Preferential behaviour and scaling in diffusive dynamics on networks

We study the fluctuation properties and return-time statistics on inhomogeneous scale-free networks using packets moving with two different dynamical rules; random diffusion (RD) and locally navigated diffusive motion with preferred edges. Scaling in the fluctuations of noise and flow occurs when the dispersion of a quantity at each node or edge increases like its mean to the power mu. We show that the occurrence of scaling in the fluctuations of the number of packets passing nodes and the number of packets flowing along edges is related to preferential behaviour in, respectively, the topology or the dynamics. Within our model, no preference leads to no scaling. When scaling does occur it is non-universal; for RD the number of packets passing a node scales with an exponent mu which increases continuously with increased acquisition time window from mu = 1/2 at small windows, to mu = 1 at long time windows. In preferentially navigated diffusive motion, busy nodes and edges have exponent mu = 1, in contrast to less busy parts of the network, where an exponent mu = 1/2 is found. Broad distributions of the return times at nodes and edges illustrate that the basis of the observed scaling is the cooperative behaviour between groups of nodes or edges. These conclusions are relevant for a large class of diffusive dynamics on networks, including packet transport with local navigation rules.