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The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Nonmonotonic betweenness behaviors are found to characterize nodes that lie close to intercommunity boundaries in the studied networks.The article by Riera-Campeny, Mehboudi, Pons, and Sanpera [Phys. Rev. E 99, 032126 (2019)2470-004510.1103/PhysRevE.99.032126] studies heat rectification in a network of harmonic oscillators which is periodically driven. Both the title and introduction stress the quantum nature of the system. Here we show that the results are more general and are equally valid for a classical system, which broadens the interest of the paper and may suggest further pathways for a basic understanding of the phenomenon.We consider a formulation for the Hopf functional differential equation which governs statistical solutions of the Navier-Stokes equations. By introducing an exponential operator with a functional derivative, we recast the Hopf equation as an integro-differential functional equation by the Duhamel principle. On this basis we introduce a successive approximation to the Hopf equation. As an illustration we take the Burgers equation and carry out the approximations to the leading order. Scale invariance of the statistical Navier-Stokes equations in d dimensions is formulated and contrasted with that of the deterministic Navier-Stokes equations. For the statistical Navier-Stokes equations, critical scale invariance is achieved for the characteristic functional of the dth derivative of the vector potential in d dimensions. The deterministic equations corresponding to this choice of the dependent variable acquire the linear Fokker-Planck operator under dynamic scaling. In three dimensions it is the vorticity gradient that behaves like a fundamental solution (more precisely, source-type solution) of deterministic Navier-Stokes equations in the long-time limit. Physical applications of these ideas include study of a self-similar decaying profile of fluid flows. Moreover, we reveal typical physical properties in the late-stage evolution by combining statistical scale invariance and the source-type solution. This yields an asymptotic form of the Hopf functional in the long-time limit, improving the well-known Hopf-Titt solution. In particular, we present analyses for the Burgers equations to illustrate the main ideas and indicate a similar analysis for the Navier-Stokes equations.Noise and fluctuations play vital roles in signal transduction in cells. Various numerical techniques for its simulation have been proposed, most of which are not efficient in cellular networks with a wide spectrum of timescales. In this paper, based on a recently developed variational technique, low-dimensional structures embedded in complex stochastic reaction dynamics are unfolded which sheds light on new design principles of efficient simulation algorithm for treating noise in the mesoscopic world. This idea is effectively demonstrated in several popular regulation models with an empirical selection of test functions according to their reaction geometry, which not only captures complex distribution profiles of different molecular species but also considerably speeds up the computation.Brownian ratchets are shown to feature a nontrivial vanishing-noise limit where the dynamics is reduced to a stochastic alternation between two deterministic circle maps (quasideterministic ratchets). Motivated by cooperative dynamics of molecular motors, here we solve exactly the problem of two interacting quasideterministic ratchets. We show that the dynamics can be described as a random walk on a graph that is specific to each set of parameters. We compute point by point the exact velocity-force V(f) function as a summation over all paths in the specific graph for each f, revealing a complex structure that features self-similarity and nontrivial continuity properties. From a general perspective, we unveil that the alternation of two simple piecewise linear circle maps unfolds a very rich variety of dynamical complexity, in particular the phenomenon of piecewise chaos, where chaos emerges from the combination of nonchaotic maps. We show convergence of the finite-noise case to our exact solution.Discrete eigenmodes of the filamentation instability in a weakly ionized current-driven plasma in the presence of a q-nonextensive electron velocity distribution is investigated. Considering the kinetic theory, Bhatnagar-Gross-Krook collision model, and Lorentz transformation relations, the generalized longitudinal and transverse dielectric permittivities are obtained. Taking into account the long-wavelength limit and diffusion frequency limit, the dispersion relations are obtained. Using the approximation of geometrical optics and linear inhomogeneity of the plasma, the real and imaginary parts of the frequency are discussed in these limits. It is shown that in the long-wavelength limit, when the normalized electron velocity is increased the growth rate of the instability increases. However, when the collision frequency is increased the growth rate of the filamentation instability decreases. In the diffusion frequency limit, results indicate that the effects of the electron velocity and q-nonextensive parameter on the growth rate of the instability are similar. Finally, it is found that when the collision frequency is increased the growth rate of the instability increases in the presence of a q-nonextensive distribution.This corrects the article DOI 10.1103/PhysRevE.100.012303.The aging process is a common phenomenon in engineering, biological, and physical systems. The hazard rate function, which characterizes the aging process, is a fundamental quantity in the disciplines of reliability, failure, and risk analysis. However, it is difficult to determine the entire hazard function accurately with limited observation data when the degradation mechanism is not fully understood. Inspired by the seminal work pioneered by Jaynes [Phys. Rev. 106, 620 (1956)PHRVAO0031-899X10.1103/PhysRev.106.620], this study develops an approach based on the principle of maximum entropy. In particular, the time-dependent hazard rate function can be established using limited observation data in a rational manner. It is shown that the developed approach is capable of constructing and interpreting many typical hazard rate curves observed in practice, such as the bathtub curve, the upside down bathtub, and so on. selleck chemicals llc The developed approach is applied to model a classical single function system and a numerical example is used to demonstrate the method.