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Register a new userStochastic sandpile model on small-world networks: scaling and crossover
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A dissipative stochastic sandpile model is constructed on one and two dimensional small-world networks with different shortcut densities $phi$ where $phi=0$ and $1$ represent a regular lattice and a random network respectively. In the small-world regime ($2^{-12} le phi le 0.1$), the critical behaviour of the model is explored studying different geometrical properties of the avalanches as a function of avalanche size $s$. For both the dimensions, three regions of $s$, separated by two crossover sizes $s_1$ and $s_2$ ($s_1<s_2$), are identified analyzing the scaling behaviour of average height and area of the toppling surface associated with an avalanche. It is found that avalanches of size $s<s_1$ are compact and follow Manna scaling on the regular lattice whereas the avalanches with size $s>s_1$ are sparse as they are on network and follow mean-field scaling. Coexistence of different scaling forms in the small-world regime leads to violation of usual finite-size scaling, in contrary to the fact that the model follows the same on the regular lattice as well as on the random network independently. Simultaneous appearance of multiple scaling forms are characterized by developing a coexistence scaling theory. As SWN evolves from regular lattice to random network, a crossover from diffusive to super-diffusive nature of sand transport is observed and scaling forms of such crossover is developed and verified.
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A dissipative sandpile model (DSM) is constructed and studied on small world networks (SWN). SWNs are generated adding extra links between two arbitrary sites of a two dimensional square lattice with different shortcut densities $phi$. Three different regimes are identified as regular lattice (RL) for $philesssim 2^{-12}$, SWN for $2^{-12}<phi< 0.1$ and random network (RN) for $phige 0.1$. In the RL regime, the sandpile dynamics is characterized by usual Bak, Tang, Weisenfeld (BTW) type correlated scaling whereas in the RN regime it is characterized by the mean field (MF) scaling. On SWN, both the scaling behaviors are found to coexist. Small compact avalanches below certain characteristic size $s_c$ are found to belong to the BTW universality class whereas large, sparse avalanches above $s_c$ are found to belong to the MF universality class. A scaling theory for the coexistence of two scaling forms on SWN is developed and numerically verified. Though finite size scaling (FSS) is not valid for DSM on RL as well as on SWN, it is found to be valid on RN for the same model. FSS on RN is appeared to be an outcome of super diffusive sand transport and uncorrelated toppling waves.
A dissipative stochastic sandpile model is constructed and studied on small world networks in one and two dimensions with different shortcut densities $phi$, where $phi=0$ represents regular lattice and $phi=1$ represents random network. The effect of dimension, network topology and specific dissipation mode (bulk or boundary) on the the steady state critical properties of non-dissipative and dissipative avalanches along with all avalanches are analyzed. Though the distributions of all avalanches and non-dissipative avalanches display stochastic scaling at $phi=0$ and mean-field scaling at $phi=1$, the dissipative avalanches display non trivial critical properties at $phi=0$ and $1$ in both one and two dimensions. In the small world regime ($2^{-12} le phi le 0.1$), the size distributions of different types of avalanches are found to exhibit more than one power law scaling with different scaling exponents around a crossover toppling size $s_c$. Stochastic scaling is found to occur for $s<s_c$ and the mean-field scaling is found to occur for $s>s_c$. As different scaling forms are found to coexist in a single probability distribution, a coexistence scaling theory on small world network is developed and numerically verified.
In the rotational sandpile model, either the clockwise or the anti-clockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anti-clockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behaviour of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anti-clockwise (or clockwise) rotational toppling rule. As the anti-clockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behaviour of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the non-existence of the Manna class.
We study the effective resistance of small-world resistor networks. Utilizing recent analytic results for the propagator of the Edwards-Wilkinson process on small-world networks, we obtain the asymptotic behavior of the disorder-averaged two-point resistance in the large system-size limit. We find that the small-world structure suppresses large network resistances: both the average resistance and its standard deviation approaches a finite value in the large system-size limit for any non-zero density of random links. We also consider a scenario where the link conductance decays as a power of the length of the random links, $l^{-alpha}$. In this case we find that the average effective system resistance diverges for any non-zero value of $alpha$.
In this paper we analyze the effect of a non-trivial topology on the dynamics of the so-called Naming Game, a recently introduced model which addresses the issue of how shared conventions emerge spontaneously in a population of agents. We consider in particular the small-world topology and study the convergence towards the global agreement as a function of the population size $N$ as well as of the parameter $p$ which sets the rate of rewiring leading to the small-world network. As long as $p gg 1/N$ there exists a crossover time scaling as $N/p^2$ which separates an early one-dimensional-like dynamics from a late stage mean-field-like behavior. At the beginning of the process, the local quasi one-dimensional topology induces a coarsening dynamics which allows for a minimization of the cognitive effort (memory) required to the agents. In the late stages, on the other hand, the mean-field like topology leads to a speed up of the convergence process with respect to the one-dimensional case.
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