## FANDOM

33.175 Seiten

 Typus KomplettPlagiat Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 112, Zeilen: 1-27
Quelle: Holmgren 2006
Seite(n): 956-957, Zeilen: p.956, right column 28-45 - p.957, left column 1-18
[Recent studies show that several complex networks have a] heterogeneous topology, i.e., some vertices have a very large number of edges, but the majority of the vertices only have a few edges. That is, the degree distribution follows a power law $P(k)\cup k^{-\gamma}$ for large k(i.e., $P(k)/ k^{-\gamma} \to 1$ when $k\to \infty$). The average degree (k) of a graph with N vertices and M edges is (k) = 2M / N.

3.4.2 Clustering Coefficient

Many complex networks exhibit an inherent tendency to cluster. In social networks this represents a circles of friends in which every member knows each other. The clustering coefficient is a local property capturing “the density” of triangles in a graph, i.e., two vertices that both are connected to a third vertex are also directly connected to each other. An $i^{th}$ vertex in a network has $k_i$ edges that connects it to $k_i$ other vertices. The maximum possible number of edges between the $k_i$ neighbours is ${k_i \choose 2}=k_i (k_i-1) / 2.$ The clustering coefficient of $i^{th}$ vertex is defined as the ratio between the number $M_i$ of edges that actually exist between these $k_i$ vertices and the maximum possible number of edges, i.e., $C_i = 2 M_i / k_i(k_i-1).$ The clustering coefficient of the whole network

$C = (1/N)\sum_{i=1}^{n}{\mathcal C}_{i} \sum_{i=1}^{n} C_i$

3.4.3 Average Path Length

The distance $l_{uv}$ between two vertices u and v is defined as the number of edges along the shortest path connecting them. The average path length $l = (l_{uv}) = [1 / N (N - 1)] \sum_{u\neq v\in V}l_{uv}$ is a measure of how a network is scattered. Sometimes, the diameter d of a graph is defined as the maximum path length between any two connected vertices in the graph. However, in other situations the concept diameter relate to the average path length, i.e., d = l.

[p. 956]

Recent studies show that several complex networks have a heterogeneous topology, i.e., some vertices have a very large number of edges, but the majority of the vertices only have a few edges. That is, the degree distribution follows a power law $P(k)\cup k^{-\gamma}$ for large k(i.e., $P(k)/ k^{-\gamma} \to 1$ when $k\to \infty$). The average degree (k) of a graph with N vertices and M edges is (k) = 2M/N.

2.2.2. Clustering Coefficient

Many complex networks exhibit an inherent tendency to cluster. In social networks this represents circles of friends in which every member knows each other. The clustering coefficient is a local property capturing “the density” of triangles in the graph, i.e., two vertices that both are connected to a third vertex are also directly connected to each other. A vertex i in the network has $k_i$ edges that connects it to

[p. 957]

$k_i$ other vertices. The maximum possible number of edges between the ki neighbors is ${k_i \choose 2}=k_i(k_i-1)/2.$ The clustering coefficient of vertex i is defined as the ratio between the number $M_i$ of edges that actually exist between these $k_i$ vertices and the maximum possible number of edges, i.e., $C_i = 2 M_i / k_i(k_i-1).$ The clustering coefficient of the whole network $C = (1/N)\sum_i {\mathcal C}_i.$

2.2.3. Average Path Length

The distance $l_{uv}$ between two vertices u and v is defined as the number of edges along the shortest path connecting them. The average path length $l = (l_{uv}) = [1 / N (N - 1)] \sum_{u\neq v\in V}l_{uv}$ is a measure of how the network is scattered. Sometimes, the diameter d of a graph is defined as the maximum path length between any two connected vertices in the graph. However, in other situations the concept diameter relate to the average path length, i.e., d = l.

 Anmerkungen The copying process continues with Nm introducing an unfortunate mistake in the formula for the clustering coefficient. Apart from this mistake, both texts are nearly identical. Sichter (Graf Isolan), Hindemith