Self Organization of Interacting Polya

We introduce a simple model which shows non-trivial self organized critical properties. The model describes a system of interacting units, modelled by Polya urns, subject to perturbations and which occasionally break down. Three equivalent formulations-stochastic, quenched and deterministic-are shown to reproduce the same dynamics. Among the novel features of the model are a non-homogeneous stationary state, the presence of a non-stationary critical phase and non-trivial exponents even in mean eld. We discuss simple interpretations in term of biological evolution and earthquake dynamics and we report on extensive numerical simulations in dimensions d = 1; 2 as well as in the random neighbors limit. Our understanding of Self Organized Criticality (SOC) 1], as a general framework for the emergence of scale-free behavior in Nature, has greatly beneetted from the introduction of simple models. Even though models such as the sandpile 1] and the Bak-Sneppen 2] are too simple to capture the complexity of natural phenomena such as earthquakess3] and biological evolutionn4,5], they have, nonetheless, identiied some basic mechanisms leading to SOC. These systems have been a starting point both for the development of more complex and realistic models of natural phenomenaa6,7], and for analytical approachess8{ 11], which have led us to a much deeper understanding of SOC. Indeed, we can now identify some basic \routes to Self Organized Criticality" such as those based on sand-pile 1], extremal dynamicss10,12], memoryy13] and network 14] models. In this Rapid Communication we propose a qualitatively diierent \route to SOC" based on a very simple model of interacting Polya urns. Its qualitative diierences with respect to other SOC models are that it is characterized by a non-homogeneous stationary state and by non trivial exponents even in the mean eld case. Furthermore, we shall show numerical evidence for the occurrence of a non-stationary self organized critical state. Moreover, the model can be formulated in three diierent but equivalent ways. This fact, on one hand allows us to use a wide variety of tools to investigate its critical properties, and on the other it bridges diierent descriptions of the same process. The model provides a general mechanism for the emergence of SOC which, as we shall discuss below, is relevant for the description of natural phenomena such as co-evolution and large scale earthquakes dynamics. Note indeed that the patterns of earthquakes activity are highly non-homogeneous and that such a system is, in principle, non-stationary. The same applies to …