The Resistance Distance Is a Diffusion Distance on a Graph
The resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing for the analysis of networks. Here, we prove th...
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MDPI AG
2025-07-01
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| Series: | Mathematics |
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| author | Ernesto Estrada |
| author_facet | Ernesto Estrada |
| author_sort | Ernesto Estrada |
| collection | DOAJ |
| description | The resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing for the analysis of networks. Here, we prove that the resistance distance is given by a difference of “mass concentrations” obtained at the vertices of a graph by a diffusive process. The nature of this diffusive process is characterized here by means of an operator corresponding to the matrix logarithm of a Perron-like matrix based on the pseudoinverse of the graph Laplacian. We prove also that this operator is indeed the Laplacian matrix of a signed version of the original graph, in which nonnearest neighbors’ “interactions” are also considered. In this way, the resistance distance is part of a family of squared Euclidean distances emerging from diffusive dynamics on graphs. |
| format | Article |
| id | doaj-art-47a0acfae75e474bb6df1278d7a4adcc |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-47a0acfae75e474bb6df1278d7a4adcc2025-08-20T03:36:31ZengMDPI AGMathematics2227-73902025-07-011315238010.3390/math13152380The Resistance Distance Is a Diffusion Distance on a GraphErnesto Estrada0Institute for Cross-Disciplinary Physics and Complex Systems (IFISC), CSIC-UIB, 07122 Palma de Mallorca, SpainThe resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing for the analysis of networks. Here, we prove that the resistance distance is given by a difference of “mass concentrations” obtained at the vertices of a graph by a diffusive process. The nature of this diffusive process is characterized here by means of an operator corresponding to the matrix logarithm of a Perron-like matrix based on the pseudoinverse of the graph Laplacian. We prove also that this operator is indeed the Laplacian matrix of a signed version of the original graph, in which nonnearest neighbors’ “interactions” are also considered. In this way, the resistance distance is part of a family of squared Euclidean distances emerging from diffusive dynamics on graphs.https://www.mdpi.com/2227-7390/13/15/2380effective resistancegraph Laplaciandiffusion on graphsmatrix functionsmatrix logarithm |
| spellingShingle | Ernesto Estrada The Resistance Distance Is a Diffusion Distance on a Graph Mathematics effective resistance graph Laplacian diffusion on graphs matrix functions matrix logarithm |
| title | The Resistance Distance Is a Diffusion Distance on a Graph |
| title_full | The Resistance Distance Is a Diffusion Distance on a Graph |
| title_fullStr | The Resistance Distance Is a Diffusion Distance on a Graph |
| title_full_unstemmed | The Resistance Distance Is a Diffusion Distance on a Graph |
| title_short | The Resistance Distance Is a Diffusion Distance on a Graph |
| title_sort | resistance distance is a diffusion distance on a graph |
| topic | effective resistance graph Laplacian diffusion on graphs matrix functions matrix logarithm |
| url | https://www.mdpi.com/2227-7390/13/15/2380 |
| work_keys_str_mv | AT ernestoestrada theresistancedistanceisadiffusiondistanceonagraph AT ernestoestrada resistancedistanceisadiffusiondistanceonagraph |