We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of the adjacency matrix of various model networks, namely, random, scale-free, and small-world networks. These distributions follow the Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via the ${\Delta }_3$ statistic of RMT as well. It follows RMT prediction of linear behavior in semilogarithmic scale with the slope being $\sim 1∕{\pi }^2$. Random and scale-free networks follow RMT prediction for very large scale. A small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.

We propose a scale-free network similar to Barabasi-Albert networks but with two different types of edges. This model is based on the idea that in many cases there are more than one kind of link in a network and when a new node enters the network both old nodes and different kinds of links compete to obtain it. The degree distribution of both the total degree and the degree of each type of edge is analyzed and found to be scale-free. Simulations are shown to confirm these results.

Empirical evidence for scale-invariant distributions in financial data has attracted the research interest of physicists. While the power-law tails of the distribution of stock returns $P\{R>x\}\sim x^{-{\zeta }_R}$ are becoming increasingly well documented, less understood are the statistics of other closely related microstructural variables such as $q_i$, the number of shares exchanged in trade $i$ (termed the trade size) and $Q_{\Delta t}\left(t\right)={\sum }_{i=1}^N{q}_i$, the total number of shares exchanged as a result of the $N=N_{\Delta t}$ trades occurring in a time interval $\Delta t$ (termed share volume). We analyze the statistical properties of trade size $q\equiv q_i$ and share volume $Q\equiv Q_{\Delta t}\left(t\right)$ by analyzing trade-by-trade data from three large databases representing three distinct markets: (i) 1000 major U.S. stocks for the $2\text{−}\mathrm{y}$ period 1994–1995, (ii) 85 major U.K. stocks for the $2\text{−}\mathrm{y}$ period 2001–2002, and (iii) 13 major Paris Bourse stocks for the $4.5\text{−}\mathrm{y}$ period 1994–1999. We find that, for all three markets analyzed, the cumulative distribution of trade size displays a power-law tail $P(q>x)\sim x^{-{\zeta }_q}$ with exponent ${\zeta }_q<2$ within the Lévy stable domain. Our analysis of the exponent estimates of ${\zeta }_q$ suggests that the exponent value is universal in the following respects: (a) ${\zeta }_q$ is consistent across stocks within each of the three markets analyzed, and also across different markets, and (b) ${\zeta }_q$ does not display any systematic dependence on market capitalization or industry sector. We next analyze the distributions of share volume $Q_{\Delta t}$ over fixed time intervals and find that for all three markets $P\{Q>x\}\sim x^{-{\zeta }_Q}$ with exponent ${\zeta }_Q<2$ within the Lévy stable domain. To test the validity for $\Delta t=1\text{day}$ of the power-law distributions found from tick-by-tick data, we analyze a fourth large database containing daily U.S. data, and confirm a value for the exponent ${\zeta }_Q$ within the Lévy stable domain.

Using the replica-exchange multicanonical Monte Carlo simulation, the intra-association of hydrophobic segments in a heteropolymer was numerically investigated by the microcanonical analysis method. We demonstrated that the microcanonical entropy shows the features of one or multiple convexes in the association transition region depending on the number and distribution of hydrophobic segments in the chain. We found that one or multiple negative specific heats imply a first-order-like transition with the coexistence of multiple phases.

In this paper, we consider the propagation of water waves in a long-wave asymptotic regime, when the bottom topography is periodic on a short length scale. We perform a multiscale asymptotic analysis of the full potential theory model and of a family of reduced Boussinesq systems parametrized by a free parameter that is the depth at which the velocity is evaluated. We obtain explicit expressions for the coefficients of the resulting effective Korteweg–de Vries (KdV) equations. We show that it is possible to choose the free parameter of the reduced model so as to match the KdV limits of the full and reduced models. Hence the reduced model is optimal regarding the embedded linear weakly dispersive and weakly nonlinear characteristics of the underlying physical problem, which has a microstructure. We also discuss the impact of the rough bottom on the effective wave propagation. In particular, nonlinearity is enhanced and we can distinguish two regimes depending on the period of the bottom where the dispersion is either enhanced or reduced compared to the flat bottom case.

Kinetic equations are derived from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for point vortex systems in an infinite plane. At the level of approximation for the Landau equation, the collision term of the kinetic equation derived coincides with that by Chavanis [Phys. Rev. E 64, 026309 (2001)]. Furthermore, we derive a kinetic equation corresponding to the Balescu-Lenard equation for plasmas using the theory of the Fredholm integral equation. For large $N$, this kinetic equation is reduced to the Landau equation above.

