Narrow Results

We extend the Sznajd Model for opinion formation by introducing persuasion probabilities for opinions. Moreover, we couple the system to an environment which mimics the application of the opinion. This results in a feedback, representing single-state opinion transitions in opposite to the two-state opinion transitions for persuading other people. We call this model opinion formation in an open community (OFOC). It can be seen as a stochastic extension of the Sznajd model for an open community, because it allows for a special choice of parameters to recover the original Sznajd model. We demonstrate the effect of feedback in the OFOC model by applying it to a scenario in which, e.g., opinion B is worse then opinion A but easier explained to other people. Casually formulated we analyzed the question, how much better one has to be, in order to persuade other people, provided the opinion is worse. Our results reveal a linear relation between the transition probability for opinion B and the influence of the environment on B.

In the present work, we propose a microscopic model of financial markets based on particle-cluster aggregation on a two-dimensional small-world information network in order to simulate the dynamics of the stock markets. "Stylized facts" of the financial market time series, such as fat-tail distribution of returns, volatility clustering and multifractality, are observed in the model. The results of the model agree with empirical data taken from historical records of the daily closures of the NYSE composite index.

The order parameters and synchronization are numerically investigated in the time delay small-world connected FitzHugh–Nagumo excitable systems. The simulations show that the order parameter continuously decreases with increasing D, the quality of the synchronization worsens for large noise intensity. As the coupling intensity goes up, the quality of the synchronization worsens also, and find that the larger rewiring probability becomes, the larger-order parameter. At the same time, the order parameter goes up, the time delay declines. We obtained the complete phase diagram for a wide range of values of noise intensity D and control parameter g.

In this paper, we present a growing complex network model, namely, the generalized corona network (GCN) which is built on a base network and a sequence of networks by using corona product of graphs. This construction generalizes several existing complex network models. We study the structural properties of the special classes of generalized corona networks and show that these networks have small diameter, the cumulative betweenness distribution follows a power-law distribution, the degree distribution decay exponentially, small average path length with the network order, high clustering coefficient and small-world behavior. Further, we obtain the spectra of the adjacency matrix and the signless Laplacian matrix of GCN when the constituting networks are regular. Also, we obtain the Laplacian spectra for all generalized corona networks. In addition, explicitly give eigenvector corresponding to the adjacency and Laplacian eigenvalues. Finally, we derive the spectral radius and the algebraic connectivity of GCN.

We study an extension of the Bornholdt spin-market model using configurable network topology. The distribution of returns was studied using the probability plot correlation coefficient and indicates three different behaviors for the tails of the distribution of returns. The price volatility was studied by fitting the magnetization with a Wiener process and a power-law behavior was found for the volatility as a function of the levels of randomness and connectivity of the network. Both parameters have opposing effects on the risk as inferred from the Shannon entropy of the magnetization. Also, we show that trading can spatially auto-organize depending on the conditions of the control space. Finally, we show that there is a range of values of the control space that renders a model that reproduces real-market data.

In this paper, two definitions about traffic congestion are proposed. We describe the traffic congestion spreading with the SEIR model of a complex small-world network. In addition, the relationships among the congestion factor, the number of average infection, the average recover rate and the infection rate are given by simulations in general traffic congestion conditions.

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. We analyze numerically the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D₀ = 0.920 ∼ 0.930 for the different network sizes and random rewiring fractions. In particular, our simulation results are compared with the numerical computations for regular networks.

When being constructed, complex dynamical networks can lose stability in the sense of Lyapunov (i. s. L.) due to positive feedback. Thus, there is much important worthiness in the theory and applications of complex dynamical networks to study the stability. In this paper, according to dissipative system criteria, we give the stability condition in general complex dynamical networks, especially, in NW small-world and BA scale-free networks. The results of theoretical analysis and numerical simulation show that the stability i. s. L. depends on the maximal connectivity of the network. Finally, we show a numerical example to verify our theoretical results.

In multi-agent system (MAS), the communication topology of agent network plays a very important role in its collaboration. Small-world networks are the networks with high local clustering and small average path length, and the communication networks of MAS can be analyzed within the frame of small-world topology. Yet the real multiagent communication networks are abundant and the classical WS small-world model is not suitable for all cases. In this paper, two new small-world network models are presented. One is based on random graph substrate and local nodes preference reconnection and the other is based on regular graph substrate and long-range nodes preference reconnection. The characteristic of the network parameter such as the clustering coefficients, average path length, and eigenvalue λ₂ and λ_n of the Laplacian matrix for these two models and WS model is studied. The consensus problem that based on these three models is also studied. An example is given and the conclusions are made in the end.

Distinguishing the long-range bonds with the regular ones, the critical temperature of the spin-lattice Gaussian model built on two typical small-world networks is studied. The results show much difference from the classical case, and thus may induce some more accurate discussion on the critical properties of the spin-lattice systems combined with the small-world networks.

