Reactive random walkers on complex networks

We introduce and study a metapopulation model of random walkersinteracting at the nodes of a complex network. The model integratesrandom relocation moves over the links of the network with localinteractions depending on the node occupation probabilities. The modelis highly versatile, as the motion of the walkers depends on thetopological properties of the nodes, such as their degree, while anygeneral nonlinear function of the occupation probability of a node canbe considered as local reaction term. In addition to this, the relativestrength of reaction and relocation can be tuned at will, depending onthe specific application being examined. We derive an analyticalexpression for the occupation probability of the walkers at equilibriumin the most general case. We show that it depends on different orderderivatives of the local reaction functions, on the degree of a node,and on the average degree of its neighbors at various distances. Forsuch a reason, reactive random walkers are very sensitive to thestructure of a network and are a powerful way to detect networkproperties such as symmetries or degree-degree correlations. As possibleapplications, we first discuss how the occupation probability ofreactive random walkers can be used to define novel measures offunctional centrality for the nodes of a network. We then illustrate hownetwork components with the same symmetries can be revealed by trackingthe evolution of reactive walkers. Finally, we show that the dynamics ofour model is influenced by the presence of degree-degree correlations,so that assortative and disassortative networks can be classified byquantitative indicators based on reactive walkers.