## Abstract

Betweenness centrality is generally regarded as a measure of others' dependence on a given node, and therefore as a measure of potential control. Closeness centrality is usually interpreted either as a measure of access efficiency or of independence from potential control by intermediaries. Betweenness and closeness are commonly assumed to be related for two reasons: first, because of their conceptual duality with respect to dependency, and second, because both are defined in terms of shortest paths. We show that the first of these ideas - the duality - is not only true in a general conceptual sense but also in precise mathematical terms. This becomes apparent when the two indices are expressed in terms of a shared dyadic dependency relation. We also show that the second idea - the shortest paths - is false because it is not preserved when the indices are generalized using the standard definition of shortest paths in valued graphs. This unveils that closeness-as-independence is in fact different from closeness-as-efficiency, and we propose a variant notion of distance that maintains the duality of closeness-as-independence with betweenness also on valued relations.

Original language | English |
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Pages (from-to) | 153-159 |

Number of pages | 7 |

Journal | Social Networks |

Volume | 44 |

DOIs | |

State | Published - Jan 1 2016 |

### Bibliographical note

Publisher Copyright:© 2015 Elsevier B.V.

### Funding

This research was partially supported by Deutsche Forschungsgemeinschaft (DFG) under grant Br 2158/6-1 .

Funders | Funder number |
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Deutsche Forschungsgemeinschaft | Br 2158/6-1 |

## Keywords

- Betweenness centrality
- Closeness centrality
- Dependency
- Derived relations
- Duality

## ASJC Scopus subject areas

- Anthropology
- Sociology and Political Science
- General Social Sciences
- General Psychology