arXiv subjects
Publications and source records attributed to J. F. F. Mendes.
We propose a construction procedure which generates a wide class of random evolving networks with fat-tailed degree distributions and an arbitrary clustering. This procedure applies the stochastic transformations of edges, which can be used as the basis of a real space renormalization group for evolving networks.
We have studied the island size distribution and spatial correlation function of an island growth model under the effect of an elastic interaction of the form $1/r^{3}$. The mass distribution $P_n(t)$ that was obtained presents a pronounced peak that widens with the increase of the total coverage of the system, $θ$. The presence of this peak is an indication of the self-organization of the system, since it demonstrates that some sizes are more frequent than others. We have treated exactly the energy of the system using periodic boundary conditions which were used in the Monte-Carlo simulations. A discussion about the effect of different factors is presented.
We find the exact critical temperature $T_c$ of the nearest-neighbor ferromagnetic Ising model on an `equilibrium' random graph with an arbitrary degree distribution $P(k)$. We observe an anomalous behavior of the magnetization, magnetic susceptibility and specific heat, when $P(k)$ is fat-tailed, or, loosely speaking, when the fourth moment of the distribution diverges in infinite networks. When the second moment becomes divergent, $T_c$ approaches infinity, the phase transition is of infinite order, and size effect is anomalously strong.
In many real growing networks the mean number of connections per vertex increases with time. The Internet, the Word Wide Web, collaboration networks, and many others display this behavior. Such a growth can be called {\em accelerated}. We show that this acceleration influences distribution of connections and may determine the structure of a network. We discuss general consequences of the acceleration and demonstrate its features applying simple illustrating examples. In particular, we show that the accelerated growth fairly well explains the structure of the Word Web (the network of interacting words of human language). Also, we use the models of the accelerated growth of networks to describe a wealth condensation transition in evolving societies.
We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent $γ=1+\ln3/\ln2$. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with $γ$ in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For the large network ($\ln N \gg 1$) the distribution tends to a Gaussian of width $\sim \sqrt{\ln N}$ centered at $\bar{\ell} \sim \ln N$. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+γ$.
We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short -- a feature known as the ``small-world'' effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc.
We obtain the exact position of the percolation threshold in intentionally damaged scale-free networks.
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, $ρ_1$, approaches a stationary value which depends on the annihilation probability ($p$); the deviation from its 'natural' value at criticality, $ρ_1^{nat}$, follows a power law: $ρ_1^{nat}-ρ_1 \sim (p-p_c)^{β_1}$ for $p>p_c$. Monte Carlo simulations yield $β_1=0.81$. A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting $β_1=1$. The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach.
We describe the anomalous phase transition of the emergence of the giant connected component in scale-free networks growing under mechanism of preferential linking. We obtain exact results for the size of the giant connected component and the distribution of vertices among connected components. We show that all the derivatives of the giant connected component size $S$ over the rate $b$ of the emergence of new edges are zero at the percolation threshold $b_c$, and $S \propto \exp\{-d(γ)(b-b_c)^{-1/2}\}$, where the coefficient $d$ is a function of the degree distribution exponent $γ$. In the entire phase without the giant component, these networks are in a ``critical state'': the probability ${\cal P}(k)$ that a vertex belongs to a connected component of a size $k$ is of a power-law form. At the phase transition point, ${\cal P}(k) \sim 1/(k\ln k)^2$. In the phase with the giant component, ${\cal P}(k)$ has an exponential cutoff at $k_c \propto 1/S$. In the simplest particular case, we present exact results for growing exponential 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 $γ$.
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.
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.
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.
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.
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.
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.
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.
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 $τ(δ)$.