Condensation transitions in a model for a directed network with weighted links

TL;DR

This paper introduces an exactly solvable model for weighted, directed networks that exhibits two types of condensation phenomena, with dynamics mapped onto a zero-range process, enabling precise analysis and generalizations.

Contribution

The paper presents a novel exactly solvable model for weighted directed networks with condensation phenomena, linked to zero-range processes, allowing detailed theoretical analysis.

Findings

Identification of two distinct condensation phases

Mapping network dynamics to zero-range processes

Theoretical conditions for condensation occurrence

Abstract

An exactly solvable model for the rewiring dynamics of weighted, directed networks is introduced. Simulations indicate that the model exhibits two types of condensation: (i) a phase in which, for each node, a finite fraction of its total out-strength condenses onto a single link; (ii) a phase in which a finite fraction of the total weight in the system is directed into a single node. A virtue of the model is that its dynamics can be mapped onto those of a zero-range process with many species of interacting particles -- an exactly solvable model of particles hopping between the sites of a lattice. This mapping, which is described in detail, guides the analysis of the steady state of the network model and leads to theoretical predictions for the conditions under which the different types of condensation may be observed. A further advantage of the mapping is that, by exploiting what is known about exactly solvable generalisations of the zero-range process, one can infer a number of generalisations of the network model and dynamics which remain exactly solvable.