Transport processes on homogeneous planar graphs with scale-free loops

We consider the role of network geometry in two types of diffusion processes: transport of constant-density information packets with queuing on nodes, and constant voltage-driven tunneling of electrons. The underlying network is a homogeneous graph with scale-free distribution of loops, which is constrained to a planar geometry and fixed node connectivity k=3. We determine properties of noise, flow and return-times statistics for both the processes on this graph and relate the observed differences to the microscopic process details. Our main findings are: (i) through the local interaction between packets queuing at the same node, long-range correlations build up in traffic streams, which are practically absent in the case of electron transport; (ii) noise fluctuations in the number of packets and in the number of tunnelings recorded at each node appear to obey the scaling laws in two distinct universality classes; (iii) the topological inhomogeneity of betweenness plays the key role in the occurrence of broad distributions of return times and in the dynamic flow. The maximum-flow spanning trees are characteristic of each process type.