Scale-free percolation

1 hour 2 mins, 916.11 MB, MPEG-4 Video 640x360, 29.97 fps, 44100 Hz, 1.97 Mbits/sec

Share this media item:

Embed this media item:

<iframe width="_width_" height="_height_" src="https://sms.cam.ac.uk/media/1936264/embed" frameborder="0" scrolling="no" allowfullscreen></iframe>

Choose size:

About this item

Available Formats

About this item

Description:	van der Hofstad, R (Technische Universität Eindhoven) Friday 20 March 2015, 10:00-11:00


Created:	2015-03-27 09:58
Collection:	Random Geometry
Publisher:	Isaac Newton Institute
Copyright:	van der Hofstad, R
Language:	eng (English)
Distribution:	World (downloadable)
Explicit content:	No
Aspect Ratio:	16:9
Screencast:	No
Bumper:	UCS Default
Trailer:	UCS Default


Abstract:	Co-authors: Mia Deijfen (Stockholm University), Gerard Hooghiemstra (Delft University of Technology) 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 Wx, 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−λWxWy/\|x−y\|α, where (a) λ is the percolation parameter, (b) \|x−y\| is the Euclidean distance between x and y, and (c) α 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(Wx>w) is regularly varying with exponent 1−τ for some τ>1. In this case, we see that the degrees are infinite a.s.\ when γ=α(τ−1)/d≤1 or α≤d, while the degrees have a power-law distribution with exponent γ when γ>1. Our main results describe phase transitions in the positivity of the percolation critical value and in the graph distances in the percolation cluster as γ varies. Our results interpolate between those proved in inhomogeneous random graphs, where a wealth of further results is known, and those in long-range percolation. We also discuss many open problems, inspired both by recent work on long-range percolation (i.e., Wx=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≠y).

Abstract:

Co-authors: Mia Deijfen (Stockholm University), Gerard Hooghiemstra (Delft University of Technology)

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 Wx, 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−λWxWy/|x−y|α, where (a) λ is the percolation parameter, (b) |x−y| is the Euclidean distance between x and y, and (c) α 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(Wx>w) is regularly varying with exponent 1−τ for some τ>1. In this case, we see that the degrees are infinite a.s.\ when γ=α(τ−1)/d≤1 or α≤d, while the degrees have a power-law distribution with exponent γ when γ>1. Our main results describe phase transitions in the positivity of the percolation critical value and in the graph distances in the percolation cluster as γ varies. Our results interpolate between those proved in inhomogeneous random graphs, where a wealth of further results is known, and those in long-range percolation. We also discuss many open problems, inspired both by recent work on long-range percolation (i.e., Wx=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≠y).

Available Formats

Format	Quality	Bitrate	Size
MPEG-4 Video *	640x360	1.97 Mbits/sec	916.11 MB	View	Download
WebM	640x360	1.24 Mbits/sec	579.53 MB	View	Download
iPod Video	480x270	529.79 kbits/sec	240.58 MB	View	Download
MP3	44100 Hz	249.76 kbits/sec	115.25 MB	Listen	Download
Auto	(Allows browser to choose a format it supports)

Streaming Media Service Upload

Scale-free percolation