We investigate analytically and numerically the dynamical properties of critical Boolean networks with power-law in-degree distributions and for two choices of update functions. When the exponent of the in-degree distribution is larger than 3, we obtain results equivalent to those obtained for networks with fixed in-degree, e.g., the number of the nonfrozen nodes scales as $N^{2/3}$ with the system size $N$. When the exponent of the distribution is between 2 and 3, the number of the nonfrozen nodes increases as $N^x$, with $x$ being between 0 and 2/3 and depending on the exponent and on the cutoff of the in-degree distribution. These and ensuing results explain various findings obtained earlier by computer simulations.

Revealing corresponding identities of a dedicated individual in several different complex systems is a common task in many areas, and this task is transferred to a node matching problem among complex networks in this paper. A feasible node matching algorithm based on network structure is proposed. Through solving node matching problems on different types of networks by our algorithm, it is revealed that the structure of the networks under study may significantly influence the final matching results. For example, it is found that higher matching precision can be obtained on random networks with moderate density of links, and the results on small-world networks are always better than those on random or regular networks. Moreover, in scale-free networks, it seems that hub nodes play dominant roles, i.e., better matching results can be expected by selecting nodes with larger degrees as the revealed matched nodes. These findings will help us design more efficient node matching algorithm in the future.

We present a method for finding the hierarchy of rational solutions of the self-focusing nonlinear Schrödinger equation and present explicit forms for these solutions from first to fourth order. We also explain their relation to the highest amplitude part of a field that starts with a plane wave perturbed by random small amplitude radiation waves. Our work can elucidate the appearance of rogue waves in the deep ocean and can be applied to the observation of rogue light pulse waves in optical fibers.

The equilibrium-restricted solid-on-solid growth model on fractal substrates is studied by introducing a fractional Langevin equation. The growth exponent $\beta$ and the roughness exponent $\alpha$ defined, respectively, by the surface width via $W\sim t^{\beta }$ and the saturated width via $W_{\text{sat}}\sim L^{\alpha }$, $L$ being the system size, were obtained by a power-counting analysis, and the scaling relation $2\alpha +d_{\text{f}}=z_{\text{RW}}$ was found to hold. The numerical simulation data on Sierpinski gasket, checkerboard fractal, and critical percolation cluster were found to agree well with the analytical predictions of the fractional Langevin equation.

We have studied the kinetics of trans to cis isomerization under the illumination of ultraviolet light, in the Langmuir monolayer of mesogenic azobenzene dimer, bis-[5-($4^{\prime }$-n-dodecyloxy benzoyloxy)-2-($4^{″}$-methylphenylazo)phenyl] adipate, at an air-water interface. We find that the trans to cis isomerization reaction of the molecules in the monolayer shows deviation from the first-order kinetics unlike those reported on Langmuir monolayers of azobenzene molecules. We attribute the deviation from first-order kinetics to the simultaneous photoisomerization of trans isomers to form cis isomers and the reverse thermal isomerization of cis isomers to form trans isomers. Our analysis of the rate of change of mole fraction of trans isomers to form cis isomers indicates a first-order kinetics for trans to cis photoisomerization reaction and a second-order kinetics for cis to trans thermal isomerization reaction. This second-order kinetics mechanism is similar to the Lindemann-Hinshelwood mechanism for the unimolecular reactions at low concentration of reactants. The formation of the activated cis isomer by collisions is a slow process as compared to the decay of the activated cis isomer to trans isomer in the liquid expanded phase. This results in the second-order kinetics for the thermal isomerization of cis isomers.

A concise method for following the evolving geometry of a moving surface using Lagrangian coordinates is described. All computations can be done in the fixed geometry of the initial surface despite the evolving complexity of the moving surface. The method is applied to three problems in nonlinear elasticity: the bulging of a thin plate under pressure (the original motivation for Föppl–von Karman theory), the buckling of a spherical shell under pressure, and the phenomenon of capillary wrinkles induced by surface tension in a thin film. In this last problem the inclusion of a gravitational potential-energy term in the total energy improves the agreement with experiment.

