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

Pablo Medina,Jaime Clark,Juan Alejandro Valdivia,José Rogan,Felipe Torres,Miguel Kiwi

doi:10.1038/s41598-018-32677-8

Abstract

The stochastic dynamics of conserved quantities is an emergent phenomena observed in many complex systems, ranging from social and to biological networks. Using an extension of the Ehrenfest urn model on a complex network, over which a conserved quantity is transported in a random fashion, we study the dynamics of many elementary packets transported through the network by means of a master equation approach and compare with the mean field approximation and stochastic simulations. By use of the mean field theory, it is possible 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 master equation. However, the master equation gives a more complete description of the stochastic system and provides a probabilistic view of the occupation number at each node. Of particular relevance is the standard deviation of the occupation number at each node, which is not uniform for a complex network. We analyze and compare different network topologies (small world, scale free, Erdos-Renyi, among others). Given the computational complexity of directly evaluating the asymptotic, or equilibrium, occupation number probability distribution, we propose a scaling relation with the number of packets in the network, that allows to construct the asymptotic probability distributions from the network with one packet. The approximation, which relies on the same matrix found in the mean field approach, becomes increasingly more accurate for a large number of packets.

Highlights

One of the most interesting and fundamental problem in physics is related to the understanding of how the reversible microphysics gives rise to irreversible thermodynamics
Following the mean field approach proposed by Clark et al.[4] one assumes that evaluating an ensemble average evolution 〈mi(t)〉 of xi(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
We have generalized the Ehrenfest urn model to a complex network of urns, in which the packets or marbles move from node to node following the network connections

Summary

Introduction

One of the most interesting and fundamental problem in physics is related to the understanding of how the reversible microphysics gives rise to irreversible thermodynamics. We observe the evolution of the average number of packets at each node 〈ni(t)〉 obtained from the master equation and 〈mi(t)〉 obtained from the mean field approach.

Results

Conclusion