Biomedical subjects

S N Dorogovtsev

Publications and source records attributed to S N Dorogovtsev.

At least 19 recordsLinked to original sources

PubMed2006

Degree-dependent intervertex separation in complex networks.

We study the mean length (l)(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, (l)(k) = A ln[N/k((gamma-1)/2)]-Ck(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 (l)(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, (l)(k)approximately A ln N-Ck. We compare our findings for growing networks with those for uncorrelated graphs.

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PubMed2006

k-core (bootstrap) percolation on complex networks: critical phenomena and nonlocal effects.

We develop the theory of the -core (bootstrap) percolation on uncorrelated random networks with arbitrary degree distributions. We show that the -core percolation is an unusual, hybrid phase transition with a jump emergence of the k-core as at a first order phase transition but also with a critical singularity as at a continuous transition. We describe the properties of the -core, explain the meaning of the order parameter for the k-core percolation, and reveal the origin of the specific critical phenomena. We demonstrate that a so-called "corona" of the k-core plays a crucial role (corona is a subset of vertices in the k-core which have exactly neighbors in the -core). It turns out that the k-core percolation threshold is at the same time the percolation threshold of finite corona clusters. The mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point. We show that a random removal of even one vertex from the k-core may result in the collapse of a vast region of the k-core around the removed vertex. The mean size of this region diverges at the critical point. We find an exact mapping of the k-core percolation to a model of cooperative relaxation. This model undergoes critical relaxation with a divergent rate at some critical moment.

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PubMed2006

k-Core organization of complex networks.

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures--k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birthpoints--the bootstrap percolation thresholds. We show that in networks with a finite mean number zeta2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if zeta2 diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.

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PubMed2005

Correlations in interacting systems with a network topology.

We study pair correlations in interacting systems placed on complex networks. We show that usually in these systems, pair correlations between interacting objects (e.g., spins), separated by a distance l, decay, on average, faster than 1/(lzl). Here zl is the mean number of the lth nearest neighbors of a vertex in a network. This behavior, in particular, leads to a dramatic weakening of correlations between second and more distant neighbors on networks with fat-tailed degree distributions, which have a divergent number z2 in the infinite network limit. In large networks of this kind, only pair correlations between the nearest neighbors are actually observable. We find the pair correlation function of the Ising model on a complex network. This exact result is confirmed by a phenomenological approach.

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PubMed2005

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 x N1/2 and const x N2/3, where is the number of vertices in a network. At the birth point of the core, we obtain the size-dependent cutoff of the distribution of the number of connections and find that its position differs from earlier estimates.

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PubMed2005

Phase transition with the Berezinskii-Kosterlitz-Thouless singularity in the Ising model on a growing network.

We consider the ferromagnetic Ising model on a highly inhomogeneous network created by a growth process. We find that the phase transition in this system is characterized by the Berezinskii-Kosterlitz-Thouless singularity, although critical fluctuations are absent and the mean-field description is exact. Below this infinite order transition, the magnetization behaves as exp((-const/square root of(Tc-T)). We show that the critical point separates the phase with the power-law distribution of the linear response to a local field and the phase where this distribution rapidly decreases. We suggest that this phase transition occurs in a wide range of cooperative models with a strong infinite-range inhomogeneity.

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PubMed2005

Complex networks created by aggregation.

We study aggregation as a mechanism for the creation of complex networks. In this evolution process vertices merge together, which increases a number of highly connected hubs. We study a range of complex network architectures produced by the aggregation. Fat-tailed (in particular, scale-free) distributions of connections are obtained for both networks with a finite number of vertices and growing networks. We observe a strong variation of a network structure with growing density of connections and find the phase transition of the condensation of edges. Finally, we demonstrate the importance of structural correlations in these networks.

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PubMed2004

Self-organization of collaboration networks.

We study collaboration networks in terms of evolving, self-organizing bipartite graph models. We propose a model of a growing network, which combines preferential edge attachment with the bipartite structure, generic for collaboration networks. The model depends exclusively on basic properties of the network, such as the total number of collaborators and acts of collaboration, the mean size of collaborations, etc. The simplest model defined within this framework already allows us to describe many of the main topological characteristics (degree distribution, clustering coefficient, etc.) of one-mode projections of several real collaboration networks, without parameter fitting. We explain the observed dependence of the local clustering on degree and the degree-degree correlations in terms of the "aging" of collaborators and their physical impossibility to participate in an unlimited number of collaborations.

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PubMed2004

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.

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PubMed2003

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 treelike 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 rho(lambda) at large /lambda/ is related to the behavior of the vertex degree distribution P(k) at large k. In particular, as P(k) approximately k(-gamma), rho(lambda) approximately /lambda/(1-2 gamma). We propose a simple approximation, which enables us to calculate spectra of various graphs analytically. We analyze spectra of various complex networks and discuss the role of vertices of low degree. We show that spectra of locally treelike 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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PubMed2003

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 networks.

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PubMed2003

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) less, similar 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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PubMed2003

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 theory.

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PubMed2002

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.

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PubMed2002

Pseudofractal scale-free web.

We find that scale-free random networks are excellently modeled by simple deterministic graphs. Our graph has a discrete degree distribution (degree is the number of connections of a vertex), which is characterized by a power law with exponent gamma=1+ln 3/ln 2. Properties of this compact structure are surprisingly close to those of growing random scale-free networks with gamma 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 a large network (ln N>>1) the distribution tends to a Gaussian of width approximately sqrt[ln N] centered at (-)l approximately ln N. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent 2+gamma.

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PubMed2001

Language as an evolving word web.

Human language may 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 interacting words in sentences are connected by edges. The empirical distribution of the number of connections of words in this network is of a peculiar form that includes two pronounced power-law regions. Here we propose a theory of the evolution of language, which treats language as a self-organizing network of interacting words. In the framework of this concept, we completely describe the observed word web structure without any fitting. We show that the two regimes in the distribution naturally emerge from the evolutionary dynamics of the word web. It follows from our theory that the size of the core part of language, the 'kernel lexicon', does not vary as language evolves.

Humans↗

PubMed2001

Anomalous percolation 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 infinity exp[-d(gamma)(b-b(c))(-1/2)], where the coefficient d is a function of the degree distribution exponent gamma. In the entire phase without the giant component, these networks are in a "critical state." The probability 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, P(k) approximately 1/(k ln k)(2). In the phase with the giant component, P(k) has an exponential cutoff at k(c) approximately 1/S. In the simplest particular case, we present exact results for growing exponential networks.