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S. N. Dorogovtsev

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

At least 73 records · Page 4Linked to original sources
arXiv2003

Potts model on complex networks

We consider the general p-state Potts model on random networks with a given degree distribution (random Bethe lattices). We find the effect of the suppression of a first order phase transition in this model when the degree distribution of the network is fat-tailed, that is, in more precise terms, when the second moment of the distribution diverges. In this situation the transition is continuous and of infinite order, and size effect is anomalously strong. In particular, in the case of $p=1$, we arrive at the exact solution, which coincides with the known solution of the percolation problem on these networks.

cond-mat.stat-mech↗
arXiv2003

Clustering of correlated networks

We obtain the clustering coefficient, the degree-dependent local clustering, and the mean clustering of networks with arbitrary correlations between the degrees of the nearest-neighbor vertices. The resulting formulas allow one to determine the nature of the clustering of a network.

cond-mat.stat-mech↗
arXiv2003

Networks with desired correlations

We discuss a simple method of constructing correlated random networks, which was recently proposed by M. Bogu~n'a and R. Pastor-Satorras (cond-mat/0306072). The result of this construction procedure is a sparse network whose degree--degree distribution asymptotically approaches a given function at large degrees. We argue that this convergence is possible if the desired function is sufficiently slowly decreasing.

cond-mat.stat-mech↗
arXiv2003

Spectra of complex networks

We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the case of networks with correlations between neighboring vertices. The tail of the density of eigenvalues $ρ(λ)$ at large $|λ|$ is related to the behavior of the vertex degree distribution $P(k)$ at large $k$. In particular, as $P(k) \sim k^{-γ}$, $ρ(λ) \sim |λ|^{1-2γ}$. We propose a simple approximation, which enables us to calculate spectra of various graphs analytically. We analyse spectra of various complex networks and discuss the role of vertices of low degree. We show that spectra of locally tree-like random graphs may serve as a starting point in the analysis of spectral properties of real-world networks, e.g., of the Internet.

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arXiv2003

Renormalization group for evolving networks

We propose a renormalization group treatment of stochastically growing networks. As an example, we study percolation on growing scale-free networks in the framework of a real-space renormalization group approach. As a result, we find that the critical behavior of percolation on the growing networks differs from that in uncorrelated nets.

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arXiv2003

Critical phenomena in networks

We develop a phenomenological theory of critical phenomena in networks with an arbitrary distribution of connections $P(k)$. The theory shows that the critical behavior depends in a crucial way on the form of $P(k)$ and differs strongly from the standard mean-field behavior. The critical behavior observed in various networks is analyzed and found to be in agreement with the theory.

cond-mat.stat-mech↗
arXiv2002

Principles of statistical mechanics of random networks

We develop a statistical mechanics approach for random networks with uncorrelated vertices. We construct equilibrium statistical ensembles of such networks and obtain their partition functions and main characteristics. We find simple dynamical construction procedures that produce equilibrium uncorrelated random graphs with an arbitrary degree distribution. In particular, we show that in equilibrium uncorrelated networks, fat-tailed degree distributions may exist only starting from some critical average number of connections of a vertex, in a phase with a condensate of edges.

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arXiv2002

Metric structure of random networks

We propose a consistent approach to the statistics of the shortest paths in random graphs with a given degree distribution. This approach goes further than a usual tree ansatz and rigorously accounts for loops in a network. We calculate the distribution of shortest-path lengths (intervertex distances) in these networks and a number of related characteristics for the networks with various degree distributions. We show that in the large network limit this extremely narrow intervertex distance distribution has a finite width while the mean intervertex distance grows with the size of a network. The size dependence of the mean intervertex distance is discussed in various situations.

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arXiv2002

Mesoscopics and fluctuations in networks

We describe fluctuations in finite-size networks with a complex distribution of connections, $P(k)$. We show that the spectrum of fluctuations of the number of vertices with a given degree is Poissonian. These mesoscopic fluctuations are strong in the large-degree region, where $P(k) \lesssim 1/N$ ($N$ is the total number of vertices in a network), and are important in networks with fat-tailed degree distributions.

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arXiv2002

Modern architecture of random graphs: Constructions and correlations

1. Basic constructions. 2. Equilibrium and nonequilibrium networks. 3. Equilibrium uncorrelated networks. 4. Nonequilibrium nongrowing scale-free nets. 5. Types of correlations. 6. When pair correlations are important. 7. When loops are important. 8. Pair degree-degree correlations in growing networks. 9. How to construct an equilibrium net with given degree-degree correlations. 10. How to construct a growing scale-free net with a given clustering (towards a real-space renormalization group for networks).

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arXiv2002

How to construct a correlated net

(a) We propose a ``static'' construction procedure for random networks with given correlations of the degrees of the nearest-neighbor vertices. This is an equilibrium graph, maximally random under the constraint that its degree-degree distribution is fixed. (b) We generalize the notion of preferential linking and introduce a new category, {\em pair preference} and a pair preference function for the attaching of edges to pairs of vertices. This allows dynamically generate equilibrium correlated networks.

cond-mat.stat-mech↗
arXiv2002

How to generate a random growing network

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.

cond-mat.stat-mech↗
arXiv2002

Ising Model on Networks with an Arbitrary Distribution of Connections

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.

cond-mat.stat-mech↗
arXiv2002

Accelerated growth of networks

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.

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arXiv2001

Pseudofractal Scale-free Web

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 $2+γ$.

cond-mat.stat-mech↗
arXiv2001

Evolution of networks

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.

cond-mat.stat-mech↗
arXiv2001

Anomalous percolating properties of growing networks

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.

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