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Weighted distances in scale free random graphs

De Julia Komjathy

Apparaît également dans la collection : Dynamics on random graphs and random maps / Dynamiques sur graphes et cartes aléatoires

In this talk I will review the recent developments on weighted distances in scale free random graphs as well as highlight key techniques used in the proofs. We consider graph models where the degree distribution follows a power-law such that the empirical variance of the degrees is infinite, such as the configuration model, geometric inhomogeneous random graphs, or scale free percolation. Once the graph is created according to the model definition, we assign i.i.d. positive edge weights to existing edges, and we are interested in the proper scaling and asymptotic distribution of weighted distances. In the infinite variance degree regime, a dichotomy can be observed in all these graph models: the edge weight distributions form two classes, explosive vs conservative weight distributions. When a distribution falls into the explosive class, typical distances converge in distribution to proper random variables. While, when a distribution falls into the conservative class, distances tend to infinity with the model size, according to a formula that captures the doubly-logarithmic graph distances as well as the precise behaviour of the distribution of edge-weights around the origin. An integrability condition decides into which class a given distribution falls. This is joint work with Adriaans, Baroni, van der Hofstad, and Lodewijks.

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.19228803
  • Citer cette vidéo Komjathy, Julia (24/10/2017). Weighted distances in scale free random graphs. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19228803
  • URL https://dx.doi.org/10.24350/CIRM.V.19228803

Bibliographie

  • Adriaans, E., & Komjathy, J. (2017). Weighted distances in scale-free configuration models. <arXiv:1709.09481> - https://arxiv.org/abs/1709.09481
  • Baroni, E., van der Hofstad, R., & Komjathy, J. (2016). Tight fluctuations of weight-distances in random graphs with infinite-variance degrees. <arXiv:1609.07269> - https://arxiv.org/abs/1609.07269

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