Random graphs are used as models for networks, in particular capturing their complexity by describing local and probabilistic rules according to which vertices are connected to each other. A quite famous model is long-range percolation in which the vertex set is either assumed to be Z^d or R^d and the probability of an edge between two arbitrary vertices asymptotically has a polynomial decay in their distances. The focus is in studying graph-theoretical properties of theaforementioned models, such as the behavior of the random walk on the graph and the shortest path between an arbitrary pair of two vertices. In particular, recent results on continuous long-range percolation are presented.
