arXiv subjects

S. N. Dorogovtsev

Publications and source records attributed to S. N. Dorogovtsev.

At least 91 records · Page 5Linked to original sources

arXiv2001

Multifractal properties of growing networks

We introduce a new family of models for growing networks. In these networks new edges are attached preferentially to vertices with higher number of connections, and new vertices are created by already existing ones, inheriting part of their parent's connections. We show that combination of these two features produces multifractal degree distributions, where degree is the number of connections of a vertex. An exact multifractal distribution is found for a nontrivial model of this class. The distribution tends to a power-law one, $Π(q) \sim q^{-γ}$, $γ=\sqrt{2}$ in the infinite network limit. Nevertheless, for finite networks's sizes, because of multifractality, attempts to interpret the distribution as a scale-free would result in an ambiguous value of the exponent $γ$.

cond-mat.stat-mech↗

arXiv2001

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 Solé) 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.

cond-mat.stat-mech↗

arXiv2001

Giant strongly connected component of directed networks

We describe how to calculate the sizes of all giant connected components of a directed graph, including the {\em strongly} connected one. Just to the class of directed networks, in particular, belongs the World Wide Web. The results are obtained for graphs with statistically uncorrelated vertices and an arbitrary joint in,out-degree distribution $P(k_i,k_o)$. We show that if $P(k_i,k_o)$ does not factorize, the relative size of the giant strongly connected component deviates from the product of the relative sizes of the giant in- and out-components. The calculations of the relative sizes of all the giant components are demonstrated using the simplest examples. We explain that the giant strongly connected component may be less resilient to random damage than the giant weakly connected one.

cond-mat.stat-mech↗

arXiv2000

Scaling properties of scale-free evolving networks: Continuous approach

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.

cond-mat↗

arXiv2000

Generic scale of the "scale-free" growing networks

We show that the connectivity distributions $P(k,t)$ of scale-free growing networks ($t$ is the network size) have the generic scale -- the cut-off at $k_{cut} \sim t^β$. The scaling exponent $β$ is related to the exponent $γ$ of the connectivity distribution, $β=1/(γ-1)$. We propose the simplest model of scale-free growing networks and obtain the exact form of its connectivity distribution for any size of the network. We demonstrate that the trace of the initial conditions -- a hump at $k_h \sim k_{cut} \sim t^β$ -- may be found for any network size. We also show that there exists a natural boundary for the observation of the scale-free networks and explain why so few scale-free networks are observed in Nature.

cond-mat.stat-mech↗

arXiv2000

Growing network with heritable connectivity of nodes

We propose a model of a growing network, in which preferential linking is combined with partial inheritance of connectivity (number of incoming links) of individual nodes by new ones. The nontrivial version of this model is solved exactly in the limit of a large network size. We demonstrate, that the connectivity distribution depends on the network size, $t$, in a {\em multifractal} fashion. When the size of the network tends to infinity, the distribution behaves as $\sim q^{-γ}\ln q$, where $γ=\sqrt{2}$. For the finite-size network, this behavior is observed for $1 \ll q \lesssim \exp(\ln ^{1/2}t) $ but the multifractality is determined by the far wider part, $1 \ll q \lesssim \sqrt t$, of the distribution function.

cond-mat.stat-mech↗

arXiv2000

WWW and Internet models from 1955 till our days and the ``popularity is attractive'' principle

We note that the model discussed in the communication of S. Bornholdt and H. Ebel (World Wide Web scaling exponent from Simon's 1955 model, cond-mat/0008465) is the particular case of the model considered and solved exactly in our paper, cond-mat/0004434. These models may be used for estimation of the order of the deviation of the scaling exponent from 2 both for the distributions of incoming links and links coming out from nodes but not for the obtaining some specific values of the exponents from the WWW growth data. We emphasize that, unlike the statement of Bornholdt and Ebel, both the network under consideration and the model of Barabási and Albert provide quite equal possibilities for individual growth. There is no great difference between them in this respect. The resulting distributions for individual nodes and arising scaling relations have been obtained in our paper, cond-mat/0004434. We discuss briefly the modern state of art in the physics of the evolving networks and the great role of the general principle -- {\em popularity is attractive} -- in the self-organization of complex communications networks, in physics of nonequilibrium phenomena, and in Nature.

cond-mat.stat-mech↗

arXiv2000

Effect of the accelerating growth of communications networks on their structure

Motivated by data on the evolution of the Internet and World Wide Web we consider scenarios of self-organization of the nonlinearly growing networks into free-scale structures. We find that the accelerating growth of the networks establishes their structure. For the growing networks with preferential linking and increasing density of links, two scenarios are possible. In one of them, the value of the exponent $γ$ of the connectivity distribution is between 3/2 and 2. In the other, $γ>2$ and the distribution is necessarily non-stationary.

