Biomedical subjects

J F F Mendes

Publications and source records attributed to J F F Mendes.

13 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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PubMed2004

Scale-free network with Boolean dynamics as a function of connectivity.

In this work we analyze scale-free networks with different power-law spectra N (k) approximately k(-gamma) under a Boolean dynamic, where the Boolean rule that each node obeys is a function of its connectivity k. This is done by using only two logical functions (AND and XOR) which are controlled by a parameter q. Using a damage spreading technique we show that the Hamming distance and the number of 1's exhibit power-law behavior as a function of q. The exponents appearing in the power laws depend on the value of gamma.

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PubMed2004

Roughness of sandpile surfaces.

We study the surface roughness of prototype models displaying self-organized criticality (SOC) and their noncritical variants in one dimension. For SOC systems, we find that two seemingly equivalent definitions of surface roughness yield different asymptotic scaling exponents. Using approximate analytical arguments and extensive numerical studies we conclude that this ambiguity is due to the special scaling properties of the nonlinear steady state surface. We also find that there is no such ambiguity for non-SOC models, although there may be intermediate crossovers to different roughness values. Such crossovers need to be distinguished from the true asymptotic behavior, as in the case of a noncritical disordered sandpile model studied by Barker and Mehta [Phys. Rev. E 61, 6765 (2000)].

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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

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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PubMed2002

Monte Carlo study of the elastic interaction in heteroepitaxial growth.

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, theta. 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 that were used in the Monte Carlo simulations. A discussion about the effect of different factors is presented.

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PubMed2001

Static critical behavior in the inactive phase of the pair contact process.

Steady-state properties in the absorbing phase of the 1d pair contact process model are investigated. It is shown that, in typical absorbing states (reached by the system's dynamic rules), the density of isolated particles rho(1), approaches a stationary value that depends on the annihilation probability (p); the deviation from its "natural" value at criticality rho(nat)(1) follows a power law: rho(nat)(1)-rho(1) approximately (p-p(c))(beta(1)) for p>p(c). Monte Carlo simulations yield beta(1)=0.81. A cluster approximation is developed for this model, qualitatively confirming the numerical results and predicting beta(1)=1. The singular behavior of the isolated particles density in the inactive phase is explained using a phenomenological approach.

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