We present an analysis of data stemming from numerical simulations of decaying magnetohydrodynamic (MHD) turbulence up to grid resolution of ${1536}^3$ points and up to Taylor Reynolds number of $\sim 1200$. The initial conditions are such that the initial velocity and magnetic fields are helical and in equipartition, while their correlation is negligible. Analyzing the data at the peak of dissipation, we show that the dissipation in MHD seems to asymptote to a constant as the Reynolds number increases, thereby strengthening the possibility of fast reconnection events in the solar environment for very large Reynolds numbers. Furthermore, intermittency of MHD flows, as determined by the spectrum of anomalous exponents of structure functions of the velocity and the magnetic field, is stronger than that of fluids, confirming earlier results; however, we also find that there is a measurable difference between the exponents of the velocity and those of the magnetic field, reminiscent of recent solar wind observations. Finally, we discuss the spectral scaling laws that arise in this flow.

Experimental and numerical results are presented for a fracture experiment carried out on a fiber-reinforced element under flexural loading, and a statistical analysis is performed for acoustic emission waiting-time distributions. By an optimization procedure, a recently proposed scaling law describing these distributions for different event magnitude scales is confirmed by both experimental and numerical data, thus reinforcing the idea that fracture of heterogeneous materials has scaling properties similar to those found for earthquakes. Analysis of the different scaling parameters obtained for experimental and numerical data leads us to formulate the hypothesis that the type of scaling function obtained depends on the level of correlation among fracture events in the system.

We propose a simple classical dynamical model of a thermoelectric (or thermochemical) heat engine based on a pair of ideal gas containers connected by two unequal scattering channels. The model is solved analytically and it is shown that a suitable combination of parameters can be chosen such that the engine operates at Carnot’s efficiency.

Molecular-dynamics simulations were performed for two opposing flat surfaces sparsely grafted with rigid polyelectrolyte chains whose lengths are smaller than their persistence lengths. The resulting force-distance dependence was analyzed theoretically in terms of two separate physical mechanisms: the pressure arising from osmotically active counterions trapped within the brush and the work required to bend the brush chains under confinement, which can be accurately characterized by a ground-state theory of rigid polymer buckling. These contributions are of the same magnitude and should be distinguishable in experiments of double-stranded DNA brushes.

We study memory effects in a kinetic roughening model. For $d=1$, a different dynamic scaling is uncovered in the memory dominated phases; the Kardar-Parisi-Zhang scaling is restored in the absence of noise. $d_c=2$ represents the critical dimension where memory is shown to smoothen the roughening front $\left(\alpha \le 0\right)$. Studies on a discrete atomistic model in the same universality class reconfirm the analytical results in the large time limit, while a different scaling behavior shows up for $t<\tau$, with $\tau$ being the memory characteristic of the atomistic model. Results can be generalized for other nonconservative systems.

Phase-separation dynamics of an asymmetric mixture of an isotropic dopant in a nematogenic fluid is presented. We show that, on steady cooling, the nucleating nematic drops move down the dopant concentration gradient, with a velocity that is dependent on the cooling rate and concentration gradient. This propulsion of the drops leads to a mechanism of droplet coarsening, where radius of a drop scales with time as $R\left(t\right)\sim t$. Various mechanisms for droplet propulsion are discussed.

Positive and negative feedback loops are often coupled to perform various functions in gene regulatory networks, acting as bistable switches, oscillators, and excitable devices. It is implied that such a system with interlinked positive and negative feedback loops is a flexible motif that can modulate itself among various functions. Here, we developed a minimal model for the system and systematically explored its dynamics and performance advantage in response to stimuli in a unifying framework. The system indeed displays diverse behaviors when the strength of feedback loops is changed. First, the system can be tunable from monostability to bistability by increasing the strength of positive feedback, and the bistability regime is modulated by the strength of negative feedback. Second, the system undergoes transitions from bistability to excitability and to oscillation with increasing the strength of negative feedback, and the reverse conversion occurs by enhancing the strength of positive feedback. Third, the system is more flexible than a single feedback loop; it can produce robust larger-amplitude oscillations over a wider stimulus regime compared with a single time-delayed negative feedback loop. Furthermore, the tunability of the system depends mainly on the topology of coupled feedback loops but less on the exact parameter values or the mode of interactions between model components. Thus, our results interpret why such a system represents a tunable motif and can accomplish various functions. These also suggest that coupled feedback loops can act as toolboxes for engineering diverse functional circuits in synthetic biology.

