Exactly solvable models of adaptive networks

TL;DR

This paper introduces an exact analytical approach using the cavity method to study phase transitions in adaptive networks, revealing how link adaptation can delay SAT-UNSAT transitions and induce a self-organized intermediate phase.

Contribution

It provides the first exact description of the phase transitions in adaptive networks using the cavity method for large deviations.

Findings

Delay of SAT-UNSAT transition due to adaptation

Existence of an intermediate self-organized phase

Agreement between analytical results and numerical simulations

Abstract

A satisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is allowed to adapt by redistributing its links, the SAT-UNSAT transition may be delayed and preceded by an intermediate phase where the structure self-organizes to satisfy the constraints. We present an analytic approach, based on the recently introduced cavity method for large deviations, which exactly describes the two phase transitions delimiting this adaptive intermediate phase. We give explicit results for random bond models subject to the connectivity or rigidity percolation transitions, and compare them with numerical simulations.