Logarithmic Clustering in Submonolayer Epitaxial Growth

We investigate submonolayer epitaxial growth with a fixed monomer flux and irreversible aggregation of adatom islands due to their effective diffusion. When the diffusivity D_k of an island of mass k is proportional to k^{-\mu}, a Smoluchowski rate equation approach predicts steady behavior for 0<\mu<1, with the concentration c_k of islands of mass k varying as k^{-(3-\mu)/2}. For \mu>1, continuous evolution occurs in which c_k(t)~(\ln t)^{-(2k-1)\mu/2}, while the total island density increases as N(t)~(\ln t)^{\mu/2}. Monte Carlo simulations support these predictions.Comment: 4 pages, 2 figure

Transition from small to large world in growing networks

We examine the global organization of growing networks in which a new vertex is attached to already existing ones with a probability depending on their age. We find that the network is infinite- or finite-dimensional depending on whether the attachment probability decays slower or faster than $(age)^{-1}$. The network becomes one-dimensional when the attachment probability decays faster than $(age)^{-2}$. We describe structural characteristics of these phases and transitions between them.Comment: 5 page

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.Comment: 5 pages, 2 figure

Effective action in DSR1 quantum field theory

We present the one-loop effective action of a quantum scalar field with DSR1 space-time symmetry as a sum over field modes. The effective action has real and imaginary parts and manifest charge conjugation asymmetry, which provides an alternative theoretical setting to the study of the particle-antiparticle asymmetry in nature.Comment: 8 page

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 $\lambda_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, $\lambda_c>0$ if $q_m>2$, and $\lambda_c=0$ if $q_m\leq2$. In both these cases the density of eigenvalues $\rho(\lambda)\to0$ as $\lambda\to\lambda_c+0$, but the limiting behaviors near $\lambda_c$ 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 $\rho(\lambda)$ near $\lambda_c$ and the long-time asymptotics of the autocorrelator and the propagator.Comment: 25 pages, 4 figure

Preferential attachment of communities: the same principle, but a higher level

The graph of communities is a network emerging above the level of individual nodes in the hierarchical organisation of a complex system. In this graph the nodes correspond to communities (highly interconnected subgraphs, also called modules or clusters), and the links refer to members shared by two communities. Our analysis indicates that the development of this modular structure is driven by preferential attachment, in complete analogy with the growth of the underlying network of nodes. We study how the links between communities are born in a growing co-authorship network, and introduce a simple model for the dynamics of overlapping communities.Comment: 7 pages, 3 figure

Robustness of planar random graphs to targeted attacks

In this paper, robustness of planar trivalent random graphs to targeted attacks of highest connected nodes is investigated using numerical simulations. It is shown that these graphs are relatively robust. The nonrandom node removal process of targeted attacks is also investigated as a special case of non-uniform site percolation. Critical exponents are calculated by measuring various properties of the distribution of percolation clusters. They are found to be roughly compatible with critical exponents of uniform percolation on these graphs.Comment: 9 pages, 11 figures. Added references.Corrected typos. Paragraph added in section II and in the conclusion. Published versio

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, $\Pi (q) \sim q^{-\gamma}$, $\gamma =\sqrt{2}$ 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 $\gamma$.Comment: 7 pages epltex, 1 figur

Mean-field scaling function of the universality class of absorbing phase transitions with a conserved field

We consider two mean-field like models which belong to the universality class of absorbing phase transitions with a conserved field. In both cases we derive analytically the order parameter as function of the control parameter and of an external field conjugated to the order parameter. This allows us to calculate the universal scaling function of the mean-field behavior. The obtained universal function is in perfect agreement with recently obtained numerical data of the corresponding five and six dimensional models, showing that four is the upper critical dimension of this particular universality class.Comment: 8 pages, 2 figures, accepted for publication in J. Phys.