Narrow Results

Prisoner's Dilemma games with two and three strategies are studied. The corresponding replicator equations, their steady states and their asymptotic stability are discussed. Local Prisoner's Dilemma games are studied using Pareto optimality. As in the case with Nash updating rule, the existence of tit for tat strategy is crucial to imply cooperation even in one dimension. Pareto updating implies less erratic behavior since the steady state configurations are mostly fixed points or at most 2-cycle. Finally, Prisoner's Dilemma game is simulated on small-world networks which are closer to real systems than regular lattices. There are no significant changes compared to the results of the regular lattice.

The Sznajd model for the opinion formation is generalized to small-world networks. This generalization destroyed the stalemate fixed point. Then a simple definition of leaders is included. No fixed points are observed. This model displays some interesting aspects in sociology. The model is investigated using time series analysis.

A modified version of susceptible-infected-recovered-susceptible (SIRS) model for the outbreaks of foot-and-mouth disease (FMD) is introduced. The model is defined on small-world networks, and a ring vaccination programme is included. This model can be a theoretical explanation for the nonlocal interactions in epidemic spreading. Ring vaccination is capable of eradicating FMD provided that the probability of infection is high enough. Also an analytical approximation for this model is studied.

Some modified versions of susceptible-infected-recovered-susceptible (SIRS) model are defined on small-world networks. Latency, incubation and variable susceptibility are separately included. Phase transitions in these models are studied. Then inhomogeneous models are introduced. In some cases, the application of the models to small-world networks is shown to increase the epidemic region.

The Sznajd model of socio-physics, with only a group of people sharing the same opinion can convince their neighbors, is applied to a scale-free random network modeled by a deterministic graph. We also study a model for elections based on the Sznajd model and the exponent obtained for the distribution of votes during the transient agrees with those obtained for real elections in Brazil and India. Our results are compared to those obtained using a Barabási–Albert scale-free network.

We investigate the frustration effects on small-world networks by studying antiferromagnetic Ising model in two dimensions. When the rewiring is constrained to those sites such that the interaction still occurs between spins in distinct sublattices and frustration does not take place, we observe that the system behaves as in previous investigations of ferromagnetic Ising model. However, when the rewiring procedure does not only produce interactions between spins in distinct sublattices, small-world configurations can effectively produce geometrical frustration and we attain a different critical behavior. In the frustrated case, the critical temperature decreases with the augment of the rewiring probability and the magnetic ordering presents two different regimes for low and high p.

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 p_s; nor for the scale-free network, when we fix the minimum node degree m and the triad formation step probability p_t.

The route guidance systems (RGS) are efficient in alleviating traffic congestion and reducing transit time of transportation networks. This paper studies the effects of RGS on performance of variably weighted small-world networks. The properties of the average shortest path length, the maximum degree, and the largest betweenness, as important indices for characterizing the network structure in complex networks, are simulated. Results show that there is an optimal guided rate of RGS to minimize the total system cost and the average shortest path length, and proper RGS can reduce the load of the node with maximum degree or largest betweenness. In addition, we found that the load distribution of nodes guided by RGS decay as the power laws which is very important for us to understand and control traffic congestion feasible.

Small-world networks, exhibiting short nodal distances and high clustering, and scale-free networks, typified by a scale-free, power-law node-degree distribution, have been shown to be widespread both in natural and artificial systems. We propose a new type of network — cluster-dense network — characterized by multiple clusters that are highly intra-connected and sparsely inter-connected. Employing two graph-theoretic measures — local density and relative density — we demonstrate that such networks are prevalent in the world of networks.

