The Manipulability of Centrality Measures An Axiomatic Approach - PowerPoint PPT Presentation

The Manipulability of Centrality Measures An Axiomatic Approach Tomek Wąs, Marcin Waniek, Talal Rahwan, Tomasz Michalak University of Warsaw, NYU Abu Dhabi

Motivation

Investigation of criminal networks

Investigation of criminal networks BOSS

Investigation of criminal networks

Investigation of criminal networks Centrality Degree Closeness Betweenness Eigenvector

Digression - Centrality measures Functions assigning value to nodes reflecting their importance

Digression - Centrality measures Functions assigning value to nodes 4 reflecting their 3 importance 2 Degree 2 4 The number of connections 4 4 1 1 1

Digression - Centrality measures Functions assigning value to nodes 1/16 reflecting their 1/18 importance Degree 1/22 1/15 1/19 The number of connections 1/17 1/16 Closeness 1 over the average 1/25 1/24 1/24 distance

Investigation of criminal networks Centrality Degree Closeness Betweenness Eigenvector

Investigation of criminal networks

Investigation of criminal networks

Investigation of criminal networks Which centrality is the hardest to manipulate? Degree Closeness Betweenness Eigenvector

Setting

Setting: Measure of Manipulability - graph distribution - evader node - centrality measure - action function

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

Setting: Graph Distribution a a a e b e b e b v v v d c d c d c

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

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

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

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

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

Setting: Action function allowed actions in graph a a a e b e b e b v v v d c d c d c

e.g.: All changes a a a e b e b e b v v v d c d c d c

e.g.: Remove Neighbors a a a e b e b e b v v v d c d c d c

e.g.: Add Between Neighbors a a a e b e b e b v v v d c d c d c

e.g.: Local changes a a a e b e b e b v v v d c d c d c

Setting: Measure of Manipulability Very easy to manipulate 1 - graph distribution - evader node - centrality measure Very hard to manipulate 0 - action function

AMAR Measure of Manipulability

Axiomatic Approach Axioms for Measure of Manipulability: - Unmanipulability - Full Manipulability - Weak Dominance - Redundant Action - Neutrality - Linearity - Normalisation

Axiomatic Approach Axioms for Measure of Manipulability: - Unmanipulability If it is certain that it is - Full Manipulability impossible to hide the - Weak Dominance evader with any subset - Redundant Action of allowed actions, then manipulability is equal to - Neutrality 0 - Linearity - Normalisation

Axiomatic Approach Axioms for Measure of Manipulability: - Unmanipulability If it is certain that any - Full Manipulability subset of actions that hides the evader - Weak Dominance according to one centrality - Redundant Action measure, hides it also - Neutrality according to the other, then the latter measure is - Linearity at least as manipulable as - Normalisation the former

Axiomatic Approach Axioms for Measure of Manipulability: - Unmanipulability - Full Manipulability Main Theorem: - Weak Dominance A measure of manipulability satisfies all seven axioms - Redundant Action if and only if it is the AMAR - Neutrality Measure of Manipulability - Linearity - Normalisation

MAR measure MAR (Minimal Actions Required) = 1 over the smallest number of actions that hides the evader or 0 if it is impossible to hide

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

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

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

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

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

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

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

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

AMAR measure Averaged Minimal Actions Required

Evaluation

Evaluation of AMAR 4 Centralities: - Degree - Closeness - Betweenness - Eigenvector

Evaluation of AMAR 4 Graph Distributions: 4 Centralities: - Random Graphs - Degree - Small-World - Closeness - Betweenness - Preferential Attachment - Eigenvector - Cellular Networks

Evaluation of AMAR 4 Centralities: 4 Graph Distributions: 4 Action functions: - Degree - Random Graphs - All changes - Closeness - Small-World - Remove neighbours - Betweenness - Preferential Attachment - Add between - Eigenvector - Cellular Networks neighbors - Local changes

Evaluation of AMAR Random Graphs - Erdős-Rényi model All changes Remove neighbours Add between neighbors Local changes

Evaluation of AMAR Small-world networks - Watts-Strogatz model All changes Remove neighbours Add between neighbors Local changes

Evaluation of AMAR Preferential attachment networks - Barabási-Albert model All changes Remove neighbours Add between neighbors Local changes

Evaluation of AMAR Cellular networks (Tsvetovat and Carley, 2005) All changes Remove neighbours Add between neighbors Local changes

Summary

Summary Manipulation of Centrality measures

Summary Manipulation of Centrality measures AMAR = Averaged Minimal Actions a e b Required v d c

Summary Manipulation of Centrality measures AMAR = Averaged Minimal Actions a e b Required v d c

Summary Manipulation of Centrality measures AMAR = Averaged Minimal Actions a e b Required v d c Thank you!