Watts-Strogatz Small-world Network Research Articles

Reentrant phase transition in a predator-prey model

We numerically investigate the six-species predator-prey game in complex networks as well as in d -dimensional regular hypercubic lattices with d=1,2,...,6 . The food-web topology of the six species contains two directed loops, each of which is composed of cyclically predating three species. As the mutation rate is lowered below the well-defined phase transition point, the Z2 symmetry related with the interchange in the two loops is spontaneously broken, and it has been known that the system develops the defensive alliance in which three cyclically predating species defend each other against the invasion of other species. In the Watts-Strogatz small-world network structure characterized by the rewiring probability alpha , the phase diagram shows the reentrant behavior as alpha is varied, indicating a twofold role of the shortcuts. In d -dimensional regular hypercubic lattices, the system also exhibits the reentrant phase transition as d is increased. We identify universality class of the phase transition and discuss the proper mean-field limit of the system.

Three structural properties reflecting the synchronizability of complex networks

During the process of adding links, we find that the synchronizability of the classical Barabási-Albert (BA) scale-free or Watts-Strogatz (WS) small-world networks can be statistically quantified by three essentially structural quantities of these networks, i.e., the eccentricity, variance of the degree distribution, and clustering coefficients. The results indicate that both the eccentricity and clustering coefficient are positively linearly correlated with synchronizability, while the variance is negatively linearly correlated. Moreover, the efficiency of some particular strategies of adding links to change the synchronizability is also investigated. This information can be used to guide us to design corresponding strategies of structure-evolving processes to manipulate the synchronizability of a given network.

Effect of incubation period on epidemic spreading in complex networks

We study the effect of incubation period on epidemic spreading in the Barabasi–Albert scale-free network and the Watts–Strogatz small world network by using a Suspectable-Incubated-Infected-Suspectable model. Our analytical investigations show that the epidemic threshold is independent of incubation period in both networks, which is verified by our large-scale simulation results. We also investigate the effect of incubation period on the epidemic dynamics in a supercritical regime. It is found that with the increase of incubation period Ω, a damped oscillation evolution of ρT (the ratio of persons in incubated state) appears and the time needed to reach a saturation value increases. Moreover, the steady value of ρT increases and approaches to an asymptotic constant with the value of Ω increasing. As a result, the infected ratio ρI decreases with the increase of Ω according to a power law.

Research of malware propagation in complex networks based on 1-D cellular automata

In this paper, based on 1-D cellular automata, the probabilistic behaviors of malware propagation in complex networks are investigated. Neighborhood and state transition functions with integrated expression are established and two models of malware propagation are proposed to evaluate the probabilistic behavior of malware propagation in various networks. We run the proposed models on nearest-neighbor coupled network （NC） and Erdos-Renyi （ER） random graph network and Watts-Strogatz（WS） small world network and Barabasi-Albert （BA） power law network respectively. Analysis and simulations show that, the proposed models describe perfectly the dynamic behaviors of propagation in the above networks. Furthermore, the proposed models describe not only the average tendency of malware propagation but also the rare events such as saturation and extinction of malware, and overcome the limitation occurring in a deterministic model based on mean-field method that describes only the average tendency of malware propagation and neglects the probabilistic event. Meanwhile, the result of simulations shows that the heterogeneity of degree distribution and local spatial interaction are key factors affecting the malware propagation and immunization.

Localizations on complex networks

We study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, such as the participation ratio, the structural entropy, and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. Whole-cell networks in the real world and the Watts-Strogatz small-world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascending-order-ranked series of the occurrence probabilities at the nodes behave generally multifractally. This characteristic can be used as a structural measure of complex networks.

Control of spiking regularity in a noisy complex neural network

The effects of spatiotemporally correlated noise on the regularity of spiking oscillations are studied in a network composed of Fitz-Hugh-Nagumo neurons. The spiking regularity of the neural network becomes the best at a moderate noise intensity, indicating the occurrence of coherence resonance (CR). The CR in a Watts-Strogatz small-world network is further improved by adding a small fraction of long-range connections. Given a set of temporal correlation constant tau and spatial correlation length lambda of the noise, there exists an optimal network topology randomness, at which the spiking oscillations show the best regularity. The optimal randomness of the network topology at different tau and lambda varies in a narrow range. Changing lambda does not affect the optimal tau for achieving the most regular spike train, whereas varying tau, the best spiking regularity emerges at different optimal lambda.

PERFORMANCE AND ROBUSTNESS OF CELLULAR AUTOMATA COMPUTATION ON IRREGULAR NETWORKS

We investigate the performances of collective task-solving capabilities and the robustness of complex networks of automata using the density and synchronization problems as typical cases. We show by computer simulations that evolved Watts–Strogatz small-world networks have superior performance with respect to several kinds of scale-free graphs. In addition, we show that Watts–Strogatz networks are as robust in the face of random perturbations, both transient and permanent, as configuration scale-free networks, while being widely superior to Barabási–Albert networks. This result differs from information diffusion on scale-free networks, where random faults are highly tolerated by similar topologies.

Kinetic-growth self-avoiding walks on small-world networks

Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz small-world networks, rewired from a two-dimensional square lattice. The maximum length L of this kind of walks is limited in regular lattices by an attrition effect, which gives finite values for its mean value 〈L 〉. For random networks, this mean attrition length 〈L 〉 scales as a power of the network size, and diverges in the thermodynamic limit (system size N ↦∞). For small-world networks, we find a behavior that interpolates between those corresponding to regular lattices and randon networks, for rewiring probability p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition length of kinetically-grown walks are finite. For p = 1, 〈L 〉 grows with system size as N1/2, diverging in the thermodynamic limit. In this limit and close to p = 1, the mean attrition length diverges as (1-p)-4. Results of approximate probabilistic calculations agree well with those derived from numerical simulations.