We study the asymmetric influential effects in the damage spreading processes, by investigating the opinion dynamics of the Krause–Hegselmann consensus model, wherein the agents sit on the nodes of small-world networks. One agent is randomly selected as the influential agent, which can influence the opinions of other agents with probability p but not vice versa. The damage consists in a sharp change of the opinion of one agent (a randomly selected agent or the influential one) in the initial random opinion configuration. We find that for small values of the influential probability p, there is a damage spreading transition at low values of the confidence bound parameter ε. Interestingly, this transition vanishes for large values of p when the influential agent is damaged in the initial state. We find as well that, there is a critical value of the confidence bound parameter above which the initial perturbation manages to propagate to the whole system. The relationship between the amount of the damage and the influential activity of the perturbed agent is also discussed.

The possible existence of small, pure carbon molecules based on small-world networks is addressed using density functional theory simulations. A ring of atoms with one or more small-world connections between pairs of non-nearest-neighbor sites was chosen for the network topology. The small-world connections are made with and without additional carbon atoms placed along the link. The energy per atom of these small-world carbon systems is compared with benchmark molecules such as the C₂₀ ring, bowl, and cage isomers, the C₆₀ Buckyball, monocyclic pure carbon rings ranging from C₄ to C₆₀, bare linear carbon chains ranging from C₂ to C₃₆, and various graphitic fragments without hydrogens. The results of the energy per atom for some of these small-world clusters provide an indication that such pure carbon molecules are reasonable for real world synthesis.

Many real-world networks show community structure characterized by dense intra-community connections and sparse inter-community links. In this paper we investigated the synchronization properties of such networks. In this work we constructed such networks in a way that they consist of a number of communities with scale-free or small-world structure. Furthermore, with a probability, the intra-community connections are rewired to inter-community links. Two synchronizability measures were considered as the eigenratio of the Laplacian matrix and the phase order parameter obtained for coupled nonidentical Kuramoto oscillators. We found a power-law relation between the eigenratio and the inter-community rewiring probability in which as the rewiring probability increased, the eigenratio decreased, and hence, the synchronizability enhanced. The phase order parameter also increased by increasing the rewiring probability. Also, small-world networks with community structure showed better synchronization properties as compared to scale-free networks with community structure.

This paper constructs a model of the consensus formation in Internet based on the directed graph after analyzing the classical models of the social consensus formation, sets up the rules for the evolvement of opinions of agents and induces the evolving algorithm of consensus in Internet. The paper presents some key parameters such as the influence area of the mainstream media, the average influence of the mainstream media, the average self-persisting ability of agents and etc. Simulation results on a small-world networks show that the less the average self-persisting capability of the agents is, the easier the guidance of the media will be. The stronger the average influence of the main stream media is, the easier the mainstream media guides the consensus. These results reflect the formation law of the network consensus and are consistent approximately with the real circumstance.

The present study quantified the degree of the small-world (SW) property defined by Watts, and evaluated its achievement level to characterize complex networks. However, because this process has a combinatorial optimization problem, we applied the chaos neural network (CNN) and the simulated annealing (SA), and confirmed their performance in terms of optimized values and numerical costs. Next, we visualized the original network and its optimized networks whose SW property was maximized or minimized by exchanging the original network topology. As a result, although CNN and SA require huge computational time, we confirmed that they can evaluate the SW property and even real SW networks still have plenty of room to enlarge their own SW property.

The El Niño-Southern Oscillation (ENSO) is the most important driver of natural climate variability and is characterized by anomalies in the sea surface temperatures (SST) over the tropical Pacific ocean. It has three phases: neutral, a warming phase or El Niño, and a cooling phase called La Niña. In this research, we modeled the climate under the three phases as a network and characterized its properties. We utilized the National Center for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) daily surface temperature reanalysis data from January 1950 to December 2016. A network associated to a month was created using the temperature spanning from the previous month to the succeeding month, for a total of three months worth of data for each network. Each site of the included data was a potential node in the network and the existence of links were determined by the strength of their relationship, which was based on mutual information. Interestingly, we found that climate networks exhibit small-world properties and these are found to be more prominent from October to April, coinciding with observations that El Niño occurrences peak from December to March. During these months, the temperature of a relatively large part of the Pacific ocean and its surrounding areas increase and the anomaly values become synchronized. This synchronization in the temperature anomalies forms links around the Pacific, increasing the clustering in the region and in effect, that of the entire network.