Nonspherical colloids can exhibit liquid-crystalline phases with different degrees of broken orientational and translational symmetry. Here we investigate hard rectangles consisting of photolithographically prepared disks standing on edge. We observe a conventional Kosterlitz-Thouless transition from isotropic to nematic with almost smectic behavior at high density. But just on the isotropic side of the isotropic to nematic transition we observe an unusual regime where short-range tetratic correlations dominate over nematic correlations. This occurs due to the proliferation of Ising-like $\pi ∕2$ grain boundaries that disrupt nematic order, but preserve tetratic correlations, at lengths shorter than the spacing between free disclinations.

We investigate the approach to stable and metastable equilibrium in both nearest-neighbor and long-range Ising models using a cluster representation. The distribution of nucleation times is determined using the Metropolis algorithm and the corresponding ${\varphi }^4$ model using Langevin dynamics. We find that the nucleation rate is suppressed at early times even after global variables such as the magnetization and energy have apparently reached their time independent values. The mean number of clusters whose size is comparable to the size of the nucleating droplet becomes time independent at about the same time that the nucleation rate reaches its constant value. We also find subtle structural differences between the nucleating droplets formed before and after apparent metastable equilibrium has been established.

The Hamiltonian mean-field model has been investigated, since its introduction about a decade ago, to study the equilibrium and dynamical properties of long-range interacting systems. Here we study the long-time behavior of long-lived, out-of-equilibrium, quasistationary dynamical states, whose lifetime diverges in the thermodynamic limit. The nature of these states has been the object of a lively debate in the recent past. We introduce a numerical tool, based on the fluctuations of the phase of the instantaneous magnetization of the system. Using this tool, we study the quasistationary states that arise when the system is started from different classes of initial conditions, showing that the new observable can be exploited to compute the lifetime of these states. We also show that quasistationary states are present not only below, but also above the critical temperature of the second-order magnetic phase transition of the model. We find that at supercritical temperatures the lifetime is much larger than at subcritical temperatures.

We analyze the exact partition function of the Ising model on a square lattice under helical boundary conditions obtained by Liaw et al. [Phys. Rev. E 73, 055101(R) (2006)]. Based on such an expression, we then extend the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive an exact asymptotic expansion of the logarithm of the partition function and its first to fourth derivatives at the critical point. From such results, we find that the shift exponent for the specific heat is $\lambda =1$ for all values of the helicity factor $d$. We also find that finite-size corrections for the free energy, the internal energy, and the specific heat of the model in a crucial way depend on the helicity factor of the lattice.

Since 1984, electric cell-substrate impedance sensing (ECIS) has been used to monitor cell behavior in tissue culture and has proven sensitive to cell morphological changes and cell motility. We have taken ECIS measurements on several cultures of noncancerous and cancerous human ovarian surface epithelial cells. By analyzing the noise in real and imaginary electrical impedance, we demonstrate that it is possible to distinguish the two cell types purely from the signatures of their electrical noise. Our measures include power-spectral exponents, Hurst and detrended fluctuation analysis, and estimates of correlation time; principal-component analysis combines all the measures. The noise from both cancerous and noncancerous cultures shows correlations on many time scales, but these correlations are stronger for the noncancerous cells.

A spontaneously active neural system that is capable of continual learning should also be capable of homeostasis of both firing rate and connectivity. Experimental evidence suggests that both types of homeostasis exist, and that connectivity is maintained at a state that is optimal for information transmission and storage. This state is referred to as the critical state. We present a simple stochastic computational Hebbian learning model that incorporates both firing rate and critical homeostasis, and we explore its stability and connectivity properties. We also examine the behavior of our model with a simulated seizure and with simulated acute deafferentation. We argue that a neural system that is more highly connected than the critical state (i.e., one that is “supercritical”) is epileptogenic. Based on our simulations, we predict that the postseizural and postdeafferentation states should be supercritical and epileptogenic. Furthermore, interventions that boost spontaneous activity should be protective against epileptogenesis.

We study the effects of inhomogeneous influence of individuals on collective phenomena. We focus analytically on a typical model of the majority rule, applied to the completely connected agents. Two types of individuals $A$ and $B$ with different influence activity are introduced. The individuals $A$ and $B$ are distributed randomly with concentrations $\nu$ and $1-\nu$ at the beginning and fixed further on. Our main result is that the location of the order-disorder transition is affected due to the introduction of the inhomogeneous influence. This result highlights the importance of inhomogeneous influence between different types of individuals during the process of opinion updating.

Recent measurements of the short-wavelength $(\sim 1–100\mathrm{nm})$ fluctuations in stacks of lipid membranes have revealed two distinct relaxations: a fast one (decay rate of $\sim 0.1{\mathrm{ns}}^{-1}$), which fits the known baroclinic mode of bulk lamellar phases, and a slower one $(\sim 1–10\mathrm{\mu }{\mathrm{s}}^{-1})$ of unknown origin. We show that the latter is accounted for by an overdamped capillary mode, depending on the surface tension of the stack and its anisotropic viscosity. We thereby demonstrate how the dynamic surface tension of membrane stacks could be extracted from such measurements.

