We introduce an analytic model for directed Watts-Strogatz small-world graphs and deduce an algebraic expression of its defining adjacency matrix. The latter is then used to calculate the small-world digraph's asymmetry index and clustering coefficient in an analytically exact fashion, valid nonasymptotically for all graph sizes. The proposed approach is general and can be applied to all algebraically well-defined graph-theoretical measures, thus allowing for an analytical investigation of finite-size small-world graphs.

We present a theoretical framework for understanding nonbinary, nonindependent percolation on networks with general degree distributions. The model incorporates a partially functional (PF) state of nodes so that both intensity and extensity of error are characterized. Two connected nodes in a PF state cannot sustain the load and therefore break their link. We give exact solutions for the percolation threshold, the fraction of giant cluster, and the mean size of small clusters. The robustness-fragility transition point for scale-free networks with a degree distribution $p_k\propto k^{-\alpha }$ is identified to be $\alpha =3$. The analysis reveals that scale-free networks are vulnerable to targeted attack at hubs: a more complete picture of their Achilles' heel turns out to be not only the hubs themselves but also the edges linking them together.

When analyzing important classes of complex interconnected systems, link directionality can hardly be neglected if a precise and effective picture of the structure and function of the system is needed. If community analysis is performed, the notion of “community” itself is called into question, since the property of having a comparatively looser external connectivity could refer to the inbound or outbound links only or to both categories. In this paper, we introduce the notions of in-, out-, and in-/out-community in order to correctly classify the directedness of the interaction of a subnetwork with the rest of the system. Furthermore, we extend the scope of community analysis by introducing the notions of in-, out-, and in-/out-pseudocommunity. They are subnetworks having strong internal connectivity but also important interactions with the rest of the system, the latter taking place by means of a minority of its nodes only. The various types of (pseudo-)communities are qualified and distinguished by a suitable set of indicators and, on a given network, they can be discovered by using a “local” searching algorithm. The application to a broad set of benchmark networks and real-world examples proves that the proposed approach is able to effectively disclose the different types of structures above defined and to usefully classify the directionality of their interactions with the rest of the system.

Integrable Hamiltonian systems are said to display nontrivial monodromy if fundamental action-angle loops defined on phase-space tori change their topological structure when the system is carried around a circuit. In an earlier paper it was shown that this topological change can occur as a result of time evolution under certain rather abstract flows in phase space. In the present paper, we show that the same topological change can occur as a result of application of ordinary forces. We also show how this dynamical phenomenon could be observed experimentally in classical or in quantum systems.

It is a challenging task to model nonequilibrium gas flows within a continuum-fluid framework. Recently some extended hydrodynamic models in the Navier-Stokes formulation have been developed for such flows. A key problem in the application of such models is that suitable boundary conditions must be specified. In the present work, a generalized second-order slip boundary condition is developed in which an effective mean-free path considering the wall effect is used. By combining this slip scheme with certain extended Navier-Stokes constitutive relation models, we obtained a method for nonequilibrium gas flows with solid boundaries. The method is applied to several rarefied gas flows involving planar or curved walls, including the Kramers' problem, the planar Poiseuille flow, the cylindrical Couette flow, and the low speed flow over a sphere. The results show that the proposed method is able to give satisfied predictions, indicating the good potential of the method for nonequilibrium flows.

Experiments on near-critical hydrogen have been conducted under magnetic compensation of gravity to investigate the Faraday instability that arises at the liquid-vapor interface under zero-gravity conditions. We investigated such instability in the absence of stabilizing gravity. Under such conditions, vibration orients the interface and can destabilize it. The experiments confirm the existence of Faraday waves and demonstrate a transition from a square to a line pattern close to the critical point. They also show a transition very close to the critical point from Faraday to periodic layering of the vapor-liquid interface perpendicular to vibration. It was seen that the Faraday wave instability is favored when the liquid-vapor density difference is large enough (fluid far from the critical point), whereas periodic layering predominates for small difference in the liquid and vapor densities (close to the critical point). It was observed for the Faraday wave instability that the wavelength of the instability decreases as one approaches the critical point. The experimental results demonstrate good agreement to the dispersion relation for zero gravity except for temperatures very close to the critical point where a transition from a square pattern to a line pattern is detected, similarly to what is observed under 1$g$ conditions.

