Theoretical Statistics and Mathematics Unit, ISI Delhi
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 $P(W_x>w)$ 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 $\gamma>1$.
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 $x\neq y$).
This is joint work with Mia Deijfen and Gerard Hooghiemstra.