Language as an Evolving Word Web

Human language can be described as a complex network of linked words. In such a treatment, each distinct word in language is a vertex of this web, and neighboring words in sentences are connected by edges. It was recently found (Ferrer and Sol\'e) that the distribution of the numbers of connections of words in such a network is of a peculiar form which includes two pronounced power-law regions. Here we treat language as a self-organizing network of interacting words. In the framework of this concept, we completely describe the observed Word Web structure without fitting.Comment: 4 pages revtex, 2 figure

Time of avalanche mixing of granular materials in a half filled rotated drum

The avalanche mixing of granular solids in a slowly rotated 2D upright drum is studied. We demonstrate that the account of the difference $\delta$ between the angle of marginal stability and the angle of repose of the granular material leads to a restricted value of the mixing time $\tau$ for a half filled drum. The process of mixing is described by a linear discrete difference equation. We show that the mixing looks like linear diffusion of fractions with the diffusion coefficient vanishing when $\delta$ is an integer part of $\pi$. Introduction of fluctuations of $\delta$ supresses the singularities of $\tau(\delta)$ and smoothes the dependence $\tau(\delta)$.Comment: 4 pages revtex (twocolumn, psfig), 2 figure

Evolution of a sandpile in a thick flow regime

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 figure

A parity conserving dimer model with infinitely many absorbing 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.p

Degree-dependent intervertex separation in complex networks

We study the mean length $\ell(k)$ of the shortest paths between a vertex of degree $k$ and other vertices in growing networks, where correlations are essential. In a number of deterministic scale-free networks we observe a power-law correction to a logarithmic dependence, $\ell(k) = A\ln [N/k^{(\gamma-1)/2}] - C k^{\gamma-1}/N + ...$ in a wide range of network sizes. Here $N$ is the number of vertices in the network, $\gamma$ is the degree distribution exponent, and the coefficients $A$ and $C$ depend on a network. We compare this law with a corresponding $\ell(k)$ dependence obtained for random scale-free networks growing through the preferential attachment mechanism. In stochastic and deterministic growing trees with an exponential degree distribution, we observe a linear dependence on degree, $\ell(k) \cong A\ln N - C k$. We compare our findings for growing networks with those for uncorrelated graphs.Comment: 8 pages, 3 figure