Watts Strogatz Research Articles

Performance of small-world feedforward neural networks for the diagnosis of diabetes

We investigate the performance of two different small-world feedforward neural networks for the diagnosis of diabetes. We use the Pima Indians Diabetic Dataset as input. We have previously shown than the Watts–Strogatz small-world feedforward neural network delivers a better classification performance than conventional feedforward neural networks. Here, we compare this performance further with the one delivered by the Newman–Watts small-world feedforward neural network, and we show that the latter is better still. Moreover, we show that Newman–Watts small-world feedforward neural networks yield the highest output correlation as well as the best output error parameters.

Emergence of ultrafast sparsely synchronized rhythms and their responses to external stimuli in an inhomogeneous small-world complex neuronal network

We consider an inhomogeneous small-world network (SWN) composed of inhibitory short-range (SR) and long-range (LR) interneurons, and investigate the effect of network architecture on emergence of synchronized brain rhythms by varying the fraction of LR interneurons plong. The betweenness centralities of the LR and SR interneurons (characterizing the potentiality in controlling communication between other interneurons) are distinctly different. Hence, in view of the betweenness, SWNs we consider are inhomogeneous, unlike the “canonical” Watts–Strogatz SWN with nearly the same betweenness centralities. For small plong, the load of communication traffic is much concentrated on a few LR interneurons. However, as plong is increased, the number of LR connections (coming from LR interneurons) increases, and then the load of communication traffic is less concentrated on LR interneurons, which leads to better efficiency of global communication between interneurons. Sparsely synchronized rhythms are thus found to emerge when passing a small critical value plong(c)(≃0.16). The population frequency of the sparsely synchronized rhythm is ultrafast (higher than 100 Hz), while the mean firing rate of individual interneurons is much lower (∼30 Hz) due to stochastic and intermittent neural discharges. These dynamical behaviors in the inhomogeneous SWN are also compared with those in the homogeneous Watts–Strogatz SWN, in connection with their network topologies. Particularly, we note that the main difference between the two types of SWNs lies in the distribution of betweenness centralities. Unlike the case of the Watts–Strogatz SWN, dynamical responses to external stimuli vary depending on the type of stimulated interneurons in the inhomogeneous SWN. We consider two cases of external time-periodic stimuli applied to sub-populations of the LR and SR interneurons, respectively. Dynamical responses (such as synchronization suppression and enhancement) to these two cases of stimuli are studied and discussed in relation to the betweenness centralities of stimulated interneurons, representing the effectiveness for transfer of stimulation effect in the whole network.

How small is it? Comparing indices of small worldliness

AbstractMany studies have attempted to determine whether an observed network exhibits a so-called “small-world structure.” Such determinations have often relied on a conceptual definition of small worldliness proposed by Watts and Strogatz in their seminal 1998 paper, but recently several quantitative indices of network small worldliness have emerged. This paper reviews and compares three such indices—the small-world quotient (Q), a small-world metric (ω), and the small-world index(SWI)—in the canonical Watts–Strogatz re-wiring model and in four real-world networks. These analyses suggest that researchers should avoid Q, and identify considerations that should guide the choice between ω and SWI.

Core–satellite graphs: Clustering, assortativity and spectral properties

Core–satellite graphs (sometimes referred to as generalized friendship graphs) are an interesting class of graphs that generalize many well known types of graphs. In this paper we show that two popular clustering measures, the average Watts–Strogatz clustering coefficient and the transitivity index, diverge when the graph size increases. We also show that these graphs are disassortative. In addition, we completely describe the spectrum of the adjacency and Laplacian matrices associated with core–satellite graphs. Finally, we introduce the class of generalized core–satellite graphs and analyze their clustering, assortativity, and spectral properties.

Rumor spreading model with noise interference in complex social networks

In this paper, a modified susceptible–infected–removed (SIR) model has been proposed to explore rumor diffusion on complex social networks. We take variation of connectivity into consideration and assume the variation as noise. On the basis of related literature on virus networks, the noise is described as standard Brownian motion while stochastic differential equations (SDE) have been derived to characterize dynamics of rumor diffusion both on homogeneous networks and heterogeneous networks. Then, theoretical analysis on homogeneous networks has been demonstrated to investigate the solution of SDE model and the steady state of rumor diffusion. Simulations both on Barabási–Albert (BA) network and Watts–Strogatz (WS) network display that the addition of noise accelerates rumor diffusion and expands diffusion size, meanwhile, the spreading speed on BA network is much faster than on WS network under the same noise intensity. In addition, there exists a rumor diffusion threshold in statistical average meaning on homogeneous network which is absent on heterogeneous network. Finally, we find a positive correlation between peak value of infected individuals and noise intensity while a negative correlation between rumor lifecycle and noise intensity overall.

