We study diffusion with a bias toward a target node in networks. This problem is relevant to efficient routing strategies in emerging communication networks like optical networks. Bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node. We investigate the scaling of the mean first passage time (MFPT) with the size of the network. We find by using theoretical analysis and computer simulations that for random regular (RR) and Erdős-Rényi networks, there exists a threshold probability, p(th), such that for p p(th), the MFPT scales logarithmically with N. The threshold value p(th) of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes. An exact solution for every value of p is given for the scaling of the MFPT in RR networks. The regime transition is also observed for the second moment of the probability distribution function, the standard deviation. For the case of scale-free (SF) networks, we present analytical bounds and simulations results showing that the MFPT scales at most as lnN to a positive power for any finite bias, which means that in SF networks even a very small bias is considerably more efficient in comparison to unbiased walk.
We investigate in detail the self-intermediate scattering function (SISF) of a lattice fluid (interacting lattice gas) with attractive nearest-neighbor interparticle interactions at a temperature slightly above the critical one by means of Monte Carlo simulations. An analytical expression is suggested to reproduce the simulation data. This expression is the generalization of the hydrodynamic limit with the wave vector, the time-dependent tracer diffusion coefficient, and the lattice geometry factor, instead of the square of the wave vector. The tracer diffusion coefficient is given by its zero wave-vector limit multiplied by the exponent of a function that contains only one fitting parameter describing its wave-vector dependence. In order to represent the time dependence of the SISF and to understand the time scales of the lattice fluid relaxation processes, we use two-and three-exponential fitting functions. The relaxation times group in three well-separated regions around 10, 100, and 1000 Monte Carlo steps and show weak concentration dependence. The analytical expression can also be used to calculate the lattice fluid dynamical structure factor.
We study the problem of a particle or message that travels as a biased random walk towards a target node in a network in the presence of traps. The bias is represented as the probability p of the particle to travel along the shortest path to the target node. The efficiency of the transmission process is expressed through the fraction f(g) of particles that succeed to reach the target without being trapped. By relating f(g) with the number S of nodes visited before reaching the target, we first show that, for the unbiased random walk, f(g) is inversely proportional to both the concentration c of traps and the size N of the network. For the case of biased walks, a simple approximation of S provides an analytical solution that describes well the behavior of f(g), especially for p>0.5. Also, it is shown that for a given value of the bias p, when the concentration of traps is less than a threshold value equal to the inverse of the mean first passage time (MFPT) between two randomly chosen nodes of the network, the efficiency of transmission is unaffected by the presence of traps and almost all the particles arrive at the target. As a consequence, for a given concentration of traps, we can estimate the minimum bias that is needed to have unaffected transmission, especially in the case of random regular (RR), Erdős-Rényi (ER) and scale-free (SF) networks, where an exact expression (RR and ER) or an upper bound (SF) of the MFPT is known analytically. We also study analytically and numerically, the fraction f(g) of particles that reach the target on SF networks, where a single trap is placed on the highest degree node. For the unbiased random walk, we find that f(g)∼N(-1/(γ-1)), where γ is the power law exponent of the SF network.
We investigate the self-intermediate scattering function (SISF) in a three-dimensional (3D) cubic lattice fluid (interacting lattice gas) with attractive nearest-neighbor interparticle interactions at a temperature slightly above the critical one by means of Monte Carlo simulations. A special representation of SISF as an exponent of the mean tracer diffusion coefficient multiplied by the geometrical factor and time is considered to highlight memory effects that are included in time and wave-vector dependence of the diffusion coefficient. An analytical expression for the diffusion coefficient is suggested to reproduce the simulation data. It is shown that the particles' mean-square displacement is equal to the time integral of the diffusion coefficient. We make a comparison with the previously considered 2D system on a square lattice. The main difference with the two-dimensional case is that the time dependence of particular characteristics of the tracer diffusion coefficient in the 3D case cannot be described by exponentially decreasing functions, but requires using stretched exponentials with rather small values of exponents, of the order of 0.2. The hydrodynamic values of the tracer diffusion coefficient (in the limit of large times and small wave vectors) defined through SIFS simulation results agree well with the results of its direct determination by the mean-square displacement of the particles in the entire range of concentrations and temperatures.
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