Nm2/Fragment 437 09

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 Typus Verschleierung Bearbeiter Hindemith Gesichtet
Untersuchte Arbeit:
Seite: 437, Zeilen: 9-19
Quelle: Koschuetzki et al 2005
Seite(n): 29-30, Zeilen: 29: 26ff; 30: 1, 13-15
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