Quelle:Nm/Koschuetzki etal 2005

Angaben zur Quelle [Bearbeiten]

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

Fragmente der Quelle:

[1.] Nm/Fragment 101 20 - Diskussion
Zuletzt bearbeitet: 2012-04-29 22:07:36 Hindemith

Fragment, Gesichtet, KomplettPlagiat, Koschuetzki etal 2005, Nm, SMWFragment, Schutzlevel sysop

Typus: KomplettPlagiat

Bearbeiter: Hindemith

Gesichtet

Untersuchte Arbeit: Seite: 101, Zeilen: 20-26	Quelle: Koschuetzki_etal_2005 Seite(n): 20, Zeilen: 12-16
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

[2.] Nm/Fragment 103 15 - Diskussion
Zuletzt bearbeitet: 2012-04-29 22:07:40 Hindemith

Fragment, Gesichtet, Koschuetzki etal 2005, Nm, SMWFragment, Schutzlevel sysop, Verschleierung

Typus: Verschleierung

Bearbeiter: Hindemith

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Untersuchte Arbeit:
Seite: 103, Zeilen: 15-27

Quelle: Koschuetzki_etal_2005
Seite(n): 22-23, Zeilen: p22: 12ff; p23: 1-3

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 
  
          ∑
          
            v
            ∈
            V
          
        d
        (
        u
        ,
        v
        )
      
    {\displaystyle \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
        )
        =
        
          1
          
              ∑
              
                v
                ∈
                V
              
            d
            (
            u
            ,
            v
            )
          
        (
        2
        )
      
    {\displaystyle C_{C}(u)=\frac{1}{\sum_{v\in V}d(u,v)}\qquad (2)}
  
          C
          
            C
          
        (
        u
        )
      
    {\displaystyle 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] 
  
          ∑
          
            v
            ∈
            V
          
        d
        (
        u
        ,
        v
        )
      
    {\displaystyle \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
        )
        =
        
          1
          
              ∑
              
                v
                ∈
                V
              
            d
            (
            u
            ,
            v
            )
          
        (
        3.2
        )
      
    {\displaystyle 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

[3.] Nm/Fragment 104 12 - Diskussion
Zuletzt bearbeitet: 2012-05-19 14:06:21 Graf Isolan

Fragment, Gesichtet, Koschuetzki etal 2005, Nm, SMWFragment, Schutzlevel sysop, Verschleierung

Typus: Verschleierung

Bearbeiter: Hindemith, Graf Isolan

Gesichtet

Untersuchte Arbeit:
Seite: 104, Zeilen: 12-24

Quelle: Koschuetzki_etal_2005
Seite(n): 29-30, Zeilen: p29: 26ff; p30: 1, 13-15

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

          δ
          
            u
            w
          
        (
        v
        )
        =
        
              σ
              
                u
                w
              
            (
            v
            )
          
            σ
            
              u
              w
            
        (
        3
        )
      
    {\displaystyle \delta_{uw}(v)=\frac{\sigma_{uw}(v)}{\sigma_{uw}}\quad (3)}
  
where 
  
          σ
          
            u
            w
          
    {\displaystyle \sigma_{uw}}
  
 denotes the number of all shortest-paths between s and t.
The ratio 
  
          δ
          
            u
            w
          
        (
        v
        )
      
    {\displaystyle \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
        )
      
    {\displaystyle C_{B}(v)}
  
 of a vertex v is given by:

          C
          
            B
          
        (
        v
        )
        =
        
          ∑
          
            u
            ≠
            v
            ∈
            V
          
          ∑
          
            w
            ≠
            v
            ∈
            V
          
          δ
          
            u
            w
          
        (
        v
        )
        
        (
        4
        )
      
    {\displaystyle 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 
  
          δ
          
            s
            t
          
        (
        v
        )
      
    {\displaystyle \delta_{st}(v)}
  
 denote the fraction of shortest paths
between s and t that contain vertex v:

          δ
          
            s
            t
          
        (
        v
        )
        =
        
              σ
              
                s
                t
              
            (
            v
            )
          
            σ
            
              s
              t
            
        (
        3.12
        )
      
    {\displaystyle \delta_{st}(v)=\frac{\sigma_{st}(v)}{\sigma_{st}}\quad (3.12)}
  
where 
  
          σ
          
            s
            t
          
    {\displaystyle \sigma_{st}}
  
 denotes the number of all shortest-path between s and t. Ratios 
  
          δ
          
            s
            t
          
        (
        v
        )
      
    {\displaystyle \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
        )
      
    {\displaystyle c_{B}(v)}
  
 of a vertex v is given by:

          c
          
            B
          
        (
        v
        )
        =
        
          ∑
          
            s
            ≠
            v
            ∈
            V
          
          ∑
          
            t
            ≠
            v
            ∈
            V
          
          δ
          
            s
            t
          
        (
        v
        )
        
        (
        3.13
        )
      
    {\displaystyle 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