General theory for extended-range percolation on simple and multiplex networks

TL;DR

This paper develops a comprehensive message passing framework to analyze extended-range percolation on simple and multiplex networks, revealing complex phase behaviors and robustness properties relevant for quantum communication.

Contribution

It introduces a general, exact theory for extended-range percolation applicable to arbitrary R on locally tree-like networks, including multiplex structures.

Findings

Extended-range percolation enhances robustness in multiplex networks.

Interdependencies increase fragility, leading to complex phase transitions.

Theoretical results match extensive Monte-Carlo simulations.

Abstract

Extended-range percolation is a robust percolation process that has relevance for quantum communication problems. In extended-range percolation nodes can be trusted or untrusted. Untrusted facilitator nodes are untrusted nodes that can still allow communication between trusted nodes if they lie on a path of distance at most R between two trusted nodes. In extended-range percolation the extended-range giant component (ERGC) includes trusted nodes connected by paths of trusted and untrusted facilitator nodes. Here, based on a message passing algorithm, we develop a general theory of extended-range percolation, valid for arbitrary values of R as long as the networks are locally tree-like. This general framework allows us to investigate the properties of extended-range percolation on interdependent multiplex networks. While the extended-range nature makes multiplex networks more robust, interdependency makes them more fragile. From the interplay between these two effects a rich phase diagram merges including discontinuous phase transitions and reentrant phases. The theoretical predictions are in excellent agreement with extensive Monte-Carlo simulations. The proposed exactly solvable model constitutes a fundamental reference for the study of models defined through properties of extended-range paths.