Synchronization criteria for a generalized complex delayed dynamical network model

It has been demonstrated that most complex networks display synchronization phenomena, and this problem has attracted great attention of various fields including science and engineering. In this paper, a generalized complex dynamical network model with time-varying delays was presented. Kronecker product was adopted to investigate this model. Moreover, several synchronization criteria were derived for both delay-independent and delay-dependent asymptotical stability. Especially, it has been shown that synchronization of such dynamical network is determined by the linear matrix inequality consisting of coupling configuration matrices, inner-coupling matrices and isolated cells. At last, illustrative examples were given to validate the above-acquired.     

Hitting time distributions in financial markets

We analyze the hitting time distributions of stock price returns in different time windows, characterized by different levels of noise present in the market. The study has been performed on two sets of data from US markets. The first one is composed by daily price of 1071 stocks trade for the 12-year period 1987-1998, the second one is composed by high frequency data for 100 stocks for the 4-year period 1995-1998. We compare the probability distribution obtained by our empirical analysis with those obtained from different models for stock market evolution. Specifically by focusing on the statistical properties of the hitting times to reach a barrier or a given threshold, we compare the probability density function (PDF) of three models, namely the geometric Brownian motion, the GARCH model and the Heston model with that obtained from real market data. We will present also some results of a generalized Heston model.     

A directed network of Greek and Roman mythology

We construct a directed network using a dictionary of Greek and Roman mythology in which the nodes represent the entries listed in the dictionary and we make directional links from an entry to other entries that appear in its explanatory part. We find that this network is clearly not a random network but a directed scale-free network in which the distributions of out-degree and in-degree follow a power-law with exponents γ_out≈3.0 and γ_in≈2.5, respectively. Also we measure several quantities which describe the topological properties of the network and compare it to that of other real networks.     

Quantitative agent-based firm dynamics simulation with parameters estimated by financial and transaction data analysis

Firm dynamics on a transaction network is considered from the standpoint of econophysics, agent-based simulations, and game theory. In this model, interacting firms rationally invest in a production facility to maximize net present value. We estimate parameters used in the model through empirical analysis of financial and transaction data. We propose two different methods (analytical method and regression method) to obtain an interaction matrix of firms. On a subset of a real transaction network, we simulate firm's revenue, cost, and fixed asset, which is the accumulated investment for the production facility. The simulation reproduces the quantitative behavior of past revenues and costs within a standard error when we use the interaction matrix estimated by the regression method, in which only transaction pairs are taken into account. Furthermore, the simulation qualitatively reproduces past data of fixed assets.     

Clustering and maximal flow in vehicular traffic through a sequence of traffic lights

We study the maximal current (maximum traffic capacity) of vehicular traffic through a sequence of traffic lights on a highway, where all signals turn on and off synchronously. The dynamical model of vehicular traffic controlled by signals is expressed in terms of a nonlinear map, where the excluded-volume effect is taken into account. The dynamical behaviors of vehicles are clarified by analyzing traffic patterns. The clustering of vehicles varies with the cycle time of signals. The maximum current is closely connected to vehicular clustering. Clustering of vehicles is controlled by varying both split and cycle time of signals. The dependence of the maximal current on both split and cycle time is derived.     

Evolutionary minority game on complex networks

In this work we investigate the dynamics of networked evolutionary minority game (NEMG) wherein each agent is allowed to evolve its strategy according to the information obtained from its neighbors in the network. We investigate four kinds of networks, including star network, regular network, random network and scale-free network. Simulation results indicate that the dynamics of the system depends crucially on the structure of the underlying network. The strategy distribution in a star network is sensitive to the precise value of the mutation magnitude L, in contrast to the strategy distribution in regular, random and scale-free networks, which is easily affected by the value of the prize-to-fine ratio R. Under a simple evolutionary scheme, the networked system with suitable parameters evolves to a high level of global coordination among its agents. In particular, the performance of the system is correlated to the clustering property of the network, where larger clustering coefficient leads to better performance.     

Novel criteria of synchronization stability in complex networks with coupling delays

Complex networks are wide spread in the real world, arising in fields as disparate as sociology, physics and biology. The information spreading through a complex network is often associated with time delays due to the finite speeds of signal transmission over a distance. Hence, complex networks with coupling delays have gained increasing attention in various fields of science and engineering today. In this paper, based on the theory of asymptotic stability of linear time-delay systems, synchronization stability in complex dynamical networks with coupling delays is investigated, and we derive novel criteria of synchronization state for both delay-independent and delay-dependent stabilities. As illustrative examples, we use the networks with coupling delays and a given coupling scheme to test the theoretical results.     

Consensus problem in multi-agent systems with physical position neighbourhood evolving network

In multi-agent system (MAS), the communication topology of agent network plays a very important role in its consensus problem. To describe the communication topologies of MAS, a class of evolving network models with the concept of physical position neighbourhood connectivity are proposed and studied in this paper. The analysis and simulation results for network parameters such as the first nonzero eigenvalue and maximal eigenvalue of graph Laplacian matrix, clustering coefficients, average distances and degree distributions for different evolving parameters of these models are presented. The dynamical behaviour of each node on the consensus problem is also studied. It was found that the time to reach consensus becomes shorter sharply with the increasing of neighbourhood depth of the nodes in these models. And it was also found that for the maximal distance preferential attachment model (Model 3), the synthetic characteristic, such as robustness to communication delay, as well as convergence speed in consensus problem, is the best in all these models.     

Dispersion of Dust Acoustic Modes and Perturbations of Plasma Flux Balance

Previous considerations of dust acoustic waves is demonstrated to be inconsistent ‐ the required equilibrium state for perturbations was not defined since balance of plasma fluxes was neglecting. The self‐consistent treatment shows that plasma flux perturbations are accompanying any collective waves propagating in dusty plasmas and can play an important role in wave dispersion, wave damping and can create instabilities. This is illustrated by the derivation of dispersion relation for dust acoustic modes taking into account the plasma flux balances and plasma flux perturbations by waves. The result of this approach shows that the dust acoustic waves with linear dependence of wave frequency on the wave number exist only in restricted range of the wave numbers. Only for wave numbers larger than some critical wave number for low frequency modes the frequency can be have approximately a linear dependence on wave number and can be called as dust acoustic wave but the phase velocity of these waves is different from that which can be obtained neglecting the flux balance and depends on grain charge variations which are determined by the balance of fluxes. The presence of plasma fluxes previously neglected is the main typical feature of dusty plasmas. The dispersion relation in the range of small wave numbers is found to be mainly determined by the change of the plasma fluxes and is quite different from that of dust acoustic type, namely it is found to have the same form as the well known dispersion relation for the gravitational instability. This result proves in general way the existence of the collective grain attractions of negatively charged grains for for large distances between them and for any source of ionization. The attraction of grains found from dispersion relation of the dust acoustic branch coincides with that found previously for pair grain interactions using some models for the ionization source. For the existing experiments the effective Jeans length for such attraction is estimated to be about 8 – 10 times larger than the ion Debye length and the effective gravitational constant for the grain attraction is estimated to be several orders of magnitude larger than the usual gravitational constant. The grain attraction at large inter‐grain distances described by the gravitationlike grain instability is considered as the simplest explanation for observed dust cloud clustering, formation of dust structures including the plasma crystals. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)