arXiv subjects

A. N. Samukhin

Publications and source records attributed to A. N. Samukhin.

At least 19 recordsLinked to original sources

arXiv2017

Real space renormalization group for Ising Spin Glass and other glassy models. I. Disordered Ising model. General formalism

Here is the first part of the summary of my work on random Ising model using real-space renormalization group (RSRG), also known as a Migdal-Kadanoff one. This approximate renormalization scheme was applied to the analysis thermodynamic properties of the model, and of probabilistic properties of a pair correlator, which is a fluctuating object in disordered systems. PACS numbers: 02.50.-r, 05.20.-y, 05.70.Fh, 64.60.ae, 75.10.-b, 75.10.Hk, 87.10.+e Keywords: statistical physics, magnetism, spin glass, neural networks

cond-mat.dis-nn↗

arXiv2008

Organization of modular networks

We examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts--``communities'' or modules. We develop an analytical approach allowing us to obtain the statistics of connected components and an intervertex distance distribution in these modular networks, and to describe their global organization and structure. In particular, we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks. We demonstrate that even a relatively small number of shortcuts unite the networks into one. In more precise terms, if the number of the interlinks is any finite fraction of the total number of connections, then the intervertex distance distribution approaches a delta-function peaked form, and so the network is united.

cond-mat.stat-mech↗

arXiv2007

Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator's resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree $q_m$ of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum $λ_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, $λ_c>0$ if $q_m>2$, and $λ_c=0$ if $q_m\leq2$. In both these cases the density of eigenvalues $ρ(λ)\to0$ as $λ\toλ_c+0$, but the limiting behaviors near $λ_c$ are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density $ρ(λ)$ near $λ_c$ and the long-time asymptotics of the autocorrelator and the propagator.

cond-mat.stat-mech↗

arXiv2005

Organization of complex networks without multiple connections

We find a new structural feature of equilibrium complex random networks without multiple and self-connections. We show that if the number of connections is sufficiently high, these networks contain a core of highly interconnected vertices. The number of vertices in this core varies in the range between $const N^{1/2}$ and $const N^{2/3}$, where $N$ is the number of vertices in a network. At the birth point of the core, we obtain the size-dependent cut-off of the distribution of the number of connections and find that its position differs from earlier estimates.

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.

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.

cond-mat.stat-mech↗

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.

cond-mat.stat-mech↗

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.

cond-mat.stat-mech↗

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).

cond-mat.stat-mech↗

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↗

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.

cond-mat.stat-mech↗

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

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

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

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↗