Maximal planar scale-free Sierpinski networks with small-world effect and power-law strength-degree correlation

Many real networks share three generic properties: they are scale-free, display a small-world effect, and show a power-law strength-degree correlation. In this paper, we propose a type of deterministically growing networks called Sierpinski networks, which are induced by the famous Sierpinski fractals and constructed in a simple iterative way. We derive analytical expressions for degree distribution, strength distribution, clustering coefficient, and strength-degree correlation, which agree well with the characterizations of various real-life networks. Moreover, we show that the introduced Sierpinski networks are maximal planar graphs.Comment: 6 pages, 5 figures, accepted by EP

Decay of Correlations in a Topological Glass

In this paper we continue the study of a topological glassy system. The state space of the model is given by all triangulations of a sphere with $N$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with other numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal between two adjacent triangles, with a temperature dependent probability. We consider the system at very low temperatures. We concentrate on several new aspects of this model: Starting from a detailed description of the stationary state, we conclude that pairs of defects (nodes with the "wrong" degree) move with very high mobility along 1-dimensional paths. As they wander around, they encounter single defects, which they then move "sideways" with a geometrically defined probability. This induces a diffusive motion of the single defects. If they meet, they annihilate, lowering the energy of the system. We both estimate the decay of energy to equilibrium, as well as the correlations. In particular, we find a decay like $t^{-0.4}$

Statistical Mechanics of Two-dimensional Foams

The methods of statistical mechanics are applied to two-dimensional foams under macroscopic agitation. A new variable -- the total cell curvature -- is introduced, which plays the role of energy in conventional statistical thermodynamics. The probability distribution of the number of sides for a cell of given area is derived. This expression allows to correlate the distribution of sides ("topological disorder") to the distribution of sizes ("geometrical disorder") in a foam. The model predictions agree well with available experimental data

A Topological Glass

We propose and study a model with glassy behavior. The state space of the model is given by all triangulations of a sphere with $n$ nodes, half of which are red and half are blue. Red nodes want to have 5 neighbors while blue ones want 7. Energies of nodes with different numbers of neighbors are supposed to be positive. The dynamics is that of flipping the diagonal of two adjacent triangles, with a temperature dependent probability. We show that this system has an approach to a steady state which is exponentially slow, and show that the stationary state is unordered. We also study the local energy landscape and show that it has the hierarchical structure known from spin glasses. Finally, we show that the evolution can be described as that of a rarefied gas with spontaneous generation of particles and annihilating collisions

Performance Evaluation of Components Using a Granularity-based Interface Between Real-Time Calculus and Timed Automata

To analyze complex and heterogeneous real-time embedded systems, recent works have proposed interface techniques between real-time calculus (RTC) and timed automata (TA), in order to take advantage of the strengths of each technique for analyzing various components. But the time to analyze a state-based component modeled by TA may be prohibitively high, due to the state space explosion problem. In this paper, we propose a framework of granularity-based interfacing to speed up the analysis of a TA modeled component. First, we abstract fine models to work with event streams at coarse granularity. We perform analysis of the component at multiple coarse granularities and then based on RTC theory, we derive lower and upper bounds on arrival patterns of the fine output streams using the causality closure algorithm. Our framework can help to achieve tradeoffs between precision and analysis time.Comment: QAPL 201

Manipulation and removal of defects in spontaneous optical patterns

Defects play an important role in a number of fields dealing with ordered structures. They are often described in terms of their topology, mutual interaction and their statistical characteristics. We demonstrate theoretically and experimentally the possibility of an active manipulation and removal of defects. We focus on the spontaneous formation of two-dimensional spatial structures in a nonlinear optical system, a liquid crystal light valve under single optical feedback. With increasing distance from threshold, the spontaneously formed hexagonal pattern becomes disordered and contains several defects. A scheme based on Fourier filtering allows us to remove defects and to restore spatial order. Starting without control, the controlled area is progressively expanded, such that defects are swept out of the active area.Comment: 4 pages, 4 figure

The Harris-Luck criterion for random lattices

The Harris-Luck criterion judges the relevance of (potentially) spatially correlated, quenched disorder induced by, e.g., random bonds, randomly diluted sites or a quasi-periodicity of the lattice, for altering the critical behavior of a coupled matter system. We investigate the applicability of this type of criterion to the case of spin variables coupled to random lattices. Their aptitude to alter critical behavior depends on the degree of spatial correlations present, which is quantified by a wandering exponent. We consider the cases of Poissonian random graphs resulting from the Voronoi-Delaunay construction and of planar, ``fat'' $\phi^3$ Feynman diagrams and precisely determine their wandering exponents. The resulting predictions are compared to various exact and numerical results for the Potts model coupled to these quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one figure added for clarification, minor re-wordings and typo cleanu

Structure and dynamics of cellular systems

This review addresses recent progress in the analysis and modelling of disordered cell structures. It focuses on planar systems, for which most research has been performed. The subject is approached from a general viewpoint rather than focusing on the specific evolution of certain systems. To this end, results of studies performed in completely different disciplines, ranging from applied sciences, biology and physics to mathematics, are gathered and discussed systematically. Special emphasis is laid on common properties of the different structures, including a critical discussion of the information contained in typically measured quantities. Efficient techniques for the simulation of typical disordered cell structures are summarized, and novel theoretical approaches to the modelling of dynamical cell structures are presented. Here, special attention is paid to the application of the methods of statistical mechanics and the resulting implications. In several systems, a breakdown of order induced by variation of an external control parameter can be observed. Its diagnostics in Voronoi tessellations is presented focusing on the connection with the underlying physics. The discussion of planar cell structures is concluded with a presentation of the peculiar properties of exceptional systems, characterized by the coexistence of two different length scales or by scale invariance. This review concludes with a presentation of recent developments in the research on three-dimensional cell structures