Many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups. This feature is known as community structure and encodes precious information about the organization and the function of the nodes. Many algorithms have been proposed but it is not yet clear how they should be tested. Recently we have proposed a general class of undirected and unweighted benchmark graphs, with heterogeneous distributions of node degree and community size. An increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links, which require suitable benchmark graphs for testing. In this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built-in community structure. We also consider the possibility that nodes belong to more communities, a feature occurring in real systems, such as social networks. As a practical application, we show how modularity optimization performs on our benchmark.

We present experimental, numerical, and theoretical studies of droplet flows in hydrodynamic networks. Using both millifluidic and microfluidic devices, we study the partitioning of monodisperse droplets in an asymmetric loop. In both cases, we show that droplet traffic results from the hydrodynamic feedback due to the presence of droplets in the outlet channels. We develop a recently-introduced phenomenological model [W. Engl et al., Phys. Rev. Lett. 95, 208304 (2005)] and successfully confront its predictions to our experimental results. This approach offers a simple way to measure the excess hydrodynamic resistance of a channel filled with droplets. We discuss the traffic behavior and the variations in the corresponding hydrodynamic resistance length $L_d$ and of the droplet mobility $\beta$, as a function of droplet interdistance and confinement for channels having circular or rectangular cross sections.

Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be $1-p_c=0.1096–8$ for the hexagonal lattice and $1-p_c=0.1092–3$ for the square-octagonal lattice, where $1-p$ denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability $1-p_c=0.110028$ for the quantum Gilbert-Varshamov bound with a zero encoding rate.

Given a network and a partition in communities, we consider the issues “how communities influence each other” and “when two given communities do communicate.” Specifically, we address these questions in the context of small-world networks, where an arbitrary quenched graph is given and long-range connections are randomly added. We prove that, among the communities, a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by M. Ostilli and J. F. F. Mendes [Phys. Rev. E 78, 031102 (2008)] $(n=1)$, which here $(n>1)$ consists in a sort of effective TAP (Thouless, Anderson, and Palmer) equations in which each community plays the role of a microscopic spin. The relative susceptibilities derived from these equations calculated at finite or zero temperature, where the method provides an effective percolation theory, give us the answers to the above issues. Unlike the case $n=1$, asymmetries among the communities may lead, via the TAP-like structure of the equations, to many metastable states whose number, in the case of negative shortcuts among the communities, may grow exponentially fast with $n$. As examples we consider the $n$ Viana-Bray communities model and the $n$ one-dimensional small-world communities model. Despite being the simplest ones, the relevance of these models in network theory, as, e.g., in social networks, is crucial and no analytic solution were known until now. Connections between percolation and the fractal dimension of a network are also discussed. Finally, as an inverse problem, we show how, from the relative susceptibilities, a natural and efficient method to detect the community structure of a generic network arises.

The influence of the tail features of the local magnetic field probability density function (PDF) on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically, we assign a quenched random field whose value is in accordance with a generic distribution that bears platykurtic and leptokurtic distributions depending on a single parameter $\tau <3$ to each site. For $\tau <5/3$, such distributions, which are basically Student-$t$ and $r$ distribution extended for all plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got the same asymptotic power-law behavior as a $\alpha$-stable Lévy distribution with $\alpha =\left(3-\tau \right)/\left(\tau -1\right)$. For every value of $\tau$, at specific temperature and width of the distribution, the system undergoes a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz $\left(\tau =2\right)$ which is accompanied with a divergent free energy per spin (at zero temperature).

It is well known that a system of grains, vibrated vertically in a cell divided into linked columns, may spontaneously move into just one of the columns due to the inelastic nature of their collisions. Here we study the behavior of a water-immersed system of spherical barium titanate particles in a rectangular cell which is divided into two columns, linked by two connecting holes, one at the top and one at the bottom of the cell. Under vibration the grains spontaneously move into just one of the columns via a gradual transfer of grains through the connecting hole at the base of the cell. We have developed numerical simulations that are able to reproduce this behavior and provide detailed information on the instability mechanism. We use this knowledge to propose a simple analytical model for this fluid-driven partition instability based on two coupled granular beds vibrated within an incompressible fluid.