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Register a new userGeneralized model for dynamic percolation
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O.Benichou
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We study the dynamics of a carrier, which performs a biased motion under the influence of an external field E, in an environment which is modeled by dynamic percolation and created by hard-core particles. The particles move randomly on a simple cubic lattice, constrained by hard-core exclusion, and they spontaneously annihilate and re-appear at some prescribed rates. Using decoupling of the third-order correlation functions into the product of the pairwise carrier-particle correlations we determine the density profiles of the environment particles, as seen from the stationary moving carrier, and calculate its terminal velocity, V_c, as the function of the applied field and other system parameters. We find that for sufficiently small driving forces the force exerted on the carrier by the environment particles shows a viscous-like behavior. An analog Stokes formula for such dynamic percolative environments and the corresponding friction coefficient are derived. We show that the density profile of the environment particles is strongly inhomogeneous: In front of the stationary moving carrier the density is higher than the average density, $rho_s$, and approaches the average value as an exponential function of the distance from the carrier. Past the carrier the local density is lower than $rho_s$ and the relaxation towards $rho_s$ may proceed differently depending on whether the particles number is or is not explicitly conserved.
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A two parameter percolation model with nucleation and growth of finite clusters is developed taking the initial seed concentration rho and a growth parameter g as two tunable parameters. Percolation transition is determined by the final static configuration of spanning clusters. A finite size scaling theory for such transition is developed and numerically verified. The scaling functions are found to depend on both g and rho. The singularities at the critical growth probability gc of a given rho are described by appropriate critical exponents. The values of the critical exponents are found to be same as that of the original percolation at all values of rho at the respective gc . The model then belongs to the same universality class of percolation for the whole range of rho.
In J. Shao et al., PRL 103, 108701 (2009) the authors claim that a model with majority rule coarsening exhibits in d=2 a percolation transition in the universality class of invasion percolation with trapping. In the present comment we give compelling evidence, including high statistics simulations on much larger lattices, that this is not correct. and that the model is trivially in the ordinary percolation universality class.
Recent studies introduced biased (degree-dependent) edge percolation as a model for failures in real-life systems. In this work, such process is applied to networks consisting of two types of nodes with edges running only between nodes of unlike type. Such bipartite graphs appear in many social networks, for instance in affiliation networks and in sexual contact networks in which both types of nodes show the scale-free characteristic for the degree distribution. During the depreciation process, an edge between nodes with degrees k and q is retained with probability proportional to (kq)^(-alpha), where alpha is positive so that links between hubs are more prone to failure. The removal process is studied analytically by introducing a generating functions theory. We deduce exact self-consistent equations describing the system at a macroscopic level and discuss the percolation transition. Critical exponents are obtained by exploiting the Fortuin-Kasteleyn construction which provides a link between our model and a limit of the Potts model.
Biased (degree-dependent) percolation was recently shown to provide new strategies for turning robust networks fragile and vice versa. Here we present more detailed results for biased edge percolation on scale-free networks. We assume a network in which the probability for an edge between nodes $i$ and $j$ to be retained is proportional to $(k_ik_j)^{-alpha}$ with $k_i$ and $k_j$ the degrees of the nodes. We discuss two methods of network reconstruction, sequential and simultaneous, and investigate their properties by analytical and numerical means. The system is examined away from the percolation transition, where the size of the giant cluster is obtained, and close to the transition, where nonuniversal critical exponents are extracted using the generating functions method. The theory is found to agree quite well with simulations. By introducing an extension of the Fortuin-Kasteleyn construction, we find that biased percolation is well described by the $qto 1$ limit of the $q$-state Potts model with inhomogeneous couplings.
We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $Lambda$ by $ell$ bonds connecting the same adjacent vertices, thereby yielding the lattice $Lambda_ell$. This relation is used to calculate the bond percolation threshold on $Lambda_ell$. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality $d ge 2$ but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the $N to infty$ limits of several families of $N$-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as $N to infty$.
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