Cooperation and defection may be considered to be two extreme responses to a social dilemma. Yet the reality is much less clear-cut. Between the two extremes lies an interval of ambivalent choices, which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool. If strategies are chosen from the unit interval, where 0 corresponds to pure defection and 1 corresponds to the maximal contribution, the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions. Here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions, and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal. However, as the cost-to-benefit ratio of cooperation increases, the characteristic threshold decreases while the corresponding steepness increases. Our observations remain valid if more complex sigmoid functions are used, thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation.

Many real-world networks interact with and depend upon other networks. We develop an analytical framework for studying a network formed by $n$ fully interdependent randomly connected networks, each composed of the same number of nodes $N$. The dependency links connecting nodes from different networks establish a unique one-to-one correspondence between the nodes of one network and the nodes of the other network. We study the dynamics of the cascades of failures in such a network of networks (NON) caused by a random initial attack on one of the networks, after which a fraction $p$ of its nodes survives. We find for the fully interdependent loopless NON that the final state of the NON does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the NON, which generalizes both the giant component of regular percolation of a single network ($n=1$) and the recently studied case of the mutual giant component of two interdependent networks ($n=2$). We also find that the mutual giant component does not depend on the topology of the NON and express it in terms of generating functions of the degree distributions of the network. Our results show that, for any $n⩾2$ there exists a critical $p=p_c>0$ below which the mutual giant component abruptly collapses from a finite nonzero value for $p⩾p_c$ to zero for $p<p_c$, as in a first-order phase transition. This behavior holds even for scale-free networks where $p_c=0$ for $n=1$. We show that, if at least one of the networks in the NON has isolated or singly connected nodes, the NON completely disintegrates for sufficiently large $n$ even if $p=1$. In contrast, in the absence of such nodes, the NON survives for any $n$ for sufficiently large $p$. We illustrate this behavior by comparing two exactly solvable examples of NONs composed of Erdős-Rényi (ER) and random regular (RR) networks. We find that the robustness of $n$ coupled RR networks of degree $k$ is dramatically higher compared to the $n$-coupled ER networks of the same average degree $\overline{k}=k$. While for ER NONs there exists a critical minimum average degree $\overline{k}={\overline{k}}_{\min }\sim \ln n$ below which the system collapses, for RR NONs $k_{\min }=2$ for any $n$ (i.e., for any $k>2$, a RR NON is stable for any $n$ with $p_c<1$). This results arises from the critical role played by singly connected nodes which exist in an ER NON and enhance the cascading failures, but do not exist in a RR NON.

The purpose of any reconstruction method is to generate realizations of two- or multiphase disordered media that honor limited data for them, with the hope that the realizations provide accurate predictions for those properties of the media for which there are no data available, or their measurement is difficult. An important example of such stochastic systems is porous media for which the reconstruction technique must accurately represent their morphology—the connectivity and geometry—as well as their flow and transport properties. Many of the current reconstruction methods are based on low-order statistical descriptors that fail to provide accurate information on the properties of heterogeneous porous media. On the other hand, due to the availability of high resolution two-dimensional (2D) images of thin sections of a porous medium, and at the same time, the high cost, computational difficulties, and even unavailability of complete 3D images, the problem of reconstructing porous media from 2D thin sections remains an outstanding unsolved problem. We present a method based on multiple-point statistics in which a single 2D thin section of a porous medium, represented by a digitized image, is used to reconstruct the 3D porous medium to which the thin section belongs. The method utilizes a 1D raster path for inspecting the digitized image, and combines it with a cross-correlation function, a grid splitting technique for deciding the resolution of the computational grid used in the reconstruction, and the Shannon entropy as a measure of the heterogeneity of the porous sample, in order to reconstruct the 3D medium. It also utilizes an adaptive technique for identifying the locations and optimal number of hard (quantitative) data points that one can use in the reconstruction process. The method is tested on high resolution images for Berea sandstone and a carbonate rock sample, and the results are compared with the data. To make the comparison quantitative, two sets of statistical tests consisting of the autocorrelation function, histogram matching of the local coordination numbers, the pore and throat size distributions, multiple-points connectivity, and single- and two-phase flow permeabilities are used. The comparison indicates that the proposed method reproduces the long-range connectivity of the porous media, with the computed properties being in good agreement with the data for both porous samples. The computational efficiency of the method is also demonstrated.

