# Quelle:Nm/Koschuetzki etal 2005

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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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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 neighbours. 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 thesis. Note, that this paragraph can also be found in other publications of Nm: Memon, Larsen, Hicks & Harkiolakis (2008) and Memon, Hicks & Larsen (2007). Henrik Legind Larsen is the thesis supervisor and David L. Hicks is the thesis committee chairman. Sichter (Hindemith), Bummelchen

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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 $\sum_{v\in V}d(u,v)$.

The problem of finding an appropriate location can be solved by computing the set of vertices with a 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 more 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 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 distance [FN 2] $\sum_{v\in V}d(u,v)$. 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 cC(u) grows with decreasing total distance of u and it is clearly a structural index.

 Anmerkungen The source is not mentioned anywhere in the thesis Sichter (Hindemith), Bummelchen

 Zuletzt bearbeitet: 2012-05-19 14:06:21 Graf Isolan

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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 s and t. 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 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:

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

[...]

[...] 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 thesis anywhere. Telling mistake: indeed, in his definition Nm writes "where $\sigma_{uw}$ denotes the number of all shortest-paths between s and t.", thus mistakenly referring to the name of the nodes in the original text. Sichter (Hindemith), Bummelchen