLIDE 1

The Manipulability of Centrality Measures

An Axiomatic Approach

Tomek Wąs, Marcin Waniek, Talal Rahwan, Tomasz Michalak University of Warsaw, NYU Abu Dhabi

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SLIDE 2

Motivation

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SLIDE 3

Investigation of criminal networks

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SLIDE 4

BOSS

Investigation of criminal networks

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SLIDE 5

Investigation of criminal networks

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SLIDE 6

Centrality

Investigation of criminal networks

Degree Closeness Betweenness Eigenvector

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SLIDE 7

Digression - Centrality measures

Functions assigning value to nodes reflecting their importance

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SLIDE 8

Digression - Centrality measures

Functions assigning value to nodes reflecting their importance The number of connections

Degree

1 1 1

2 2

3

4 4 4 4

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SLIDE 9

Digression - Centrality measures

Functions assigning value to nodes reflecting their importance The number of connections 1 over the average distance

Degree

1/24

1/25

1/19

1/22

1/18

1/16

1/15

1/17

1/16

Closeness

1/24

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SLIDE 10

Centrality

Investigation of criminal networks

Degree Closeness Betweenness Eigenvector

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SLIDE 11

Investigation of criminal networks

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SLIDE 12

Investigation of criminal networks

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SLIDE 13

Which centrality is the hardest to manipulate?

Investigation of criminal networks

Degree Closeness Betweenness Eigenvector

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SLIDE 14

Setting

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SLIDE 15

Setting: Measure of Manipulability

  • graph distribution
  • evader node
  • centrality measure
  • action function
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SLIDE 16

Setting: Graph Distribution

(B) Watts-Strogatz Small World Network (A) Erdős-Rényi Random Graphs (C) Barabási–Albert Preferential Attachment Network

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SLIDE 17

Setting: Graph Distribution

e b a d c v e b a d c v e b a d c v

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SLIDE 18

Setting: Evader & Actions

e b a d c v node Degree Closeness v I (4) I (1 / 6) a IV (2) IV (1 / 8) b II (3) II (1 / 7) c IV (2) VI (1 / 9) d IV (2) IV (1 / 8) e II (3) II (1 / 7)

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SLIDE 19

Setting: Evader & Actions

e b a d c v node Degree Closeness v I (4) I (1 / 6) a IV (2) IV (1 / 8) b II (3) II (1 / 7) c IV (2) VI (1 / 9) d IV (2) IV (1 / 8) e II (3) II (1 / 7)

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SLIDE 20

Setting: Evader & Actions

e b a d c v node Degree Closeness v I (4) I (1 / 6) a IV (2) IV (1 / 8) b II (3) II (1 / 7) c IV (2) VI (1 / 9) d IV (2) IV (1 / 8) e II (3) II (1 / 7)

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SLIDE 21

Setting: Evader & Actions

e b a d c v node Degree Closeness v I (4) I (1 / 6) a IV (2) IV (1 / 8) b II (3) II (1 / 7) c IV (2) VI (1 / 9) d IV (2) IV (1 / 8) e II (3) II (1 / 7)

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SLIDE 22

Setting: Evader & Actions

e b a d c v node Degree Closeness v II (3) II (1 / 7) a II (3) II (1 / 7) b I (4) I (1 / 6) c VI (1) VI (1 / 10) d V (2) V (1 / 9) e II (3) II (1 / 7)

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SLIDE 23

Setting: Action function

allowed actions in graph

e b a d c v e b a d c v e b a d c v

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SLIDE 24

e.g.: All changes

e b a d c v e b a d c v e b a d c v

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SLIDE 25

e.g.: Remove Neighbors

e b a d c v e b a d c v e b a d c v

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SLIDE 26

e.g.: Add Between Neighbors

e b a d c v e b a d c v e b a d c v

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SLIDE 27

e.g.: Local changes

e b a d c v e b a d c v e b a d c v

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SLIDE 28

Setting: Measure of Manipulability

1 Very easy to manipulate Very hard to manipulate

  • graph distribution
  • evader node
  • centrality measure
  • action function
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SLIDE 29

AMAR Measure of Manipulability

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SLIDE 30

Axiomatic Approach

Axioms for Measure of Manipulability:

  • Unmanipulability
  • Full Manipulability
  • Weak Dominance
  • Redundant Action
  • Neutrality
  • Linearity
  • Normalisation
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SLIDE 31

Axiomatic Approach

Axioms for Measure of Manipulability:

  • Unmanipulability
  • Full Manipulability
  • Weak Dominance
  • Redundant Action
  • Neutrality
  • Linearity
  • Normalisation

