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We investigate the multifractals of the normalized first passage time on one-dimensional small-world network with both reflecting and absorbing barriers. The multifractals is estimated from the distribution of the normalized first passage time charactrized by the random walk on the small-world network with three fractions of edges rewired randomly. Particularly, our estimate is the fractal dimension D_0 = 0.917, 0.926, 0.930 for lattice points L = 80 and a randomly rewired fraction p = 0.2. The numerical result is found to disappear multifractal properties in the regime p> p_c, where p_c is the critical rewired fraction.
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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.
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 $alpha$.
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.
We study the decay process for the reaction-diffusion process of three species on the small-world network. The decay process is manipulated from the deterministic rate equation of three species in the reaction-diffusion system. The particle density and the global reaction rate on a two dimensional small-world network adding new random links is discussed numerically, and the global reaction rate before and after the crossover is also found by means of the Monte Carlo simulation. The time-dependent global reaction rate scales as a power law with the scaling exponent 0.66 at early time regime while it scales with -0.50 at long time regime, in all four cases of the added probability $p=0.2-0.8$. Especially, our result presented is compared with the numerical calculation of regular networks.
A lattice of three-state stochastic phase-coupled oscillators introduced by Wood it et al. exhibits a phase transition at a critical value of the coupling parameter $a$, leading to stable global oscillations (GO). On a complete graph, upon further increase in $a$, the model exhibits an infinite-period (IP) phase transition, at which collective oscillations cease and discrete rotational ($C_3$) symmetry is broken. In the case of large negative values of the coupling, Escaff et al. discovered the stability of travelling-wave states with no global synchronization but with local order. Here, we verify the IP phase in systems with long-range coupling but of lower connectivity than a complete graph and show that even for large positive coupling, the system sometimes fails to reach global order. The ensuing travelling-wave state appears to be a metastable configuration whose birth and decay (into the previously described phases) are associated with the initial conditions and fluctuations.
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