Upper and lower bounds for the q-entropy of network models with application to network model selection

Graphs are important tools for modeling data in different biological, social and technological domains. The measurement of their complexity has theoretical and practical applications in many areas such as pattern recognition, graph clustering, network inference and network analysis. A widely used measure is the q-entropy. While this measure has extensively been studied for Erdos–Renyi (ER) random graphs, only few results have been reported for real-world networks that are typically scale-free or small-world graphs. In this paper, we study the q-entropy of scale-free graphs (generated by the Barabasi–Albert model) and small-world graphs (generated by the Watts–Strogatz model). We derive upper and lower bounds for the q-entropy and confirm our analytical studies by numerical simulations. We study the effect of the network parameters on the q-entropy for both scale-free and small-world graphs which yields to reveal some novel and interesting insights. For example, we show that the q-entropy of a Barabasi–Albert graph is usually larger than the q-entropy of a small-world graph with the same size and density, independent of the randomness degree of the small-world graph. We use the derived bounds as characteristics of real-world networks as well as network models. Therefore, based on the derived bounds, we propose a network model selection algorithm, that given a real-world network and a set of models, determines which of the models most likely generates the real-world network.

The most common friend first immunization**Project supported by the National Natural Science Foundation of China (Grant No. 61263019), the Program for International Science and Technology Cooperation Projects of Gansu Province, China (Grant No. 144WCGA166), and the Program for Longyuan Young Innovation Talents and the Doctoral Foundation of Lanzhou University of Technology, China.

In this paper, a standard susceptible-infected-recovered-susceptible(SIRS) epidemic model based on the Watts–Strogatz (WS) small-world network model and the Barabsi–Albert (BA) scale-free network model is established, and a new immunization scheme — “the most common friend first immunization” is proposed, in which the most common friend’s node is described as being the first immune on the second layer protection of complex networks. The propagation situations of three different immunization schemes — random immunization, high-risk immunization, and the most common friend first immunization are studied. At the same time, the dynamic behaviors are also studied on the WS small-world and the BA scale-free network. Moreover, the analytic and simulated results indicate that the immune effect of the most common friend first immunization is better than random immunization, but slightly worse than high-risk immunization. However, high-risk immunization still has some limitations. For example, it is difficult to accurately define who a direct neighbor in the life is. Compared with the traditional immunization strategies having some shortcomings, the most common friend first immunization is effective, and it is nicely consistent with the actual situation.

Adaptive synchronization for complex networks with probabilistic time-varying delays

This paper addresses adaptive synchronization problem for complex networks with probabilistic time-varying delays. Using delay partitioning method and integral inequalities, combining with reciprocal convex combination approach technique, some delay-dependent synchronization stability criteria are derived. Finally, two numerical examples including Watts–Strogatz random complex networks and regular networks are provided to illustrate the importance of our theoretical analysis.

Generating and Analyzing Spatial Social Networks

In this paper, we propose a class of models for generating spatial versions of three classic networks: Erdos-Renyi (ER), Watts-Strogatz (WS), and Barabasi-Albert (BA). We assume that nodes have geographical coordinates, are uniformly distributed over an m × m Cartesian space, and long-distance connections are penalized. Our computational results show higher clustering coefficient, assortativity, and transitivity in all three spatial networks, and imperfect power law degree distribution in the BA network. Furthermore, we analyze a special case with geographically clustered coordinates, resembling real human communities, in which points are clustered over k centers. Comparison between the uniformly and geographically clustered versions of the proposed spatial networks show an increase in values of the clustering coefficient, assortativity, and transitivity, and a lognormal degree distribution for spatially clustered ER, taller degree distribution and higher average path length for spatially clustered WS, and higher clustering coefficient and transitivity for the spatially clustered BA networks.

Using Graph and Vertex Entropy to Compare Empirical Graphs with Theoretical Graph Models

Over the years, several theoretical graph generation models have been proposed. Among the most prominent are: the Erdős–Renyi random graph model, Watts–Strogatz small world model, Albert–Barabási preferential attachment model, Price citation model, and many more. Often, researchers working with real-world data are interested in understanding the generative phenomena underlying their empirical graphs. They want to know which of the theoretical graph generation models would most probably generate a particular empirical graph. In other words, they expect some similarity assessment between the empirical graph and graphs artificially created from theoretical graph generation models. Usually, in order to assess the similarity of two graphs, centrality measure distributions are compared. For a theoretical graph model this means comparing the empirical graph to a single realization of a theoretical graph model, where the realization is generated from the given model using an arbitrary set of parameters. The similarity between centrality measure distributions can be measured using standard statistical tests, e.g., the Kolmogorov–Smirnov test of distances between cumulative distributions. However, this approach is both error-prone and leads to incorrect conclusions, as we show in our experiments. Therefore, we propose a new method for graph comparison and type classification by comparing the entropies of centrality measure distributions (degree centrality, betweenness centrality, closeness centrality). We demonstrate that our approach can help assign the empirical graph to the most similar theoretical model using a simple unsupervised learning method.

