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 Autor D. Koschützki, K.A. Lehmann, L. Peeters, S. Richter, D. Tenfelde- Podehl, O. Zlotowski Titel Chapter 3 Centrality Indices Sammlung Network Analysis: Methodological Foundations Herausgeber Ulrik Brandes, Thomas Erlebach Ort Berlin Heidelberg Verlag Springer Jahr 2005 Seiten 16-61 ISBN 978-3-540-24979-5 ISSN 0302-9743 URL http://books.google.de/books?id=TTNhSm7HYrIC Literaturverz. no Fußnoten no Fragmente 3

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 Zuletzt bearbeitet: 2014-01-11 22:28:47 WiseWoman

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The degree centrality Cd(v) of a vertex v is simply defined as the degree d(v) of v if the considered graph is undirected. The degree centrality is, e.g., applicable whenever the graph represents something like a voting result. These networks represent a static situation and we are interested in the vertex that has the most direct votes or that can reach most other vertices directly. The degree centrality is a local measure, because the centrality value of a vertex is only determined by the number of its neighbors. The most simple centrality is the degree centrality cD(v) of a vertex v that is simply defined as the degree d(v) of v if the considered graph is undirected. [...] The degree centrality is, e.g., applicable whenever the graph represents something like a voting result. These networks represent a static situation and we are interested in the vertex that has the most direct votes or that can reach most other vertices directly. The degree centrality is a local measure, because the centrality value of a vertex is only determined by the number of its neighbors.
 Anmerkungen The source is not mentioned anywhere in the paper. Sichter (Hindemith), WiseWoman

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We denote the sum of the distances from a vertex u ∈ V to any other vertex in a graph G = (V,E) as the total distance $\sum_{u\in V}d(u,v)$, where d(u,v) is shortest [sic] distance between the nodes u and v The problem of finding an appropriate location can be solved by computing the set of vertices with minimum total distance.

In SNA literature, a centrality measure based on this concept is called closeness. The focus lies here, for example, on measuring the closeness of a person to all other people in the network. People with a small total distance are considered as most important as those with high total distance. The most commonly employed definition of closeness is the reciprocal of the total distance:

$C_{C}(u)=\frac{1}{\sum_{v\in V}d(u,v)}\qquad (2)$

$C_{C}(u)$ grows with decreasing total distance of u, therefore it is also known as a structural index.

We denote the sum of the distances from a vertex u ∈ V to any other vertex in a graph G = (V,E) as the total distance2 $\sum_{v\in V}d(u,v)$. The problem of finding an appropriate location can be solved by computing the set of vertices with minimum total distance. [...]

In social network analysis a centrality index based on this concept is called closeness. The focus lies here, for example, on measuring the closeness of a person to all other people in the network. People with a small total distance are considered as more important as those with a high total distance. [...] The most commonly employed definition of closeness is the reciprocal of the total distance

[page 23]

$C_{C}(u)=\frac{1}{\sum_{v\in V}d(u,v)}\qquad (3.2)$

In our sense this definition is a vertex centrality, since '$C_{C}(u$) grows with decreasing total distance of u and it is clearly a structural index.

2 In [273], Harary used the term status to describe a status of a person in an organization or a group. In the context of communication networks this sum is also called transmission number.

 Anmerkungen The source is not mentioned anywhere in the paper. Sichter (Hindemith), WiseWoman

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Let $\delta_{uw}(v)$ denotes the fraction of shortest paths between u and w that contain vertex v:

$\delta_{uw}(v)=\frac{\sigma_{uw}(v)}{\sigma_{uw}}\quad (3)$

where $\sigma_{uw}$ denotes the number of all shortest-paths between u and w. The ratio $\delta_{uw}(v)$ can be interpreted as the probability that vertex v is involved into any communication between u and w. Note, that the measure implicitly assumes that all communication is conducted along shortest paths. Then the betweenness centrality $C_{B}(v)$ of a vertex v is given by:

$C_{B}(v)=\sum_{u\neq v\in V}\sum_{w\neq v\in V}\delta_{uw}(v)\quad (4)$

Any pair of vertices u and w without any shortest path from u to w will add zero to the betweenness centrality of every other vertex in the network.

Let $\delta_{st}(v)$ denote the fraction of shortest paths between s and t that contain vertex v:

$\delta_{st}(v)=\frac{\sigma_{st}(v)}{\sigma_{st}}\quad (3.12)$

where $\sigma_{st}$ denotes the number of all shortest-path [sic] between s and t. Ratios $\delta_{st}(v)$ can be interpreted as the probability that vertex v is involved into any communication between s and t. Note, that the index implicitly assumes that all communication is conducted along shortest paths. Then the betweenness centrality $c_{B}(v)$ of a vertex v is given by:

[page 30]

$c_{B}(v)=\sum_{s\neq v\in V}\sum_{t\neq v\in V}\delta_{st}(v)\quad (3.13)$

[...]

Any pair of vertices s and t without any shortest path from s to t just will add zero to the betweenness centrality of every other vertex in the network.

 Anmerkungen The definitions given here are, of course, standard and don't require a citation. However, the interpreting and explaining text is taken from the source word for word. The source is not mentioned in the paper anywhere. Sichter (Hindemith), WiseWoman