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Steady state properties in the absorbing phase of the $1d$ pair contact process (PCP) model are investigated. It is shown that, in typical absorbing states (reached by the system's dynamic rules), the density of isolated particles, $\rho_1$, approaches a stationary value which depends on the annihilation probability ($p$); the deviation from its 'natural' value at criticality, $\rho_1^{nat}$, follows a power law: $\rho_1^{nat}-\rho_1 \sim (p-p_c)^{\beta_1}$ for $p>p_c$. Monte Carlo simulations yield $\beta_1=0.81$. A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting $\beta_1=1$. The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach., Comment: 4 pages revtex (twocolumn, psfig), 3 EPS figures

The effect of power-law aging on a contact process is studied by simulation and using a mean-field approach. We find that the system may approach its stationary state in a nontrivial, nonmonotonous way. For the particular value of the aging exponent, $\alpha=1$, we observe a rich set of behaviors: depending on the process parameters, the relaxation to the stationary state proceeds as $1/\ln t$ or via a power law with a nonuniversal exponent. Simulation results suggest that for $0, Comment: 4 pages revtex (twocolumn, psfig), 3 figures

We study the growth of a network with aging of sites. Each new site of the network is connected to some old site with probability proportional (i) to the connectivity of the old site as in the Barab\'asi-Albert's model and (ii) to ${\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\alpha}},$ where $\ensuremath{\tau}$ is the age of the old site. We find both from simulation and analytically that the network shows scaling behavior only in the region $\ensuremath{\alpha}l1.$ When $\ensuremath{\alpha}$ increases from $\ensuremath{-}\ensuremath{\infty}$ to $0,$ the exponent $\ensuremath{\gamma}$ of the distribution of connectivities $[P(k)\ensuremath{\propto}{k}^{\ensuremath{-}\ensuremath{\gamma}}$ for large $k]$ grows from $2$ to the value for the network without aging. The ensuing increase of $\ensuremath{\alpha}$ to $1$ causes $\ensuremath{\gamma}$ to grow to $\ensuremath{\infty}.$ For $\ensuremath{\alpha}g1,$ the distribution $P(k)$ is exponentional.

We consider the Bak--Sneppen model near zero dimension where the avalanche exponent $\tau$ is close to 1 and the exponents $\mu$ and $\sigma$ are close to 0. We demonstrate that $\tau-1=\mu-\sigma=\exp\{-\mu^{-1}-\gamma+...\}$ in this limit, where $\gamma$ is the Euler's constant. The avalanche hierarchy equation is rewritten in a form that makes possible to find the relation between the critical exponents $\sigma$ and $\mu$ with high accuracy. We obtain new, more precise, values of the critical exponents for the 1D and 2D Bak-Sneppen model and for the 1D anisotropic Bak-Sneppen model., Comment: 4 pages revtex (twocolumn, epsfig), 1 figure. Submitted to Phys. Rev. E, Rapid Comm

We solve a one-dimensional sandpile problem analytically in a thick flow regime when the pile evolution may be described by a set of linear equations. We demonstrate that, if an income flow is constant, a space periodicity takes place while the sandpile evolves even for a pile of only one type of particles. Hence, grains are piling layer by layer. The thickness of the layers is proportional to the input flow of particles $r_0$ and coincides with the thickness of stratified layers in a two-component sandpile problem which were observed recently. We find that the surface angle $\theta$ of the pile reaches its final critical value ($\theta_f$) only at long times after a complicated relaxation process. The deviation ($\theta_f - \theta $) behaves asymptotically as $(t/r_{0})^{-1/2}$. It appears that the pile evolution depends on initial conditions. We consider two cases: (i) grains are absent at the initial moment, and (ii) there is already a pile with a critical slope initially. Although at long times the behavior appears to be similar in both cases, some differences are observed for the different initial conditions are observed. We show that the periodicity disappears if the input flow increases with time., Comment: 14 pages, 7 figures

