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Generalized Erdős numbers (GENs) offer a novel way to measure node closeness in complex networks. This approach improves predictions for network dynamics like random walks and epidemic spread.

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Area of Science:

  • Network Science
  • Complex Systems Analysis
  • Graph Theory

Background:

  • Identifying relationships and central nodes is crucial in complex networks.
  • Existing measures may not fully capture nuanced pairwise relationships in weighted networks.

Purpose of the Study:

  • To introduce and evaluate Generalized Erdős Numbers (GENs) as a measure of topological proximity in weighted networks.
  • To develop new centrality measures based on GENs and assess their correlation with established metrics.
  • To demonstrate the utility of GENs in predicting network dynamics.

Main Methods:

  • Utilized Generalized Erdős Numbers (GENs) to quantify topological proximity.
  • Defined an importance matrix based on GENs and analyzed its leading eigenvector.
  • Developed personalized centrality measures derived from GENs.
  • Compared GENs-based predictions with existing measures for random walks and epidemic spreading.

Main Results:

  • GENs exhibit desirable properties like real-valuedness, non-locality, and asymmetry.
  • The square of the leading eigenvector of the GENs-based importance matrix strongly correlates with PageRank.
  • A personalized centrality measure based on GENs shows good correlation with other established measures.
  • GENs better predict asymmetries in random walk dynamics and mean infection time in epidemic spreading compared to other closeness measures.

Conclusions:

  • GENs provide a valuable and interpretable measure of node importance and topological proximity in weighted networks.
  • GENs-based centrality measures offer a promising alternative to existing methods.
  • The proposed GENs measure enhances the understanding and prediction of network dynamics, particularly in contexts involving resource sharing and spread phenomena.