## Fast algorithms for determining (generalized) core groups in social networks.(English)Zbl 1284.05252

Summary: The structure of a large network (graph) can often be revealed by partitioning it into smaller and possibly more dense sub-networks that are easier to handle. One of such decompositions is based on “$$k$$-cores”, proposed in 1983 by Seidman. Together with connectivity components, cores are one among few concepts that provide efficient decompositions of large graphs and networks. In this paper we propose an efficient algorithm for determining the cores decomposition of a given network with complexity $$\mathcal O(m)$$, where $$m$$ is the number of lines (edges or arcs). In the second part of the paper the classical concept of $$k$$-core is generalized in a way that uses a vertex property function instead of degree of a vertex. For local monotone vertex property functions the corresponding generalized cores can be determined in $$\mathcal O(m\cdot\max(\Delta,\log n))$$ time, where $$n$$ is the number of vertices and $$\Delta$$ is the maximum degree. Finally the proposed algorithms are illustrated by the analysis of a collaboration network in the field of computational geometry.

### MSC:

 05C82 Small world graphs, complex networks (graph-theoretic aspects) 05A18 Partitions of sets 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C85 Graph algorithms (graph-theoretic aspects) 68R10 Graph theory (including graph drawing) in computer science 68W40 Analysis of algorithms 91D30 Social networks; opinion dynamics

### Keywords:

core; large network; decomposition; graph algorithm

### Software:

LaNet-vi; Pajek; Pajek datasets
Full Text:

### References:

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