The evolution of domain structure is investigated in a two-dimensional voter model with three states under cyclic dominance. The study focus on the dynamics of vortices, defined by the points where three states (domains) meet. We can distinguish vortices and antivortices which walk randomly and annihilate each other. The domain wall motion can create vortex-antivortex pairs at a rate which is increased by the spiral formation due to the cyclic dominance. This mechanism is contrasted with a branching annihilating random walk (BARW) in a particle antiparticle system with density dependent pair creation rate. Numerical estimates for the critical indices of the vortex density ($\beta=0.29(4)$) and of its fluctuation ($\gamma=0.34(6)$) improve an earlier Monte Carlo study [Tainaka and Itoh, Europhys. Lett. 15, 399 (1991)] of the three-state cyclic voter model in two dimensions., Comment: 5 pages, 6 figures, to appear in PRE

We study critical spreading in a surface-modified directed percolation model in which the left- and right-most sites have different occupation probabilities than in the bulk. As we vary the probability for growth at an edge, the critical exponents switch from the compact directed percolation class to ordinary directed percolation. We conclude that the nonuniversality observed in models with multiple absorbing configurations cannot be explained as a simple surface effect., Comment: 4 pages, Revtex, 5 figures postscript

Preferential attachment is a standard mechanism producing power-laws in growing networks. Thanks to its simplicity this mechanism is realized in most of the models of scale-free network, but, unfortunately, it rather mimics scale-free networks and not explains them. Optimization based mechanisms have a much greater potential to explain the evolution of scale-free networks. We consider one of the simplest optimization based models generating power-law growing networks. Our model is defined as follows. At each time step, a new node is created and connected to m previous nodes in the network, which are selected to minimize the product sαr, where s is the birth time of the node and r is a random number drawn from some distribution. In the case of complete optimization, the networks generated by this model have a power-law degree distribution with the exponent γ = 1 + 1/α for a wide range of the random number distributions. For partial optimization, including a finite fraction of nodes in a network, we observe an exponential degree distribution.

We analytically explore the scaling properties of a general class of nested subgraphs in complex networks, which includes the $K$-core and the $K$-scaffold, among others. We name such class of subgraphs $K$-nested subgraphs due to the fact that they generate families of subgraphs such that $...S_{K+1}({\cal G})\subseteq S_K({\cal G})\subseteq S_{K-1}({\cal G})...$. Using the so-called {\em configuration model} it is shown that any family of nested subgraphs over a network with diverging second moment and finite first moment has infinite elements (i.e. lacking a percolation threshold). Moreover, for a scale-free network with the above properties, we show that any nested family of subgraphs is self-similar by looking at the degree distribution. Both numerical simulations and real data are analyzed and display good agreement with our theoretical predictions., 6 pages, 4 figures

We examine the global organization of growing networks in which a new vertex is attached to already existing ones with a probability depending on their age. We find that the network is infinite- or finite-dimensional depending on whether the attachment probability decays slower or faster than $(age)^{-1}$. The network becomes one-dimensional when the attachment probability decays faster than $(age)^{-2}$. We describe structural characteristics of these phases and transitions between them., 5 pages

We consider the Potts model with competing two-step interactions and spin values 1, 2, 3 on a semi-infinite Cayley tree of order 2. We describe ground states and verify the Peierls condition for the model. Using a contour argument we show the existence of three different Gibbs measures associated with translation invariant ground states.

We propose and study a model where, for the first time, two aspects are present: parity conservation and infinitely many absorbing states. Whereas steady-state simulations show that the static critical behaviour is not affected by the presence of multiple absorbing configurations, the influence of the initial state associated with the presence of slowly decaying memory effects is clearly displayed in time dependent simulations. We report results of a detailed investigation of the dependence of critical spreading exponents on the initial particle density., Comment: 4 pages, 3 figures.ps