The interest in learning how innovations spread in our society has led to the development of a variety of theoretical-computational models to describe the mechanisms that govern the diffusion of new ideas among people. In this paper, the diffusion of innovations is addressed with the use of the Axelrod’s cultural model where an agent is represented by a cultural vector of $F$ features, in which each feature can take on $Q$ integer states. The innovation or new idea is introduced in the population by setting a single feature of a single agent to a new state ($Q+1$) in the initial configuration. Particularly, we focus on the effect of the small-world topology on the dynamics of the innovation adoption. Our results indicate that the innovation spreads sublinearly ($\sim t^{1∕2}$) in a regular one-dimensional lattice of connectivity $K=4$, whereas the innovation spreads linearly ($\sim t$) when a nonvanishing fraction of the short-ranged links are replaced by long-range ones. In addition, we find that the small-world topology prevents the emergence of complete order in the thermodynamic limit. For systems of finite size, however, the introduction of long range links causes the dynamics to reach a final ordered state much more rapidly than for the regular lattice.

Exchange rates are important indicators of the economic power of countries, directly affected by the international trading patterns and relations. Since almost every pair of countries in the globalized world are economically and financially related, exchange rates can be evaluated as nodes of a global financial network to make meaningful inferences.

In this study, a financial network approach is conducted by evaluating the movements of the most traded 35 currencies against gold between years 2005 and 2017. Using graph theory and statistical methods, the analysis of economic relations between currencies is carried out, supported with geographical and cultural inferences. A risk map of currencies is generated through the portfolio optimization. Another approach of applying various threshold levels for correlations to determine connections between currencies is also employed. Results indicate that there exists a saddle point for correlation threshold as 0.9 which results in a robust network topology that is highly modular and clustered, also dominantly displaying small-world and scale-free properties.

Much empirical evidence shows that many real-world networks fall into the broad class of small-world networks and have a modular structure. The modularity has been revealed to have an important effect on cascading failure in isolated networks. However, the corresponding results for interdependent modular small-world networks remain missing. In this paper, we investigate the relationship between cascading failures and the intra-modular rewiring probabilities and inter-modular connections under different coupling preferences, i.e. random coupling with modules (RCWM), assortative coupling in modules (ACIM) and assortative coupling with modules (ACWM). The size of the largest connected component is used to evaluate the robustness from global and local perspectives. Numerical results indicate that increasing intra-modular rewiring probabilities and inter-modular connections can improve the robustness of interdependent modular small-world networks under intra-attacks and inter-attacks. Meanwhile, experiments on three coupling strategies demonstrate that ACIM has a better effect on preventing the cascading failures compared with RCWM and ACWM. These results can be helpful to allocate and optimize the topological structure of interdependent modular small-world networks to improve the robustness of such networks.

The emergence and evolution of cooperation in complex natural, social and economical systems is an interdisciplinary topic of recent interest. This paper focuses on the cooperation on complex networks using the approach of evolutionary games. In particular, the phenomenon of diversity-optimized cooperation is briefly reviewed and the effect of network clustering on cooperation is treated in detail. For the latter, a general type of public goods games is used with the result that, for fixed average degree and degree distributions in the underlying network, a high clustering coefficient can promote cooperation. Basic quantities such as the cooperator and defector clusters, mean payoffs of cooperators and defectors along their respective boundaries, the fraction of cooperators for different classes as well as the mean payoffs of hubs in scale-free networks are also investigated. Since strong clustering is typical in many social networks, these results provide insights into the emergence of cooperation in such networks.

A model for HIV transmission in homosexual populations is proposed taking into consideration different preventive attitudes, blood screening and effects of social networks. The equilibrium points of the system are calculated with and without blood screening and their stabilities are analyzed. By using these analytical results and numerical simulations, some evolving aspects of the epidemic are discussed.