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This study introduces a novel method using absorbing Markov chains to generalize betweenness and closeness centralities beyond shortest paths. This approach reveals complex changes in network node importance, particularly for nodes near community boundaries.

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Area of Science:

  • Network theory
  • Graph analysis
  • Computational mathematics

Background:

  • Node centrality measures like betweenness and closeness are crucial in network analysis.
  • These measures traditionally rely on shortest paths (geodesics).
  • Limitations exist in capturing node importance when considering non-shortest paths.

Purpose of the Study:

  • To develop a generalized method for betweenness and closeness centralities.
  • To interpolate these centralities from geodesic to unrestricted path considerations.
  • To analyze how node importance rankings change with path length restrictions.

Main Methods:

  • Utilizing absorbing Markov chains to model path exploration.
  • Developing a continuous interpolation framework for centrality measures.
  • Applying the method to analyze four real-world networks.

Main Results:

  • The interpolation method successfully bridges geodesic centralities to current-betweenness and resistance-closeness.
  • Node centrality rankings exhibit complex, parameter-dependent changes.
  • Nonmonotonic betweenness behavior is observed for nodes near intercommunity boundaries.

Conclusions:

  • The developed method offers a flexible way to assess node importance across different path considerations.
  • Findings highlight the nuanced role of nodes at community interfaces.
  • This generalization provides deeper insights into network structure and dynamics.