Generating and analyzing spatial social networks

In this paper, we propose a class of models for generating spatial versions of three classic networks: Erdos-Renyi (ER), Watts-Strogatz (WS), and Barabasi-Albert (BA). We assume that nodes have geographical coordinates, are uniformly distributed over an m × m Cartesian space, and long-distance connections are penalized. Our computational results show higher clustering coefficient, assortativity, and transitivity in all three spatial networks, and imperfect power law degree distribution in the BA network. Furthermore, we analyze a special case with geographically clustered coordinates, resembling real human communities, in which points are clustered over k centers. Comparison between the uniformly and geographically clustered versions of the proposed spatial networks show an increase in values of the clustering coefficient, assortativity, and transitivity, and a lognormal degree distribution for spatially clustered ER, taller degree distribution and higher average path length for spatially clustered WS, and higher clustering coefficient and transitivity for the spatially clustered BA networks.

Cascading failures with local load redistribution in interdependent Watts–Strogatz networks

Cascading failures of loads in isolated networks have been studied extensively over the last decade. Since 2010, such research has extended to interdependent networks. In this paper, we study cascading failures with local load redistribution in interdependent Watts–Strogatz (WS) networks. The effects of rewiring probability and coupling strength on the resilience of interdependent WS networks have been extensively investigated. It has been found that, for small values of the tolerance parameter, interdependent networks are more vulnerable as rewiring probability increases. For larger values of the tolerance parameter, the robustness of interdependent networks firstly decreases and then increases as rewiring probability increases. Coupling strength has a different impact on robustness. For low values of coupling strength, the resilience of interdependent networks decreases with the increment of the coupling strength until it reaches a certain threshold value. For values of coupling strength above this threshold, the opposite effect is observed. Our results are helpful to understand and design resilient interdependent networks.

A general stochastic model for studying time evolution of transition networks

Abstract We consider a class of complex networks whose nodes assume one of several possible states at any time and may change their states from time to time. Such networks represent practical networks of rumor spreading, disease spreading, language evolution, and so on. Here, we derive a model describing the dynamics of this kind of network and a simulation algorithm for studying the network evolutionary behavior. This model, derived at a microscopic level, can reveal the transition dynamics of every node. A numerical simulation is taken as an “experiment” or “realization” of the model. We use this model to study the disease propagation dynamics in four different prototypical networks, namely, the regular nearest-neighbor (RN) network, the classical Erdos–Renyi (ER) random graph, the Watts–Strogatz small-world (SW) network, and the Barabasi–Albert (BA) scalefree network. We find that the disease propagation dynamics in these four networks generally have different properties but they do share some common features. Furthermore, we utilize the transition network model to predict user growth in the Facebook network. Simulation shows that our model agrees with the historical data. The study can provide a useful tool for a more thorough understanding of the dynamics networks.

Interplay between signaling network design and swarm dynamics

AbstractDistributed information transfer is of paramount importance to the effectiveness of dynamic collective behaviors, especially when a swarm is confronted with complex environmental circumstances. Recently, the signaling network of interaction underlying such effective information transfers has been revealed in the particular case of bird flocks governed by a topological interaction. Such biological systems are known to be evolutionary optimized, but are also constrained by the very nature of the signaling mechanisms—owing to intrinsic limitations in sensory modalities—enabling communication among individuals. Here, we propose that artificial swarm design can be tackled from the angle of signaling network design. To this aim, we use different network models to investigate the impact of some network structural properties on the effectiveness of a specific emergent swarming behavior, namely global consensus. Two new network models are introduced, which together with the well-known Watts–Strogatz model form the basis for an analysis of the relationship between clustering, shortest path and speed to consensus. A network-theoretic approach combined with spectral graph theory tools are used to propose some signaling network design principles. Eventually, one key design principle—a concomitant reduction in clustering and connecting path—is successfully tested on simulations of swarms of self-propelled particles.

Using adjacency matrices to lay out larger small-world networks

Many networks exhibit small-world properties. The structure of a small-world network is characterized by short average path lengths and high clustering coefficients. Few graph layout methods capture this structure well which limits their effectiveness and the utility of the visualization itself. Here we present an extension to our novel graphTPP layout method for laying out small-world networks using only their topological properties rather than their node attributes. The Watts–Strogatz model is used to generate a variety of graphs with a small-world network structure. Community detection algorithms are used to generate six different clusterings of the data. These clusterings, the adjacency matrix and edgelist are loaded into graphTPP and, through user interaction combined with linear projections of the adjacency matrix, graphTPP is able to produce a layout which visually separates these clusters. These layouts are compared to the layouts of two force-based techniques. graphTPP is able to clearly separate each of the communities into a spatially distinct area and the edge relationships between the clusters show the strength of their relationship. As a secondary contribution, an edge-grouping algorithm for graphTPP is demonstrated as a means to reduce visual clutter in the layout and reinforce the display of the strength of the relationship between two communities.