If it is certain that it is impossible to hide the evader with any subset

  • f allowed actions, then

manipulability is equal to

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SLIDE 32

Axiomatic Approach

Axioms for Measure of Manipulability:

  • Unmanipulability
  • Full Manipulability
  • Weak Dominance
  • Redundant Action
  • Neutrality
  • Linearity
  • Normalisation

If it is certain that any subset of actions that hides the evader according to one centrality measure, hides it also according to the other, then the latter measure is at least as manipulable as the former

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SLIDE 33

Axiomatic Approach

Axioms for Measure of Manipulability:

  • Unmanipulability
  • Full Manipulability
  • Weak Dominance
  • Redundant Action
  • Neutrality
  • Linearity
  • Normalisation

Main Theorem: A measure of manipulability satisfies all seven axioms if and only if it is the AMAR Measure of Manipulability

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SLIDE 34

MAR measure

MAR (Minimal Actions Required) = 1 over the smallest number of actions that hides the evader

  • r 0 if it is impossible to hide
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SLIDE 35

MAR measure

node Degree v I (4) a IV (2) b II (3) c IV (2) d IV (2) e II (3) e b a d c v

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SLIDE 36

MAR measure

node Degree v I (4) a IV (2) b II (3) c IV (2) d IV (2) e II (3) e b a d c v

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SLIDE 37

MAR measure

e b a d c v node Degree v I (4) a IV (2) b II (3) c IV (2) d IV (2) e II (3)

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SLIDE 38

Impact set

e b a d c v node Degree v III (2) a V (1) b I (3) c V (1) d III (2) e I (3)

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SLIDE 39

Impact set

e b a d c v node Degree v III (2) a V (1) b I (3) c V (1) d III (2) e I (3)

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SLIDE 40

MAR measure

e b a d c v node Degree Closeness v I (4) I (1 / 6) a IV (2) IV (1 / 8) b II (3) II (1 / 7) c IV (2) VI (1 / 9) d IV (2) IV (1 / 8) e II (3) II (1 / 7)

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SLIDE 41

MAR measure

e b a d c v node Degree Closeness v I (4) III (1 / 8) a IV (2) IV (1 / 8) b II (3) I (1 / 7) c IV (2) V (1 / 10) d IV (2) VI (1 / 11) e II (3) I (1 / 7)

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SLIDE 42

MAR measure

e b a d c v node Degree Closeness v I (4) III (1 / 8) a IV (2) IV (1 / 8) b II (3) I (1 / 7) c IV (2) V (1 / 10) d IV (2) VI (1 / 11) e II (3) I (1 / 7)

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SLIDE 43

AMAR measure Averaged Minimal Actions Required

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SLIDE 44

Evaluation

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SLIDE 45

Evaluation of AMAR 4 Centralities:

  • Degree
  • Closeness
  • Betweenness
  • Eigenvector
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SLIDE 46

Evaluation of AMAR

4 Centralities:

  • Degree
  • Closeness
  • Betweenness
  • Eigenvector

4 Graph Distributions:

  • Random Graphs
  • Small-World
  • Preferential Attachment
  • Cellular Networks
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SLIDE 47

Evaluation of AMAR

4 Centralities:

  • Degree
  • Closeness
  • Betweenness
  • Eigenvector

4 Graph Distributions:

  • Random Graphs
  • Small-World
  • Preferential Attachment
  • Cellular Networks

4 Action functions:

  • All changes
  • Remove

neighbours

  • Add between

neighbors

  • Local changes
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SLIDE 48

Evaluation of AMAR

All changes Remove neighbours Add between neighbors Local changes

Random Graphs - Erdős-Rényi model

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SLIDE 49

Evaluation of AMAR

All changes Remove neighbours Add between neighbors Local changes

Small-world networks - Watts-Strogatz model

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SLIDE 50

Evaluation of AMAR

All changes Remove neighbours Add between neighbors Local changes

Preferential attachment networks - Barabási-Albert model

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SLIDE 51

Evaluation of AMAR

All changes Remove neighbours Add between neighbors Local changes

Cellular networks (Tsvetovat and Carley, 2005)

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SLIDE 52

Summary

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SLIDE 53

Summary

Manipulation of Centrality measures

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SLIDE 54

Summary

AMAR = Averaged Minimal Actions Required

e b a d c v

Manipulation of Centrality measures

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SLIDE 55

Summary

AMAR = Averaged Minimal Actions Required

e b a d c v

Manipulation of Centrality measures

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SLIDE 56

Summary

AMAR = Averaged Minimal Actions Required

e b a d c v

Thank you! Manipulation of Centrality measures