Self Organization of Interacting Polya Urns

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 field. 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. PACS. 64.60.Ht Dynamic critical phenomena – 64.60.Lx Self-organized criticality; avalanche effect Our understanding of Self Organized Criticality (SOC) [1], as a general framework for the emergence of scale-free behavior in Nature, has greatly benefitted 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 earthquakes[3] and biological evolution[4,5], they have, nonetheless, identified 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 phenomena[6,7], and for analytical approaches[8,9,10,11], 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 sandpile [1], extremal dynamics[10,12], memory[13] and network [14] models. In this Rapid Communication we propose a qualitatively different “route to SOC” based on a very simple model of interacting Polya urns. Its qualitative differences 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 field 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 different 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 different descriptions of the same process. All these features can well be relevant in the description of natural phenomena. The model indeed provides a general framework for the emergence of SOC which, as we shall discuss, can be applied both to coevolution and to 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 our ecosystem, which is in a nonstationary state where ever fitter species replace less fit ones. We consider a system of interacting Polya urns arranged on a d-dimensional lattice. A Polya urn is a simple model to study e.g. the occurrence of accidents[16]. Each urn contains initially b black balls and 1 white one. As in sandpile models, at each time step we randomly select a site and attempt to add a “grain of sand”, i.e. a white ball, to the corresponding urn. A ball is drawn from the selected urn: If the ball is white the attempt is successful and a new white ball is added to the urn. If it is black a “fatal accident” occurs: The urn becomes unstable and it “topples”. The toppling mechanism is as follows: 1) the urn is reset to 1 white ball and b black ones and 2) for each white ball of the original urn a similar attempt to add a white ball is made on a randomly chosen nearest neighbor urn. In this way, white balls released by an unstable urn can provoke some “fatal accident” in nearby urns (addition of white ball to already unstable urns leaves them unstable but it increases the number of white balls in it). The process stops when all balls are redistributed provoking no further toppling. A new attempt to add a ball to a randomly chosen urn is made, at the next time-step, and the process goes on. Dissipation of balls at the boundary, as in the sandpile[1], can also be considered to allow the system to relax to a stationary state. Actually, in order to keep the same definition of the model both in finite dimensions and in the random neighbor version, we consider here “bulk dissipation” modifying step 2) into: 2’) with probability λ all white balls disappear, otherwise 2) applies. 2 Matteo Marsili, Angelo Valleriani: Self Organization of Interacting Polya Urns