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Proceedings Paper

Free zero-range processes on networks
Author(s): L. Bogacz; Z. Burda; W. Janke; B. Waclaw
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Paper Abstract

A free zero-range process (FRZP) is a simple stochastic process describing the dynamics of a gas of particles hopping between neighboring nodes of a network. We discuss three different cases of increasing complexity: (a) FZRP on a rigid geometry where the network is fixed during the process, (b) FZRP on a random graph chosen from a given ensemble of networks, (c) FZRP on a dynamical network whose topology continuously changes during the process in a way which depends on the current distribution of particles. The case (a) provides a very simple realization of the phenomenon of condensation which manifests as the appearance of a condensate of particles on the node with maximal degree. A particularly interesting example is the condensation on scalefree networks. Here we will model it by introducing a single-site inhomogeneity to a k-regular network. This simplified situation can be easily treated analytically and, on the other hand, shows quantitatively the same behavior as in the case of scale-free networks. The case (b) is very interesting since the averaging over typical ensembles of graphs acts as a kind of homogenization of the system which makes all nodes identical from the point of view of the FZRP. In effect, the partition function of the steady state becomes invariant with respect to the permutations of the particle occupation numbers. This type of symmetric systems has been intensively studied in the literature. In particular, they undergo a phase transition to the condensed phase, which is caused by a mechanism of spontaneous symmetry breaking. In the case (c), the distribution of particles and the dynamics of network are coupled to each other. The strength of this coupling depends on the ratio of two time scales: for changes of the topology and of the FZRP. We will discuss a specific example of that type of interaction and show that it leads to an interesting phase diagram. The case (b) mentioned above can be viewed as a limiting case where the typical time scale of topology fluctuations is much larger than that of the FZRP.

Paper Details

Date Published: 15 June 2007
PDF: 11 pages
Proc. SPIE 6601, Noise and Stochastics in Complex Systems and Finance, 66010V (15 June 2007); doi: 10.1117/12.726304
Show Author Affiliations
L. Bogacz, Jagellonian Univ. (Poland)
Z. Burda, Jagellonian Univ. (Poland)
W. Janke, Univ. Leipzig (Germany)
B. Waclaw, Jagellonian Univ. (Poland)
Univ. Leipzig (Germany)

Published in SPIE Proceedings Vol. 6601:
Noise and Stochastics in Complex Systems and Finance
János Kertész; Stefan Bornholdt; Rosario N. Mantegna, Editor(s)

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