We studied distribution of zeros of the partition function of the ferromagnetic Ising model near the Yang-Lee edge on a family of Sierpinski gasket lattices whose members are labeled by an integer $b$ ($2\le b<\infty$). The obtained exact results on the first seven members of this family show that, for $b\ge 4$, associated correlation length diverges more slowly than any power law when distance $\delta h$ from the edge tends to zero, ${\xi }_{\mathrm{YL}}\sim \exp \left[\ln \left(b\right)\sqrt{\left|\ln \left(\delta h\right)\right|/\ln \left({\lambda }_0\right)}\right]$, ${\lambda }_0$ being a decreasing function of $b$. We suggest a possible scenario for the emergence of the usual power-law behavior in the limit of very large $b$ when fractal lattices become almost compact.

We consider two coupled particles moving along a periodic substrate potential with negligible inertia effects (overdamped limit). Even when the particles are identical and the substrate spatially symmetric, a sinusoidal external driving of appropriate amplitude and frequency may lead to spontaneous symmetry breaking in the form of a permanent directed motion of the dimer. Thermal noise restores ergodicity and thus zero net velocity, but entails arbitrarily fast diffusion of the dimer for sufficiently weak noise. Moreover, upon application of a static bias force, the dimer exhibits a motion opposite to that force (absolute negative mobility). The key requirement for all these effects is a nonconvex interaction potential of the two particles.

We fluidize a granular medium with localized stirring in a split-bottom shear cell. We probe the mechanical response of quiescent regions far from the main flow by observing the vertical motion of cylindrical probes rising, sinking, and floating in the grains. First, we find that the probe motion suggests that the granular material behaves in a liquid-like manner: high-density probes sink and low-density probes float at the depth given by Archimedes’ law. Second, we observe that the drag force on moving probes scales linearly with their velocity, which allows us to define an effective viscosity for the system. This effective viscosity is inversely proportional to the rotation rate of the disk which drives the split bottom flow. Moreover, the apparent viscosity depends on radius and mass of the probe: despite the linear dependence of the drag forces on sinking speed of the probe, the granular medium is not simply Newtonian, but exhibits a more complex rheology. The decrease of viscosity with filling height of the cell, combined with the poor correlation between local strain rate and viscosity, suggests that the fluid-like character of the material is set by agitations generated in the stirred region: the relation between applied stress and observed strain rate in one location depends on the strain rate in another location. We probe the nature of the granular fluctuations that we believe mediates these nonlocal interactions by characterizing the small and random up and down motion that the probe experiences. These Gaussian fluctuations exhibit a mix of diffusive and subdiffusive behavior at short times and saturate at a value of roughly 1/10th of a grain diameter longer times, consistent with the picture of a random walker in a potential well. The product of crossover time and effective viscosity is constant, evidencing a direct link between fluctuations and viscosity.

When a liquid melt is cooled, a glass or phase transition can be obtained depending on the cooling rate. Yet, this behavior has not been clearly captured in energy-landscape models. Here, a model is provided in which two key ingredients are considered in the landscape, metastable states and their multiplicity. Metastable states are considered as in two level system models. However, their multiplicity and topology allows a phase transition in the thermodynamic limit for slow cooling, while a transition to the glass is obtained for fast cooling. By solving the corresponding master equation, the minimal speed of cooling required to produce the glass is obtained as a function of the distribution of metastable states.

