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We introduce a Generalized k-core (Gk-core) algorithm that generalizes leaf removal. This method progressively prunes networks to reveal a hierarchy of nested Gk-cores, offering new insights into network structure.

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Area of Science:

  • Network science
  • Statistical physics

Background:

  • The leaf removal algorithm is a standard method for analyzing network structures.
  • Generalizing this algorithm can provide deeper insights into complex network properties.

Purpose of the Study:

  • To introduce and define the Generalized k-core (Gk-core) and its decomposition.
  • To develop an analytical framework for Gk-core percolation in random networks.
  • To apply the Gk-core decomposition to real-world networks.

Main Methods:

  • Introduction of the k-leaf removal algorithm, a generalization of leaf removal.
  • Progressive removal of vertices with degree less than k and their neighbors.
  • Development of an analytical framework using the configuration model for arbitrary degree distributions.
  • Derivation of rate equations to confirm structural characteristics.

Main Results:

  • The k-leaf removal algorithm defines a Generalized k-core (Gk-core) subgraph.
  • A hierarchy of progressively nested Gk-cores is obtained by varying k.
  • An analytical framework for Gk-core percolation in undirected uncorrelated networks is presented.
  • The algorithm is validated through rate equations and applied to real-world networks.

Conclusions:

  • The Generalized k-core decomposition provides a novel method for analyzing network structure.
  • The framework accurately describes Gk-core percolation for various network types.
  • The algorithm is effective for dissecting the hierarchical organization of real-world networks.