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Scale-free percolation

Friday 20th March 2015 - 10:00 to 11:00
INI Seminar Room 1
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 $W_x$, 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 $1-{\mathrm{e}}^{-\lambda W_xW_y/|x-y|^{\alpha}}$, 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 $\mathbb{P}(W_x>w)$ is regularly varying with exponent $1-\tau$ for some $\tau>1$. In this case, we see that the degrees are infinite a.s.\ when $\gamma =\alpha(\tau-1)/d \leq 1$ or $\alpha\leq d$, while the degrees have a power-law distribution with exponent $\gamma$ when $\gamma>1$. Our main results describe phase transitions in the positivity of the percolation critical value and in the graph distances in the percolation cluster as $\gamma$ 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., $W_x=1$ 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 $x\neq y$).

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons