Variance and Covariance of Distributions on Graphs
| Autor: | Samuel Martin-Gutierrez, Renaud Lambiotte, Karel Devriendt |
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| Rok vydání: | 2022 |
| Předmět: |
FOS: Computer and information sciences
Physics - Physics and Society Computational Mathematics 60B99 05C12 05C69 90C35 05C82 05C85 Applied Mathematics Probability (math.PR) FOS: Mathematics FOS: Physical sciences Applications (stat.AP) Physics and Society (physics.soc-ph) Statistics - Applications Mathematics - Probability Theoretical Computer Science |
| Zdroj: | SIAM Review. 64:343-359 |
| ISSN: | 1095-7200 0036-1445 |
| DOI: | 10.1137/20m1361328 |
| Popis: | We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts. Comment: 38 pages, 9 figures |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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