A diffuse interface model is developed to investigate the growth and coarsening in driven thin films under the influence of strong surface energy anisotropy, misfit strain, mass deposition, and diffusion. Two high-order regularizations are used to remove the ill-posedness introduced by the strong anisotropy: a Willmore (or bending energy) regularization and a simplified linear regularization. For similar effective growth rates and regularization strengths, the two regularizations undergo similar dynamics and coarsening, as peaks (antikinks) and valleys (kinks) of the corrugated surface are removed as the film grows. Competition among the influence of elastic strains, surface energy anisotropy and the rate of deposition lead to a change in the coarsening events compared to that observed in unstressed thin films. As the rate of deposition is increased, the primary coarsening events change from a combination of both antikink-ternary and binary events at low deposition rates, to binary events at moderate rates, and finally to kink-ternary events at high rates. The kink-ternary events are the only coarsening events observed in growing unstressed films at all deposition rates. Accordingly, the coarsening dynamics of two-dimensional (2D) stressed films is much faster than for unstressed films at low deposition rates. In addition, deep grooves may form in the film due to the Asaro-Tiller-Grinfeld instability and may lead to the breakup of the film into islands. At higher deposition rates, the evolution of stressed films is found to become limited by the surface energy anisotropy and the regularizations, which inhibits groove formation. The corresponding coarsening events and rates then tend toward those observed in unstressed films. Preliminary results of 3D thin films indicate that coarsening processes in elastically stressed films are also much faster than their unstressed counterparts at low deposition rates. In three dimensions, however, the coarsening events are also found to involve saddles.

We consider a free energy, within the framework of the Onsager approximation, for a spatially and orientationally inhomogeneous distribution of hard rods confined on a spherical surface. These rods interact with each other though the excluded-volume interaction, forming a textured nematic structure on the spherical surface at high surface coverage. Our numerical solution to the model shows that the splay state, where on average rods line up in parallel to the longitudes on the spherical surface, is the only stable state. Other types of textures that have recently been suggested were also tested and all yield higher free energy than that of a ground splay state. We also provide a study of the disorder-splay transition, which is shown to have first-order characteristics.

The distribution of counterions and the electrostatic interaction between two similarly charged dielectric slabs is studied in the strong coupling limit. Dielectric inhomogeneities and discreteness of charge on the slabs have been taken into account. It is found that the amount of dielectric constant difference between the slabs and the environment, and the discreteness of charge on the slabs have opposing effects on the equilibrium distribution of the counterions. At small interslab separations, increasing the amount of dielectric constant difference increases the tendency of the counterions toward the middle of the intersurface space between the slabs and the discreteness of charge pushes them to the surfaces of the slabs. In the limit of point charges, independent of the strength of dielectric inhomogeneity, counterions distribute near the surfaces of the slabs. The interaction between the slabs is attractive at low temperatures and its strength increases with the dielectric constant difference. At room temperature, the slabs may completely attract each other, reach to an equilibrium separation, or have two equilibrium separations with a barrier in between, depending on the system parameters.

The immune system is one of the most complex signal processing machineries in biology. The adaptive immune system, consisting of B and T lymphocytes, is activated in response to a large spectrum of pathogen antigens. B cells recognize and bind the antigen through B-cell receptors (BCRs) and this is fundamental for B-cell activation. However, the system response is dependent on BCR-antigen affinity values that span several orders of magnitude. Moreover, the ability of the BCR to discriminate between affinities at the high end (e.g., ${10}^9{M}^{-1}$–${10}^{10}{M}^{-1}$) challenges the formulation of a mathematical model able to robustly separate these affinity-dependent responses. Queuing theory enables the analysis of many related processes, such as those resulting from the stochasticity of protein binding and unbinding events. Here we define a network of queues, consisting of BCR early signaling states and transition rates related to the propensity of molecular aggregates to form or disassemble. By considering the family of marginal distributions of BCRs in a given signaling state, we report a significant separation (measured as Jensen-Shannon divergence) that arises from a broad spectrum of antigen affinities.

