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We study the Kleinberg problem of navigation in Small World networks when the underlying lattice is stretched along a preferred direction. Extensive simulations confirm that maximally efficient navigation is attained when the length $r$ of long-range links is taken from the distribution $P({bf r})sim r^{-alpha}$, when the exponent $alpha$ is equal to 2, the dimension of the underlying lattice, regardless of the amount of anisotropy, but only in the limit of infinite lattice size, $Ltoinfty$. For finite size lattices we find an optimal $alpha(L)$ that depends strongly on $L$. The convergence to $alpha=2$ as $Ltoinfty$ shows interesting power-law dependence on the anisotropy strength.
We study bond percolation on several four-dimensional (4D) lattices, including the simple (hyper) cubic (SC), the SC with combinations of nearest neighbors and second nearest neighbors (SC-NN+2NN), the body-centered cubic (BCC), and the face-centered
We study bond percolation on the simple cubic (SC) lattice with various combinations of first, second, third, and fourth nearest-neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond threshol
The locations of multicritical points on many hierarchical lattices are numerically investigated by the renormalization group analysis. The results are compared with an analytical conjecture derived by using the duality, the gauge symmetry and the re
We investigate the critical properties of the Ising model in two dimensions on {it directed} small-world lattice with quenched connectivity disorder. The disordered system is simulated by applying the Monte Carlo update heat bath algorithm. We calcul
Chase-Escape is a simple stochastic model that describes a predator-prey interaction. In this model, there are two types of particles, red and blue. Red particles colonize adjacent empty sites at an exponential rate $lambda_{R}$, whereas blue particl