'Advanced Network Analysis / Pajek: Large ... - Vladimir Batagelj

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V. Batagelj: Analysis and visulization of large networks with Pajek 53 ✬ ✫ Cores and generalized cores The notion of core was introduced by Seidman in 1983. Let G = (V, E) be a graph. A subgraph H = (W, E|W ) induced by the set W is a k-core or a core of order k iff ∀v ∈ W : deg H(v) ≥ k, and H is a maximal subgraph with this property. The core of maximum order is also called the main core. The core number of vertex v is the highest order of a core that contains this vertex. The degree deg(v) can be: in-degree, out-degree, in-degree + out-degree, etc., determining different types of cores. Methodenforum der Fakultät für Sozialwissenschaften, Universität Wien, 21-22. 6. 2007 ▲ ❙ ▲ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ✩ ✪
V. Batagelj: Analysis and visulization of large networks with Pajek 54 ✬ ✫ Properties of cores From the figure, representing 0, 1, 2 and 3 core, we can see the following properties of cores: • The cores are nested: i < j =⇒ Hj ⊆ Hi • Cores are not necessarily connected subgraphs. An efficient algorithm for determining the cores hierarchy is based on the following property: If from a given graph G = (V, E) we recursively delete all vertices, and edges incident with them, of degree less than k, the remaining graph is the k-core. For details see the paper. Methodenforum der Fakultät für Sozialwissenschaften, Universität Wien, 21-22. 6. 2007 ▲ ❙ ▲ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ✩ ✪
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'Advanced Network Analysis / Pajek: Large ... - Vladimir Batagelj

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