We discuss the nature of nonequilibrium phase transitions in the Hamiltonian mean-field model using detailed numerical simulations of the Vlasov equation and molecular dynamics. Starting from fixed magnetization water bag initial distributions and varying the energy, the states obtained after a violent relaxation undergo a phase transition from magnetized to nonmagnetized states when going from lower to higher energies. The phase transitions are either first order or are composed of a cascade of phase reentrances. This result is at variance with most previous results in the literature mainly based on the Lynden-Bell theory of violent relaxation. The latter is a rough approximation and, consequently, is not suited for an accurate description of nonequilibrium phase transition in long-range interacting systems.

We characterize the distributions of size and duration of avalanches propagating in complex networks. By an avalanche we mean the sequence of events initiated by the externally stimulated excitation of a network node, which may, with some probability, then stimulate subsequent excitations of the nodes to which it is connected, resulting in a cascade of excitations. This type of process is relevant to a wide variety of situations, including neuroscience, cascading failures on electrical power grids, and epidemiology. We find that the statistics of avalanches can be characterized in terms of the largest eigenvalue and corresponding eigenvector of an appropriate adjacency matrix that encodes the structure of the network. By using mean-field analyses, previous studies of avalanches in networks have not considered the effect of network structure on the distribution of size and duration of avalanches. Our results apply to individual networks (rather than network ensembles) and provide expressions for the distributions of size and duration of avalanches starting at particular nodes in the network. These findings might find application in the analysis of branching processes in networks, such as cascading power grid failures and critical brain dynamics. In particular, our results show that some experimental signatures of critical brain dynamics (i.e., power-law distributions of size and duration of neuronal avalanches) are robust to complex underlying network topologies.

A chlorine–iodine–malonic-acid Turing system involving a local concentration-dependent diffusivity (LCDD) has fundamental significance for physical, chemical, and biological systems with inhomogeneous medium. We investigated such a system by both numerical computation and mathematical analysis. Our research reveals that a variable local diffusivity has an evident effect on regulating the Turing patterns for different modes. An intrinsic square-root law is given by λ ∼ ($c_1+c_{2}k$)${}^{1/2}$, which relates the pattern wavelength (λ) with the LCDD coefficient ($k$). This law indicates that the system pattern has the properties of an equivalent Turing pattern. The current study confirms that, for the Turing system with LCDD, the system pattern form retains the basic characteristics of a traditional Turing pattern in a wide range of LCDD coefficients.

We study the random concatenation of slightly different two-dimensional Hamiltonian maps with a mixed phase space. We consider a regular island whose fixed point is identical for all maps. Trajectories of the concatenated maps near this fixed point are no longer confined to invariant tori. We derive a stochastic model for the distance from the fixed point, which turns out to be a biased random walk with multiplicative noise. We give an analytical prediction of the survival probability of trajectories inside the regular island, which asymptotically is the product of a power law and an exponential. We confirm these results numerically for the parametrically perturbed standard map.

We study incompressible magnetohydrodynamic turbulence in both two and three dimensions, with an emphasis on the number of degrees of freedom $N$. This number is estimated in terms of the magnetic Prandtl number $\mathrm{Pr}$, kinetic Reynolds number $\mathrm{Re}$, and magnetic Reynolds number $\mathrm{Rm}$. Here $\mathrm{Re}$ and $\mathrm{Rm}$ are dynamic in nature, defined in terms of the kinetic and magnetic energy dissipation rates (or averages of the velocity and magnetic field gradients), viscosity and magnetic diffusivity, and the system size. It is found that for the two-dimensional case, $N$ satisfies $N⩽\mathrm{Pr}{\mathrm{Re}}^{3/2}+{\mathrm{Rm}}^{3/2}$ for $\mathrm{Pr}>1$ and $N⩽{\mathrm{Re}}^{3/2}+{\mathrm{Pr}}^{-1}{\mathrm{Rm}}^{3/2}$ for $\mathrm{Pr}⩽1$. In three dimensions, on the other hand, $N$ satisfies $N⩽{(\mathrm{Pr}{\mathrm{Re}}^{3/2}+{\mathrm{Rm}}^{3/2})}^{3/2}$ for $\mathrm{Pr}>1$ and $N⩽{({\mathrm{Re}}^{3/2}+{\mathrm{Pr}}^{-1}{\mathrm{Rm}}^{3/2})}^{3/2}$ for $\mathrm{Pr}⩽1$. In the limit $\mathrm{Pr}\to 0$, ${\mathrm{Re}}^{3/2}$ dominates ${\mathrm{Pr}}^{-1}{\mathrm{Rm}}^{3/2}$, and the present estimate for $N$ appropriately reduces to ${\mathrm{Re}}^{9/4}$ as in the case of usual Navier-Stokes turbulence. For $\mathrm{Pr}\approx 1$, our results imply the classical spectral scaling of the energy inertial range and dissipation wave number (in the form of upper bounds). These bounds are consistent with the existing predictions in the literature for turbulence with or without Alfvén wave effects. We discuss the possibility of solution regularity, with an emphasis on the two-dimensional case in the absence of either one or both of the dissipation terms.