This study is devoted to the experimental and theoretical investigation of aerodynamic drop breakup phenomena. We show that the phenomena of drop impact onto a rigid wall, drop binary collisions, and aerodynamic drop deformation are similar if the correct scaling is applied. Then we use observations of the deforming drop to estimate the evolution of the film thickness of the bag, the value that determines the size of the fine child drops produced by bag breakup. This prediction of film thickness, based on film kinematics, is validated for the initial stage by direct drop thickness measurements and at the latest stage by the data obtained from the velocity of hole expansion in the film. It is shown that the film thickness correlates well with the dimensionless position of the bag apex.

We analyze the so-called Kovacs effect in the one-dimensional Ising model with Glauber dynamics. We consider small enough temperature jumps, for which a linear response theory has been recently derived. Within this theory, the Kovacs hump is directly related to the monotonic relaxation function of the energy. The analytical results are compared with extensive Monte Carlo simulations, and an excellent agreement is found. Remarkably, the position of the maximum in the Kovacs hump depends on the fact that the true asymptotic behavior of the relaxation function is different from the stretched exponential describing the relevant part of the relaxation at low temperatures.

The efficiency of crystal growth in alloys is limited by the morphological instability, which is caused by a positive feedback between the interface deformation and the diffusive flux of solute at the front of the phase transition. Usually this phenomenon is described in the framework of the normal diffusion equation, which stems from the linear relation between time and the mean squared displacement of molecules $\langle x^2\left(t\right)\rangle \sim K_{1}t$ ($K_1$ is the classical diffusion coefficient) that is characteristic of Brownian motion. However, in some media (e.g., in gels and porous media) the random walk of molecules is hindered by obstacles, which leads to another power law, $\langle x^2\left(t\right)\rangle \sim K_{\alpha }{t}^{\alpha }$, where $0<\alpha \le 1$. As a result, the diffusion is anomalous, and it is governed by an integro-differential equation including a fractional derivative in time variable, i.e., a memory. In the present work, we investigate the stability of a directional solidification front in the case of an anomalous diffusion. Linear stability of a moving planar directional solidification front is studied, and a generalization of the Mullins-Sekerka stability criterion is obtained. Also, an asymptotic nonlinear long-wave evolution equation of Sivashinsky's type, which governs the cellular structures at the interface, is derived.

In many aspects the onset of a chronic disease resembles a phase transition in a complex dynamic system: Quantitative changes accumulate largely unnoticed until a critical threshold is reached, which causes abrupt qualitative changes of the system. In this study we examine a special case, the onset of type-1 diabetes (T1D), a disease that results from loss of the insulin-producing pancreatic islet β cells. Within each islet, the β cells are electrically coupled to each other via gap-junctional channels. This intercellular coupling enables the β cells to synchronize their insulin release, thereby generating the multiscale temporal rhythms in blood insulin that are critical to maintaining blood glucose homeostasis. Using percolation theory we show how normal islet function is intrinsically linked to network connectivity. In particular, the critical amount of β-cell death at which the islet cellular network loses site percolation is consistent with laboratory and clinical observations of the threshold loss of β cells that causes islet functional failure. In addition, numerical simulations confirm that the islet cellular network needs to be percolated for β cells to synchronize. Furthermore, the interplay between site percolation and bond strength predicts the existence of a transient phase of islet functional recovery after onset of T1D and introduction of treatment, potentially explaining the honeymoon phenomenon. Based on these results, we hypothesize that the onset of T1D may be the result of a phase transition of the islet β-cell network.

Homogeneous turbulence in rotating shear flows is studied by means of pseudospectral direct numerical simulation and analytical spectral linear theory (SLT). The ratio of the Coriolis parameter to shear rate is varied over a wide range by changing the rotation strength, while a constant moderate shear rate is used to enable significant contributions to the nonlinear interscale energy transfer and to the nonlinear intercomponental redistribution terms. In the destabilized and neutral cases, in the sense of kinetic energy evolution, nonlinearity cannot saturate the growth of the largest scales. It permits the smallest scale to stabilize by a scale-by-scale quasibalance between the nonlinear energy transfer and the dissipation spectrum. In the stabilized cases, the role of rotation is mainly nonlinear, and interacting inertial waves can affect almost all scales as in purely rotating flows. In order to isolate the nonlinear effect of rotation, the two-dimensional manifold with vanishing spanwise wave number is revisited and both two-component spectra and single-point two-dimensional energy components exhibit an important effect of rotation, whereas the SLT as well as the purely two-dimensional nonlinear analysis are unaffected by rotation as stated by the Proudman theorem. The other two-dimensional manifold with vanishing streamwise wave number is analyzed with similar tools because it is essential for any shear flow. Finally, the spectral approach is used to disentangle, in an analytical way, the linear and nonlinear terms in the dynamical equations.