cond-mat.stat-mech↗

arXiv2000

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 $δ$ between the angle of marginal stability and the angle of repose of the granular material leads to a restricted value of the mixing time $τ$ 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 $δ$ is an integer part of $π$. Introduction of fluctuations of $δ$ supresses the singularities of $τ(δ)$ and smoothes the dependence $τ(δ)$.

cond-mat.stat-mech↗

arXiv2000

Anomalous Behavior of the Contact Process with Aging

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, $α=1$, 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 $0<α<1$, the absorbing-state phase transition is in the universality class of directed percolation.

cond-mat.stat-mech↗

arXiv2000

Scaling Behaviour of Developing and Decaying Networks

We find that a wide class of developing and decaying networks has scaling properties similar to those that were recently observed by Barabási and Albert in the particular case of growing networks. The networks considered here evolve according to the following rules: (i) Each instant a new site is added, the probability of its connection to old sites is proportional to their connectivities. (ii) In addition, (a) new links between some old sites appear with probability proportional to the product of their connectivities or (b) some links between old sites are removed with equal probability.

cond-mat↗

arXiv2000

Structure of Growing Networks: Exact Solution of the Barabasi--Albert's Model

We generalize the Barabási--Albert's model of growing networks accounting for initial properties of sites and find exactly the distribution of connectivities of the network $P(q)$ and the averaged connectivity $\bar{q}(s,t)$ of a site $s$ in the instant $t$ (one site is added per unit of time). At long times $P(q) \sim q^{-γ}$ at $q \to \infty$ and $\bar{q}(s,t) \sim (s/t)^{-β}$ at $s/t \to 0$, where the exponent $γ$ varies from 2 to $\infty$ depending on the initial attractiveness of sites. We show that the relation $β(γ-1)=1$ between the exponents is universal.

cond-mat↗

arXiv2000

Bak-Sneppen model near zero dimension

We consider the Bak--Sneppen model near zero dimension where the avalanche exponent $τ$ is close to 1 and the exponents $μ$ and $σ$ are close to 0. We demonstrate that $τ-1=μ-σ=\exp\{-μ^{-1}-γ+...\}$ in this limit, where $γ$ 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 $σ$ and $μ$ 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.

cond-mat.stat-mech↗

arXiv2000

Evolution of reference networks with aging

We study the growth of a reference network with aging of sites defined in the following way. 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 Barabási-Albert's model and (ii) to $τ^{-α}$, where $τ$ is the age of the old site. We consider $α$ of any sign although reasonable values are $0 \leq α\leq \infty$. We find both from simulation and analytically that the network shows scaling behavior only in the region $α< 1$. When $α$ increases from $-\infty$ to 0, the exponent $γ$ of the distribution of connectivities ($P(k) \propto k^{-γ}$ for large $k$) grows from 2 to the value for the network without aging, i.e. to 3 for the Barabási-Albert's model. The following increase of $α$ to 1 makes $γ$ to grow to $\infty$. For $α>1$ the distribution $P(k)$ is exponentional, and the network has a chain structure.

cond-mat↗

arXiv1999

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 $θ$ of the pile reaches its final critical value ($θ_f$) only at long times after a complicated relaxation process. The deviation ($θ_f - θ$) 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.

cond-mat.mtrl-sci↗

arXiv1999

Exactly solvable analogy of small-world networks

We present an exact description of a crossover between two different regimes of simple analogies of small-world networks. Each of the sites chosen with a probability $p$ from $n$ sites of an ordered system defined on a circle is connected to all other sites selected in such a way. Every link is of a unit length. Thus, while $p$ changes from 0 to 1, an averaged shortest distance between a pair of sites changes from $\bar{\ell} \sim n$ to $\bar{\ell} = 1$. We find the distribution of the shortest distances $P(\ell)$ and obtain a scaling form of $\bar{\ell}(p,n)$. In spite of the simplicity of the models under consideration, the results appear to be surprisingly close to those obtained numerically for usual small-world networks.

cond-mat.stat-mech↗

arXiv1999

How sandpiles spill: Sandpile problem in a thick flow regime

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 $θ$ of the pile reaches it's final critical value ($θ_f$) from lower values only at long times. The deviation ($θ_f - θ$) behaves as $(t/r_{0})^{-1/2}$.

cond-mat.mtrl-sci↗

arXiv1998

Avalanche mixing of granular solids in a rotating 2D drum: diffusion and fractionality

The dynamics of the avalanche mixing in a slowly rotated 2D upright drum is studied in the situation where the difference $δ$ between the angle of marginal stability and the angle of repose of the granular material is finite. An analytical solution of the problem is found for a half filled drum, that is the most interesting case. The mixing is described by a simple linear difference equation. We show that the mixing looks like linear diffusion of fractions under consideration with the diffusion coefficient vanishing when $δ$ is an integer part of $π$. The characteristic mixing time tends to infinity in these points. A full dependence of the mixing time on $δ$ is calculated and predictions for an experiment are made.

cond-mat↗