We have investigated the effect of the initial condition on the spreading exponents of the one-dimensional pair contact process (PCP) and threshold transfer process (TTP).The non-order field was found to exhibit critical fluctuations, relaxing to its natural value with the same power-law as the order parameter field. We argue that this slow relaxation, which was not taken into account in earlier studies of these models, is responsible for the continuously changing survival probability exponent. High precision numerical simulations show evidence of a(slight) dependence of the location of the transition point on the initial concentration, in the case of PCP. The damage spreading (DS) point and the spreading exponents coincide with those of the ordinary critical point in both cases., 11 pages, 11 figures (included)

Scaling behavior of scale-free evolving networks arising in communications, citations, collaborations, etc. areas is studied. We derive universal scaling relations describing properties of such networks and indicate limits of their validity. We show that main properties of scale-free evolving networks may be described in frames of a simple continuous approach. The simplest models of networks, which growth is determined by a mechanism of preferential linking, are used. We consider different forms of this preference and demonstrate that the range of types of preference linking producing scale-free networks is wide. We obtain also scaling relations for networks with nonlinear, accelerating growth and describe temporal evolution of arising distributions. Size-effects - cut-offs of these distributions - implement restrictions for observation of power-law dependences. The main characteristic of interest is so-called degree distribution, i.e., distribution of a number of connections of nodes. A scaling form of the distribution of links between pairs of individual nodes for the growing network of citations is also studied. We describe effects that produce difference of nodes. ``Aging'' of nodes changes exponents of distributions. Appearence of a single ``strong'' node changes dramatically the degree distribution of a network. If its strength exceeds some threshold value, the strong node captures a finite part of all links of a network. We show that permanent random damage of a growing scale-free network - permanent deleting of some links - change radically values of the scaling exponents. We describe the arising rich phase diagram. Results of other types of permanent damage are described., 21 pages revtex (twocolumn), 9 figures

We investigate the kinetics of submonolayer epitaxial growth which is driven by a fixed flux of monomers onto a substrate. Adatoms diffuse on the surface, leading to irreversible aggregation of islands. We also account for the effective diffusion of islands, which originates from hopping processes of their constituent adatoms, on the kinetics. When the diffusivity of an island of mass k scales as k^{-mu}, the (mean-field) Smoluchowski rate equations predicts steady behavior for 0=1, a quasi-static approximation to the rate equations predicts slow continuous evolution in which the island density increases as ln t^{mu/2}. A more refined matched asymptotic expansion reveals unusual multiple-scale mass dependence for the island size distribution. Our theory also describes basic features of epitaxial growth in a more faithful model of growing circular islands. For epitaxial growth in an initial population of monomers and no external flux, a scaling approach predicts power-law island growth and a mass distribution with a behavior distinct from that of the non-zero flux system. Finally, we extend our results to one- and two-dimensional substrates. The physically-relevant latter case exhibits only logarithmic corrections compared to the mean-field predictions., 12 pages, 5 figures, RevTeX 2-column format to be submitted to Phys. Rev. B

We investigate submonolayer epitaxial growth with a fixed monomer flux and irreversible aggregation of adatom islands due to an effective island diffusion, with a diffusivity for an mass k island proportional to \({\kappa ^{ - \mu }}\). For \(0 \le \mu < 1\), there is a steady state, while for \(\mu \ge 1\), continuously evolving logarithmic islanding occurs in which the island density grows extremely slowly, as \({(lnt)^{\mu /2}}\). In the latter regime, the island size distribution exhibits complex, but universal, multiple-scale mass dependence which we account for theoretically.

We obtain an analytical solution of a one-dimensional sandpile problem in a thick flow regime, when it can be formulated in terms of linear equations. It is shown that a space periodicity takes place during the sandpile evolution even for a sandpile of only one type of particles. Similar periodicity was observed previo usly for many component sandpiles. Space periods are proportional to an input flow of particles $r_0$. We find that the surface angle $\theta$ of the pile reaches it's final critical value ($\theta_f$) from lower values only at long times. The deviation ($\theta_f - \theta$) behaves as $(t/r_{0})^{-1/2}$., Comment: 4 pages revtex (twocolumn), 3 figures