We have performed nonequilibrium classical and quantum-mechanical molecular dynamics simulations that follow the interpenetration of deuterium-tritium (DT) and carbon (C) components through an interface initially in hydrostatic and thermal equilibrium. We concentrate on the warm, dense matter regime with initial densities of 2.5–5.5 g/cm${}^3$ and temperatures from 10 to 100 eV. The classical treatment employs a Yukawa pair-potential with the parameters adjusted to the plasma conditions, and the quantum treatment rests on an orbital-free density functional theory at the Thomas-Fermi-Dirac level. For times greater than about a picosecond, the component concentrations evolve in accordance with Fick's law for a classically diffusing fluid with the motion, though, described by the mutual diffusion coefficient of the mixed system rather than the self-diffusion of the individual components. For shorter times, microscopic processes control the clearly non-Fickian dynamics and require a detailed representation of the electron probability density in space and time.

Stationary solutions for the $1+1$ cubic nonlinear Schrödinger equation (NLS) modeling attractive Bose-Einstein condensates (BECs) in a small potential are obtained via a form of nonlinear perturbation. The focus here is on perturbations to the bright soliton solutions due to small potentials which either confine or repel the BECs: under arbitrary piecewise continuous potentials, we obtain the general representation for the perturbation theory of the bright solitons. Importantly, we do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This is useful, in that it allows us to consider perturbations of bright solitons of arbitrary size. In some cases, exact solutions can be recovered, and these agree with known results from the literature. Several special cases are considered which involve confining potentials of specific relevance to BECs. We make several observations on the influence of the small potentials on the behavior of the perturbed bright solitons. The results demonstrate the difference between perturbed bright solitons in the attractive NLS and those results found in the repulsive NLS for dark solitons, as discussed by Mallory and Van Gorder, [Phys. Rev. E 88, 013205 (2013)]. Extension of these results to more spatial dimensions is mentioned.

The nature of the interactions among self-propelled agents (SPA), i.e., topological versus metric or a combination of both types, is a relevant open question in the field of self-organization phenomena. We studied the critical behavior of a Vicsek-like system of SPA given by a group of agents moving at constant speed and interacting among themselves under the action of a topological rule: each agent aligns itself with the average direction of its seven nearest neighbors, independent of the distance, under the influence of some noise. Based on both stationary and dynamic measurements, we provide strong evidence that both types of interactions are manifestations of the same phenomenon, which defines a robust universality class. Also, the cluster size distribution evaluated at the critical point shows a power-law behavior, and the exponent corresponding to the topological model is in excellent agreement with that of the metric one, further reinforcing our claim. Furthermore, we found that with topological interactions the average distance of influence between agents undergoes large fluctuations that diverge at the critical noise, thus providing clues about a mechanism that could be implemented by the agents to change their moving strategy.

For short duration impulse loadings, elastic granular chains are known to support solitary waves, while elastoplastic chains have recently been shown to exhibit two force decay regimes [Pal, Awasthi, and Geubelle, Granular Matter 15, 747 (2013).]. In this work, the dynamics of monodisperse elastic and elastoplastic granular chains under a wide range of loading conditions is studied, and two distinct response regimes are identified in each of them. In elastic chains, a short loading duration leads to a single solitary wave propagating down the chain, while a long loading duration leads to the formation of a train of solitary waves. A simple model is developed to predict the peak force and wave velocity for any loading duration and amplitude. In elastoplastic chains, wave trains form even for short loading times due to a mechanism distinct from that in elastic chains. A model based on energy balance predicts the decay rate and transition point between the two decay regimes. For long loading durations, loading and unloading waves propagate along the chain, and a model is developed to predict the contact force and particle velocity.

