Fermi-Dirac statistics and traffic in complex networks

We propose an idealized model for traffic in a network, in which many particles move randomly from node to node, following the network's links, and it is assumed that at most one particle can occupy any given node. This is intended to mimic the finite forwarding capacity of nodes in communication networks, thereby allowing the possibility of congestion and jamming phenomena. We show that the particles behave like free fermions, with appropriately defined energy-level structure and temperature. The statistical properties of this system are thus given by the corresponding Fermi-Dirac distribution. We use this to obtain analytical expressions for dynamical quantities of interest, such as the mean occupation of each node and the transport efficiency, for different network topologies and particle densities. We show that the subnetwork of free nodes always fragments into small isolated clusters for a sufficiently large number of particles, implying a communication breakdown at some density for all network topologies. These results are compared to direct simulations.