Extended-range percolation in complex networks

TL;DR

This paper develops a theoretical framework for extended-range percolation in complex networks, addressing communication scenarios where messages can be transmitted despite gaps, and reveals new critical behaviors in scale-free networks.

Contribution

It introduces a novel extended-range percolation model applicable to classical and quantum networks, with exact results for infinite networks and a message-passing approach for real-world networks.

Findings

Exact results for infinite uncorrelated networks.

New critical behavior in scale-free networks.

Effective message-passing formulation for sparse networks.

Abstract

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.