This paper presents a detailed analysis on the dynamics of a two-dimensional delayed small-world network under delayed state feedback control. On the basis of stability switch criteria, the equilibrium is studied, and the stability conditions are determined. This study shows that with properly chosen delay and gain in the delayed feedback path, the controlled small-world delayed network may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or the multistability solutions via three types of codimension two bifurcations. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are determined by using the normal form theory and center manifold theorem. In addition, the study shows that the controlled model exhibits period-doubling bifurcations which lead eventually to chaos; and the chaos can also directly occur via the bifurcations from the quasi-periodic solutions. The results show that the delayed feedback is an effective approach in order to generate or annihilate complex behaviors in practical applications.

In a small-world network of mainly attractively coupled nonidentical neurons, we show that a small fraction of phase-repulsive couplings is able to strongly improve synchronization for certain values of the link strength, and long-range connection probability. By means of a spectral analysis we relate the observed dynamical behavior with the structural properties of the network.

We present a time-delayed SIS model on complex networks to study epidemic spreading. We found that the existence of delay will affect, and oftentimes enhance, both outbreak and prevalence of infectious diseases in the networks. For small-world networks, we found that the epidemic threshold and the delay time have a power-law relation. For scale-free networks, we found that for a given transmission rate, the epidemic prevalence has an exponential form, which can be analytically obtained, and it decays as the delay time increases. We confirm all results by sufficient numerical simulations.

The response of a random and modular network to the simultaneous presence of two frequencies is considered. The competition for controlling the dynamics of the network results in different behaviors, such as frequency changes or permanent synchronization frustration, which can be directly related to the network structure. From these observations, we propose a new method for detecting overlapping communities in structured networks.

The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any "d" dimensional network. It shows that to stabilize a "d" dimensional delayed small-world network, at least a "d – 1" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems.

In this paper, we mainly discuss effects of partial time delay on temporal dynamics of Watts–Strogatz (WS) small-world neuronal networks by controlling two parameters. One is the time delay $\tau$ and the other is the probability of partial time delay $p_{\mathrm{\text{delay}}}$. Temporal dynamics of WS small-world neuronal networks are discussed with the aid of temporal coherence and mean firing rate. With the obtained simulation results, it is revealed that for small time delay $\tau$, the probability $p_{\mathrm{\text{delay}}}$ could weaken temporal coherence and increase mean firing rate of neuronal networks, which indicates that it could improve neuronal firings of the neuronal networks while destroying firing regularity. For large time delay $\tau$, temporal coherence and mean firing rate do not have great changes with respect to $p_{\mathrm{\text{delay}}}$. Time delay $\tau$ always has great influence on both temporal coherence and mean firing rate no matter what is the value of $p_{\mathrm{\text{delay}}}$. Moreover, with the analysis of spike trains and histograms of interspike intervals of neurons inside neuronal networks, it is found that the effects of partial time delays on temporal coherence and mean firing rate could be the result of locking between the period of neuronal firing activities and the value of time delay $\tau$. In brief, partial time delay could have great influence on temporal dynamics of the neuronal networks.

We investigate the multifractals of the first passage time on a one-dimensional small-world network with reflecting and absorbing barriers. The multifractals can be obtained from the distribution of the first passage time at which the random walker arrives for the first time at an absorbing barrier after starting from an arbitrary initial site. Our simulation is found to estimate the fractal dimension D₀ = 0.920 ~ 0.930 for the different network sizes and random rewiring fractions. In particular, the multifractal structure breaks down into a small-world network, when the rewiring fraction p is larger than the critical value p_c = 0.3. Our simulation results are compared with the numerical computations for regular networks.

A model of particle-cluster aggregation on the rewired small-world network is studied. In this model, particles can adjoin to the end of the rewired long-range edges. The rewired long-range edge is controlled by two parameters, the reconnecting probability and distance controlling parameter. The simulation results indicate that, with the increasing of the length and the number of rewired long-range edges, the patterns become diffuse and the fractal dimensions of the aggregates decrease. The range of the value of dimension and the feature of patterns obtained in the model are consistent with the amyloid deposits.

The fractal and small-word properties are two important properties of complex networks. In this paper, we propose a new random rewiring method to transform fractal networks into small-world networks. We theoretically prove that the proposed method can retain the degree of all nodes (hence the degree distribution) and the connectivity of the network. Further, we also theoretically prove that our method also retains the tree structure of tree graphs. Our method can transform many different types of fractal networks into small-world networks while the degree distribution and connectivity of these networks remain unchanged, demonstrating the generality of small-world networks. In addition, the method also works for other types of complex networks. The rewiring method proposed in this paper can be used in a broader range of applications of network analysis.

The connectivity structure in cortical networks defines how information is transmitted and processed, and it is a source of the complex spatiotemporal patterns of network's development, and the process of creation and deletion of connections is continuous in the whole life of the organism. In this paper, we study how neural activity influences the growth process in neural networks. By using a two-dimensional activity-dependent growth model we demonstrated the neural network growth process from disconnected neurons to fully connected networks. For making quantitative investigation of the network's activity influence on its topological properties we compared it with the random growth network not depending on network's activity. By using the random graphs theory methods for the analysis of the network's connections structure it is shown that the growth in neural networks results in the formation of a well-known “small-world” network.