Two-dimensional small-world networks: Navigation with local information

A navigation process is studied on a variant of the Watts-Strogatz small-world network model embedded on a square lattice. With probability , each vertex sends out a long-range link, and the probability of the other end of this link falling on a vertex at lattice distance away decays as r(-a). Vertices on the network have knowledge of only their nearest neighbors. In a navigation process, messages are forwarded to a designated target. For alpha < 3 and alpha not equal to 2, a scaling relation is found between the average actual path length and , where is the average length of the additional long range links. Given pL > 1, a dynamic small world effect is observed, and the behavior of the scaling function at large enough is obtained. At alpha = 2 and 3, this kind of scaling breaks down, and different functions of the average actual path length are obtained. For alpha > 3, the average actual path length is nearly linear with network size.

Effect of spatially correlated noise on coherence resonance in a network of excitable cells

We study the effect of spatially correlated noise on coherence resonance (CR) in a Watts-Strogatz small-world network of Fitz Hugh-Nagumo neurons, where the noise correlation decays exponentially with distance between neurons. It is found that CR is considerably improved just by a small fraction of long-range connections for an intermediate coupling strength. For other coupling strengths, an abrupt change in CR occurs following the drastic fracture of the clustered structures in the network. Our study shows that spatially correlated noise plays a significant role in the phenomenon of CR reinforcing the role of the clustered structure of the system.

Immunization dynamics on a two-layer network model

We introduced a two-layer network model for the study of the immunization dynamics in epidemics. Spreading of an epidemic is modeled as an excitatory process in a Watts–Strogatz small-world network (infection layer) while immunization by prevention of the disease as a dynamic process in a Barabási–Albert scale-free network (prevention layer). It is shown that prevention indeed turns periodic rages of an epidemic into small fluctuations, and in a certain situation, actually plays an adverse role and helps the disease survive. We argue that the presence of two different characteristic time scales contributes to the immunization dynamics observed.

SZNAJD MODEL WITH SYNCHRONOUS UPDATING ON COMPLEX NETWORKS

We analyze the evolution of Sznajd Model with synchronous updating in several complex networks. Similar to the model on square lattice, we have found a transition between the state with nonconsensus and the state with complete consensus in several complex networks. Furthermore, by adjusting the network parameters, we find that a large clustering coefficient does not favor development of a consensus. In particular, in the limit of large system size with the initial concentration p =0.5 of opinion +1, a consensus seems to be never reached for the Watts–Strogatz small-world network, when we fix the connectivity k and the rewiring probability ps; nor for the scale-free network, when we fix the minimum node degree m and the triad formation step probability pt.

Heterogeneity and Network Structure in the Dynamics of Diffusion: Comparing Agent-Based and Differential Equation Models

When is it better to use agent-based (AB) models, and when should differential equation (DE) models be used? Whereas DE models assume homogeneity and perfect mixing within compartments, AB models can capture heterogeneity across individuals and in the network of interactions among them. AB models relax aggregation assumptions, but entail computational and cognitive costs that may limit sensitivity analysis and model scope. Because resources are limited, the costs and benefits of such disaggregation should guide the choice of models for policy analysis. Using contagious disease as an example, we contrast the dynamics of a stochastic AB model with those of the analogous deterministic compartment DE model. We examine the impact of individual heterogeneity and different network topologies, including fully connected, random, Watts-Strogatz small world, scale-free, and lattice networks. Obviously, deterministic models yield a single trajectory for each parameter set, while stochastic models yield a distribution of outcomes. More interestingly, the DE and mean AB dynamics differ for several metrics relevant to public health, including diffusion speed, peak load on health services infrastructure, and total disease burden. The response of the models to policies can also differ even when their base case behavior is similar. In some conditions, however, these differences in means are small compared to variability caused by stochastic events, parameter uncertainty, and model boundary. We discuss implications for the choice among model types, focusing on policy design. The results apply beyond epidemiology: from innovation adoption to financial panics, many important social phenomena involve analogous processes of diffusion and social contagion.

Heterogeneity and Network Structure in the Dynamics of Diffusion: Comparing Agent-Based and Differential Equation Models

When is it better to use agent based (AB) models, and when should differential equation (DE) models be used? Where DE models assume homogeneity and perfect mixing within compartments, AB models can capture heterogeneity in agent attributes and in the network of interactions among them. Using contagious disease as an example, we contrast the dynamics of AB models with those of the corresponding mean-field DE model, specifically, comparing the standard SEIR model - a nonlinear DE - to an explicit AB model of the same system. We examine both agent heterogeneity and the impact of different network structures, including fully connected, random, Watts-Strogatz small world, scale-free, and lattice networks. Surprisingly, in many conditions the AB and DE dynamics are quite similar. Differences between the DE and AB models are not statistically significant on key metrics relevant to public health, including diffusion speed, peak load on health services infrastructure and total disease burden. We explore the conditions under which the AB and DE dynamics differ, and consider implications for managing infectious disease. The results extend beyond epidemiology: from innovation adoption to the spread of rumor and riot to financial panics, many important social phenomena involve analogous processes of diffusion and social contagion.