The frozen-wave instability which appears at a liquid-vapor interface when a harmonic vibration is applied in a direction tangential to it has been less studied until now. The present paper reports experiments on hydrogen (H${}_2$) in order to study this instability when the temperature is varied near its critical point for various gravity levels. Close to the critical point, a liquid-vapor density difference and surface tension can be continuously varied with temperature in a scaled, universal way. The effect of gravity on the height of the frozen waves at the interface is studied by performing the experiments in a magnetic facility where effective gravity that results from the coupling of the Earth's gravity and magnetic forces can be varied. The stability diagram of the instability is obtained. The experiments show a good agreement with an inviscid model [Fluid Dyn. 21, 849 (1987)], irrespective of the gravity level. It is observed in the experiments that the height of the frozen waves varies weakly with temperature and increases with a decrease in the gravity level, according to a power law with an exponent of 0.7. It is concluded that the wave height becomes of the order of the cell size as the gravity level is asymptotically decreased to zero. The interface pattern thus appears as a bandlike pattern of alternate liquid and vapor phases, a puzzling phenomenon that was observed with CO${}_2$ and H${}_2$ near their critical point in weightlessness [Beysens et al., Acta Astron. 61, 1002 (2007); Europhys. Lett. 86, 16003 (2009)].

We present a two-dimensional lattice model of self-propelled spins that can change direction only upon collision with another spin. We show that even with ballistic motion and minimal cooperativity, these spins display robust flocking behavior at nearly all densities, forming long bands of stripes. The structural transition in this system is not a thermodynamic phase transition, but it can still be characterized by an order parameter, and we demonstrate that if this parameter is studied as a dynamical variable rather than a steady-state observable, we can extract a detailed picture of how the flocking mechanism varies with density.

Since the lattice Boltzmann method originally carries out the simulations on the regular Cartesian lattices, curved boundaries are often approximated as a series of stair steps. The most commonly employed technique for resolving curved-boundary problems is extrapolating or interpolating macroscopic properties of boundary nodes. Previous investigations have indicated that using more than one equation for extrapolation or interpolation in boundary conditions potentially causes abrupt changes in particle distributions. Therefore, a curved-boundary treatment is introduced to improve computational accuracy of the conventional stair-shaped approximation used in lattice Boltzmann simulations by using a unified equation for extrapolation of macroscopic variables. This boundary condition is not limited to fluid flow and can be extended to potential fields. The proposed treatment is tested against several well-established problems and the solutions order of accuracy is evaluated. Numerical results show that the present treatment is of second-order accuracy and has reliable stability characteristics.

In this work we study opinion formation on a fully connected population participating of a public debate with two distinct choices, where the agents may adopt three different attitudes (favorable to either one choice or to the other, or undecided). The interactions between agents occur by pairs and are competitive, with couplings that are either negative with probability $p$ or positive with probability $1-p$. This bimodal probability distribution of couplings produces a behavior similar to the one resulting from the introduction of Galam's contrarians in the population. In addition, we consider that a fraction $d$ of the individuals are intransigent, that is, reluctant to change their opinions. The consequences of the presence of contrarians and intransigents are studied by means of computer simulations. Our results suggest that the presence of inflexible agents affects the critical behavior of the system, causing either the shift of the critical point or the suppression of the ordering phase transition, depending on the groups of opinions to which the intransigents belong. We also discuss the relevance of the model for real social systems.

Cells often exhibit different and stable phenotypes from the same DNA sequence. Robustness and plasticity of such cellular states are controlled by diverse transcriptional and epigenetic mechanisms, among them the modification of biochemical marks on chromatin. Here, we develop a stochastic model that describes the dynamics of epigenetic marks along a given DNA region. Through mathematical analysis, we show the emergence of bistable and persistent epigenetic states from the cooperative recruitment of modifying enzymes. We also find that the dynamical system exhibits a critical point and displays, in the presence of asymmetries in recruitment, a bifurcation diagram with hysteresis. These results have deep implications for our understanding of epigenetic regulation. In particular, our study allows one to reconcile within the same formalism the robust maintenance of epigenetic identity observed in differentiated cells, the epigenetic plasticity of pluripotent cells during differentiation, and the effects of epigenetic misregulation in diseases. Moreover, it suggests a possible mechanism for developmental transitions where the system is shifted close to the critical point to benefit from high susceptibility to developmental cues.