We propose and study a random graph model on the hypercubic lattice that interpolates between models of scale-free random graphs and long-range percolation. In our model, each vertex x has a weight W_x, where the weights of different vertices are i.i.d. random variables. Given the weights, the edge between x and y is, independently of all other edges, occupied with probability 1-e^{-\lambda W_xW_y/|x-y|^{\alpha}}, where
(a) \lambda is the percolation parameter,
(b) |x-y| is the Euclidean distance between x and y, and
(c) \alpha is a long-range parameter.
The most interesting behavior can be observed when the random weights have a power-law distribution, i.e., when P(W_x>w) is regularly varying with exponent 1-\tau for some \tau>1. In this case, we see that the degrees are infinite a.s. when \gamma =\alpha(\tau-1)/d <1 or \alpha\leq d, while the degrees have a power-law distribution with exponent \gamma when \gamma>1.
The main results describe phase transitions in the positivity of the percolation critical value and in the graph distances in the percolation cluster as \gamma varies. We discuss open problems, inspired both by work on long-range percolation (i.e., W_x=1 for every x), and on inhomogeneous random graphs (i.e., the model on the complete graph of size n and where |x-y|=n for every x\neq y).
We also discuss relations to recent results on geometric inhomogeneous random graphs, which is a close of models alike scale-free percolation, but then on a torus with a finite number of nodes.
[This is joint work with Mia Deijfen and Gerard Hooghiemstra.]
About the speaker
About the speaker
Remco van der Hofstad received his Ph.D. at the University of Utrecht in 1997, under the supervision of Frank den Hollander and Richard Gill. Since then, he worked at McMaster University in Hamilton, Canada, Microsoft Research, and Delft University of Technology. He is currently Full Professor in probability at Eindhoven University of Technology. Further, jointly with Frank den Hollander, he is responsible for the 'Random Spatial Structures' Program at Eurandom. Remco is also former acting scientific director of Eurandom. Remco works on statistical mechanics in high dimensions, on random graph models for complex networks and stochastic processes on them, and the interplay between these topics. He is further interested in exploring applications of probability in chemistry, electrical engineering and computer science with scientists from these fields. Remco received the Prix Henri Poincaré 2003 jointly with Gordon Slade, the Rollo Davidson Prize 2007, is a laureate of the 'Innovative Research VIDI Scheme' 2003 and 'Innovative Research VICI Scheme' 2008, and was elected in the Royal Netherlands Academy of Arts and Sciences in 2018. He is also one of the 11 co-applicants of the Dutch Gravitation program NETWORKS. This 23 M euro program, a collaboration between the universities of Amsterdam, Leiden and Eindhoven, and the Center for Mathematics and Computer Science in Amsterdam, aims at combining stochastics and algorithmics to solve network problems. Remco has lectured in most of the major summer schools in probability, including Saint-Flour 2017, Cornell 2012, PIMS-CRM in Montreal 2015, and has taught some 20 more shorter and longer courses throughout the world. He is author of some 150 papers and 2 books, with two more books in preparation. He has supervised 20 Ph.D. students, of whom 15 have completed their Ph.D., and 15 postdocs. Finally, Remco is editor in chief of the 'Network Pages', an interactive website by the networks community for everyone interested in networks.
Remco van der Hofstad received his Ph.D. at the University of Utrecht in 1997, under the supervision of Frank den Hollander and Richard Gill. Since then, he worked at McMaster University in Hamilton, Canada, Microsoft Research, and Delft University of Technology. He is currently Full Professor in probability at Eindhoven University of Technology. Further, jointly with Frank den Hollander, he is responsible for the 'Random Spatial Structures' Program at Eurandom. Remco is also former acting scientific director of Eurandom. Remco works on statistical mechanics in high dimensions, on random graph models for complex networks and stochastic processes on them, and the interplay between these topics. He is further interested in exploring applications of probability in chemistry, electrical engineering and computer science with scientists from these fields. Remco received the Prix Henri Poincaré 2003 jointly with Gordon Slade, the Rollo Davidson Prize 2007, is a laureate of the 'Innovative Research VIDI Scheme' 2003 and 'Innovative Research VICI Scheme' 2008, and was elected in the Royal Netherlands Academy of Arts and Sciences in 2018. He is also one of the 11 co-applicants of the Dutch Gravitation program NETWORKS. This 23 M euro program, a collaboration between the universities of Amsterdam, Leiden and Eindhoven, and the Center for Mathematics and Computer Science in Amsterdam, aims at combining stochastics and algorithmics to solve network problems. Remco has lectured in most of the major summer schools in probability, including Saint-Flour 2017, Cornell 2012, PIMS-CRM in Montreal 2015, and has taught some 20 more shorter and longer courses throughout the world. He is author of some 150 papers and 2 books, with two more books in preparation. He has supervised 20 Ph.D. students, of whom 15 have completed their Ph.D., and 15 postdocs. Finally, Remco is editor in chief of the 'Network Pages', an interactive website by the networks community for everyone interested in networks.