Even more interesting than the intricate organization of complex networks is the dynamical behavior of systems underlain by such structures. Among the many types of dynamics, one particularly interesting category involves the evolution of trails left by moving agents progressing through random walks and dilating processes in a complex network. The emergence of trails is present in many dynamical process, such as pedestrian traffic, information flow, and metabolic pathways. Important problems related to trails include the reconstruction of the trail and the identification of its source, when complete knowledge of the trail is missing. In addition, the following of trails in multiagent systems represents a particularly interesting situation related to pedestrian dynamics and swarming intelligence. The present work addresses these three issues while taking into account permanent and transient marks left in the visited nodes. Different topologies are considered for trail reconstruction and trail source identification, including four complex network models and four real networks, namely, the Internet, the U.S. airlines network, an email network, and the scientific collaboration network of complex network researchers. Our results show that the topology of the network influences trail reconstruction, source identification, and agent dynamics.

Results of systematic studies of discrete dark solitons (DDSs) in the one-dimensional discrete nonlinear Schrödinger equation with the cubic-quintic on-site nonlinearity are reported. The model may be realized as an array of optical waveguides made of an appropriate non-Kerr material. First, regions free of the modulational instability are found for staggered and unstaggered cw states, which are then used as the background supporting DDS. Static solitons of both on-site and inter-site types are constructed. Eigenvalue spectra which determine the stability of DDSs against small perturbations are computed in a numerical form. For on-site solitons with the unstaggered background, the stability is also examined by dint of an analytical approximation, that represents the dark soliton by a single lattice site at which the field is different from cw states of two opposite signs that form the background of the DDS. Stability regions are identified for the DDSs of three types: unstaggered on-site, staggered on-site, and staggered inter-site; all unstaggered inter-site dark solitons are unstable. A remarkable feature of the model is coexistence of stable DDSs of the unstaggered and staggered types. The predicted stability is verified in direct simulations; it is found that unstable unstaggered DDSs decay, while unstable staggered ones tend to transform themselves into moving dark breathers. A possibility of setting DDS in motion is studied too. Analyzing the respective Peierls-Nabarro potential barrier, and using direct simulations, we infer that unstaggered DDSs cannot move, but their staggered counterparts can be readily set in motion.

In their textbook, Suzuki and Varga [Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems (Springer, Berlin, 1998)] present the stochastic variational method with the correlated Gaussian basis in a very exhaustive way. However, the Fourier transform of these functions and their application to the management of a relativistic kinetic energy operator are missing and cannot be found in the literature. In this paper we present these interesting formulas. We also give a derivation for formulations concerning central potentials.

We discuss the order parameter correlation function in the vicinity of continuous phase transitions using a two-parameter scaling form $G\left(k\right)=k_c^{-2}g(k\xi ,k∕k_c)$, where $k$ is the wave vector and $\xi$ is the correlation length, and the interaction-dependent nonuniversal momentum scale $k_c$ remains finite at the critical fixed point. The correlation function describes the entire critical regime and captures the classical to critical crossover. One-parameter scaling is recovered only in the limit $k∕k_c\to 0$. We present an approximate calculation of $g(x,y)$ for the Ising universality class using the functional renormalization group.

We investigate a simple dynamical model of a microtubule that evolves by attachment of guanosine triphosphate (GTP) tubulin to its end, irreversible conversion of GTP to guanosine diphosphate (GDP) tubulin by hydrolysis, and detachment of GDP at the end of a microtubule. As a function of rates of these processes, the microtubule can grow steadily or its length can fluctuate wildly. In the regime where detachment can be neglected, we find exact expressions for the tubule and GTP cap length distributions, as well as power-law length distributions of GTP and GDP islands. In the opposite limit of instantaneous detachment, we find the time between catastrophes, where the microtubule shrinks to zero length, and determine the size distribution of avalanches (sequence of consecutive GDP detachment events). We obtain the phase diagram for general rates and verify our predictions by numerical simulations.

The $1∕f^{\alpha }$ noise displayed by the fluctuation of the $n$th unfolded eigenvalue, where $n$ plays the role of a discrete time, was recently characterized for the classical Gaussian ensembles of $N\times N$ random matrices. It is investigated here for the $\beta$-Hermite ensemble by wavelet analysis of Monte Carlo simulated series both as a function of $\beta$ and of $N$. When $\beta$ decreases from 1 to 0, for a given and large enough $N$, the evolution from a $1∕f$ noise at $\beta =1$ Gaussian orthogonal ensemble (GOE) to a $1∕f^2$ noise at $\beta =0$ Gaussian diagonal ensemble (GDE) is heterogeneous with a $\sim 1∕f^2$ noise at the finest scales and a $\sim 1∕f$ noise at the coarsest ones.