Diameter of a long-range percolation graph for the critical exponent $s=d$

Many real-world networks exhibit the small-world phenomenon: their typical distances are much smaller than their sizes. A natural way to model this phenomenon is a long-range percolation graph on the lattice $Z^d$, in which edges are added between far-away vertices with probability falling off to the $s$-th power of the Euclidean distance. How does the resulting graph distance scale with the Euclidean distance? The question has been intensely studied in the past and the answer depends on the exponent $s$ in the connection probabilities, for which five regimes of behavior have been identified. In this talk I will give an overview on past results and then discuss our recent progress on the critical regime $s=d$. (Joint work with Marek Biskup.)