Generalization of the Ehrenfest urn model to a complex network

Abstract

The Ehrenfest urn model is extended to a complex directed network, over which a conserved quantity is transported in a random fashion. The evolution of the conserved number of packets in each urn, or node of the network, is illustrated by means of a stochastic simulation. Using mean-field theory we were able to compute an approximation to the ensemble-average evolution of the number of packets in each node which, in the thermodynamic limit, agrees quite well with the results of the stochastic simulation. Using this analytic approximation we are able to find the asymptotic dynamical state of the system and the time scale to approach the equilibrium state, for different networks. The study is extended to large scale-free and small-world networks, in which the relevance of the connectivity distribution and the topology of the network for the distribution of time scales of the system is apparent. This analysis may contribute to the understanding of the transport properties in real networks subject to a perturbation, e.g., the asymptotic state and the time scale required to approach it.

The Stochastic Transport Dynamics of a Conserved Quantity on a Complex Network

“…The “Ehrenfest urn” over a complex network, as generalized by Clark et al . 4 , describes the transport of N packets between the M nodes of a directed network. At a given time t a random packet, which is at a node i , is chosen to move to one of the k i nodes to which node i is connected, i.e., in its outgoing set .…”

“…Following the mean field approach proposed by Clark et al . 4 one assumes that evaluating an ensemble average evolution 〈 m i ( t )〉 of x i ( t ), in the thermodynamic limit, is equivalent to assume that all the N packets move to a new node in a time N , so that the evolution equation for the ensemble average is …”

“…Here we elaborate on a generalization of the Ehrenfest model by Clark et al . 4 , in which a number of urns are interconnected as a complex network, and the marbles or packets can only jump to urns to which the first one has a directed connection in a conservative manner.…”

“…The generalization of the “Ehrenfest urn” to a complex network, as proposed by Clark et al . 4 , is an example of the nontrivial transport that can occur on a complex network 38 – 41 , and should have similar properties to traffic in cities, electric networks, etc.…”

“…In this manuscript we construct the master equation that describes the evolution of the occupation number probability at each urn, and then compare the results with a stochastic simulation of the network of urns and the mean field approach proposed by 4 . The mean field theory approximation to the ensemble average evolution of the number of packets in each node agrees quite well with the results of the master equation, particularly in the thermodynamic limit.…”