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Centrality measures for node-weighted networks via line graphs and the matrix exponential

Abstract

This paper is concerned with the identification of important nodes in node-weighted graphs by applying matrix functions, in particular the matrix exponential. Many tools that use an adjacency matrix for a graph have been developed to study the importance of the nodes in unweighted or edge-weighted networks. However, adjacency matrices for node-weighted graphs have not received much attention. The present paper proposes using a line graph associated with a node-weighted graph to construct an edge-weighted graph that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node increases, the importance of this node in the graph increases as well, provided that the adjacency matrix is suitably scaled. Applications to real-life problems are presented.

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Acknowledgments

The authors would like to thank a referee for comments that lead to clarifications of the presentation.

Funding

This work was supported in part by NSF grants DMS-1729509 and DMS-1720259.

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Correspondence to Lothar Reichel.

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De la Cruz Cabrera, O., Matar, M. & Reichel, L. Centrality measures for node-weighted networks via line graphs and the matrix exponential. Numer Algor 88, 583–614 (2021). https://doi.org/10.1007/s11075-020-01050-0

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Keywords

  • Network analysis
  • Node weight
  • Node importance
  • Line graph
  • Matrix exponential