Large-scale particle-in-cell calculations of the plasma wake interactions of two negatively charged grains smaller than the Debye length are carried out using the coptic code over a wide range of subsonic plasma flow velocities. In plasmas with the temperature ratio $T_e/T_i=100$, it is found that a single grain's oscillatory wake disappears for flow Mach numbers $M$ less than approximately 0.3, which is the parameter regime where Landau damping is expected to be strong. Neutral collisions suppress potential oscillations above $M=0.3$, but not the trailing attractive potential peak caused by ion focusing. The transverse (grain-aligning) force on a downstream particle in the wake of another is obtained rigorously from the code in three-dimensional simulations. It shows general agreement with the force that would be deduced from the single-grain wake potential gradient. Except for relatively large grains in the nonlinear collisional regime, the grain-aligning force is very small for slow flow.

We consider the nearest-neighbor spacing distributions of mixed random matrix ensembles interpolating between different symmetry classes or between integrable and nonintegrable systems. We derive analytical formulas for the spacing distributions of $2\times 2$ or $4\times 4$ matrices and show numerically that they provide very good approximations for those of random matrices with large dimension. This generalizes the Wigner surmise, which is valid for pure ensembles that are recovered as limits of the mixed ensembles. We show how the coupling parameters of small and large matrices must be matched depending on the local eigenvalue density.

In a recent work [Schneider et al., Proc. Natl. Acad. Sci. USA 108, 3838 (2011)], the authors proposed a simple measure for network robustness under malicious attacks on nodes. Using a greedy algorithm, they found that the optimal structure with respect to this quantity is an onion structure in which high-degree nodes form a core surrounded by rings of nodes with decreasing degree. However, in real networks the failure can also occur in links such as dysfunctional power cables and blocked airlines. Accordingly, complementary to the node-robustness measurement ($R_n$), we propose a link-robustness index ($R_l$). We show that solely enhancing $R_n$ cannot guarantee the improvement of $R_l$. Moreover, the structure of an $R_l$-optimized network is found to be entirely different from that of an onion network. In order to design robust networks that are resistant to a more realistic attack condition, we propose a hybrid greedy algorithm that takes both the $R_n$ and $R_l$ into account. We validate the robustness of our generated networks against malicious attacks mixed with both nodes and links failure. Finally, some economical constraints for swapping the links in real networks are considered, and significant improvement in both aspects of robustness is still achieved.

Turbulent convection in quasi-one-dimensional geometry is studied by means of high-resolution direct numerical simulations within the framework of Rayleigh-Taylor turbulence. Geometrical confinement has dramatic effects on the dynamics of the turbulent flow, inducing a transition from superdiffusive to subdiffusive evolution of the mixing layer and arresting the growth of kinetic energy. A nonlinear diffusion model is shown to reproduce accurately the above phenomenology. The model is used to predict, without free parameters, the spatiotemporal evolution of the heat flux profile and the dependence of the Nusselt number on the Rayleigh number.