The Value Strength Aided Information Diffusion in Socially-Aware Mobile Networks

Socially-aware mobile networks have become ubiquitous in our daily life, which create the phenomenon of the socially aware mobile big data. Due to the movement of mobile entities, traditional transmission mechanisms are not conducive to the spreading of information. Thus, how to improve the information diffusion performances in socially aware mobile networks has become a hot research topic. However, most of the existing works focus their attention on the information-spreading dynamics rather than on the network structure and nodes’ social attributes. In this paper, we proposed the concept of the value strength, social strength as well as a time-varying graph (TVG)-based mobility model from the perspective of the network science, based on which a forwarding nodes’ selection scheme was presented. It is beneficial in terms of improving the propagation efficiency and information coverage ratio of mobile networks. Furthermore, sufficient experiments were conducted to verify the diffusion performances both over a range of complex network topologies, such as the Watts–Strogatz small-world network, the Barabasi–Albert scale-free network, the real-world Flickr network, and over a TVG-based mobile network. Indeed, the definition of the value/social strength plays a critical role in selecting forwarding links and nodes both for static-topology networks as well as for socially aware mobile networks.

Impact of small-world network topology on the conventional artificial neural network for the diagnosis of diabetes

Artificial intelligent systems have been widely used for diagnosis of diseases. Due to their importance, new approaches are attempted consistently to increase the performance of these systems. In this study, we introduce a new approach for diagnosis of diabetes based on the Small-World Feed Forward Artificial Neural Network (SW- FFANN). We construct the small-world network by following the Watts–Strogatz approach, and use this architecture for classifying the diabetes, and compare its performance with that of the regular or the conventional FFANN. We show that the classification performance of the SW-FFANN is better than that of the conventional FFANN. The SW-FFANN approach also results in both the highest output correlation and the best output error parameters. We also perform the accuracy analysis and show that SW-FFANN approach exhibits the highest classifier performance.

Quantum transport with long-range steps on Watts–Strogatz networks

We study transport dynamics of quantum systems with long-range steps on the Watts–Strogatz network (WSN) which is generated by rewiring links of the regular ring. First, we probe physical systems modeled by the discrete nonlinear schrödinger (DNLS) equation. Using the localized initial condition, we compute the time-averaged occupation probability of the initial site, which is related to the nonlinearity, the long-range steps and rewiring links. Self-trapping transitions occur at large (small) nonlinear parameters for coupling [Formula: see text] (1), as long-range interactions are intensified. The structure disorder induced by random rewiring, however, has dual effects for [Formula: see text] and inhibits the self-trapping behavior for [Formula: see text]. Second, we investigate continuous-time quantum walks (CTQW) on the regular ring ruled by the discrete linear schrödinger (DLS) equation. It is found that only the presence of the long-range steps does not affect the efficiency of the coherent exciton transport, while only the allowance of random rewiring enhances the partial localization. If both factors are considered simultaneously, localization is greatly strengthened, and the transport becomes worse.

Mapping the [formula omitted]-voter model: From a single chain to complex networks

We propose and compare six different ways of mapping the modified q-voter model to complex networks. Considering square lattices, Barabási–Albert, Watts–Strogatz and real Twitter networks, we ask the question if always a particular choice of the group of influence of a fixed size q leads to different behavior at the macroscopic level. Using Monte Carlo simulations we show that the answer depends on the relative average path length of the network and for real-life topologies the differences between the considered mappings may be negligible.

When local and global clustering of networks diverge

The average Watts–Strogatz clustering coefficient and the network transitivity are widely used descriptors for characterizing the transitivity of relations in real-world graphs (networks). These indices are bounded between zero and one, with low values indicating poor transitivity and large ones indicating a high proportion of closed triads in the graphs. Here, we prove that these two indices diverge for windmill graphs when the number of nodes tends to infinity. We also give evidence that this divergence occurs in many real-world networks, especially in citation and collaboration graphs. We obtain analytic expressions for the eigenvalues and eigenvectors of the adjacency and the Laplacian matrices of the windmill graphs. Using this information we show the main characteristics of two dynamical processes when taking place on windmill graphs: synchronization and epidemic spreading. Finally, we show that many of the structural and dynamical properties of a real-world citation network are well reproduced by the appropriate windmill graph, showing the potential of these graphs as models for certain classes of real-world networks.