Self-organized criticality on complex networks : Sandpile model, scaling and universality

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Sandpile is a generic model to study self-organized criticality (SOC) which provides a general mechanism for the emergence of complex behavior in many physio-hemical processes. On the other hand, the topology of many interacting systems can be modeled by complex networks which are usually compact, highly disordered, and maximally heterogeneous structures. The situation gets more intriguing when a complex dynamical process like SOC occurs on the top of a complex network. In order to investigate such situations, various sandpile models have been developed on several complex networks. Starting from the regular lattice small world network (SWN) has been developed adding shortcuts with a certain density. Dissipative versions of both the deterministic and the stochastic sandpile models have been studied on SWN. The steady-state critical properties of these newly developed sandpile models are characterized studying distribution functions of various avalanche properties. Three regimes of SWNs are identified: regular lattice regime (low shortcuts density), small world regime (intermediate shortcuts density), and random network regime (high shortcuts density). In the regular lattice regime, the sandpile dynamics is characterized by the respective scaling behaviour that usually occur on the regular lattice such as, Bak, Tang, and Wiesenfeld (BTW)-type correlated scaling for the deterministic model and the stochastic Manna scaling for the stochastic model. Whereas, in the random network regime, the dynamics is characterized by mean-field scaling for both the models. Interestingly, on small world regime, both the scaling behavior are found to coexist. In SWN regime, it is possible to identify certain characteristic size, area or time of avalanches below which the avalanche properties follow usual scaling on regular lattice and above which they obey mean-field scaling. Novel scaling forms of such characteristic properties of avalanches are developed analyzing several geometrical quantities of the toppling surface associated with an avalanche.
Supervisor: S. B. Santra