  Classical percolation theory underlies many processes of information transfer
along the links of a network. In these standard situations, the requirement for
two nodes to be able to communicate is the presence of at least one
uninterrupted path of nodes between them. In a variety of more recent data
transmission protocols, such as the communication of noisy data via
error-correcting repeaters, both in classical and quantum networks, the
requirement of an uninterrupted path is too strict: two nodes may be able to
communicate even if all paths between them have interruptions/gaps consisting
of nodes that may corrupt the message. In such a case a different approach is
needed. We develop the theoretical framework for extended-range percolation in
networks, describing the fundamental connectivity properties relevant to such
models of information transfer. We obtain exact results, for any range $R$, for
infinite random uncorrelated networks and we provide a message-passing
formulation that works well in sparse real-world networks. The interplay of the
extended range and heterogeneity leads to novel critical behavior in scale-free
networks.

Statistical Mechanics