Watts-Strogatz Network Research Articles

Analysis of transmission dynamics of dengue fever on a partially degenerated weighted network

In this paper, we propose a partially degenerated weighted network dynamical model for dengue fever transmission to study its spatial transmission dynamics, in which population mobility are characterized by the weighted graph Laplacian diffusion. Firstly, we establish the comparison principle for general reaction–diffusion differential equations defined on finite weighted network. Next, the well-posedness of solutions is established for the model. Then the basic reproduction number of the model is calculated, and then the stability of disease-free and endemic equilibria is investigated by means of the upper and lower solutions method and the construction of Lyapunov function. Furthermore, the uniform persistence of the model also is demonstrated. Finally, we apply the generalized weighted graph to the Watts–Strogatz network and present several numerical examples to verify theoretical results and obtain some interesting conclusions: the peak numbers of infected human population and infected mosquito populations depending on the node degree, even though the equations for the mosquito population in the model has no diffusion term.

Reinforcement learning-based pinning control for synchronization suppression in complex networks

Synchronization in complex networks is a ubiquitous and important phenomenon with implications in various fields. Excessive synchronization may lead to undesired consequences, making desynchronization techniques essential. Exploiting the Proximal Policy Optimization algorithm, this work studies reinforcement learning-based pinning control strategies for synchronization suppression in global coupling networks and two types of irregular coupling networks: the Watts-Strogatz small-world networks and the Barabási-Albert scale-free networks. We investigate the impact of the ratio of controlled nodes and the role of key nodes selected by the LeaderRank algorithm on the performance of synchronization suppression. Numerical results demonstrate the effectiveness of the reinforcement learning-based pinning control strategy in different coupling schemes of the complex networks, revealing a critical ratio of the pinned nodes and the superior performance of a newly proposed hybrid pinning strategy. The results provide valuable insights for suppressing and optimizing network synchronization behavior efficiently.

Dynamic properties of rumor propagation model induced by Lévy noise on social networks

Abstract Social networks are inevitably subject to disruptions from the physical world, such as sudden internet outages that sever local connections and impede information flow. While Gaussian white noise, commonly used to simulate stochastic disruptions, only fluctuates within a narrow range around its mean and fails to capture large-scale variations, Lévy noise can effectively compensate for this limitation. Therefore, a susceptible–infected–removed rumor propagation model with Lévy noise is constructed on homogeneous and heterogeneous networks, respectively. Then, the existence of a global positive solution and the asymptotic path-wise of the solution are derived on heterogeneous networks, and the sufficient conditions of rumor extinction and persistence are investigated. Subsequently, theoretical results are verified through numerical calculations and the sensitivity analysis related to the threshold is conducted on the model parameters. Through simulation experiments on Watts–Strogatz (WS) and Barabási–Albert networks, it is found that the addition of noise can inhibit the spread of rumors, resulting in a stochastic resonance phenomenon, and the optimal noise intensity is obtained on the WS network. The validity of the model is verified on three real datasets by particle swarm optimization algorithm.

Modelling the impact of human behavior using a two-layer Watts-Strogatz network for transmission and control of Mpox

PurposeThis study aims to evaluate the effectiveness of mitigation strategies and analyze the impact of human behavior on the transmission of Mpox. The results can provide guidance to public health authorities on comprehensive prevention and control for the new Mpox virus strain in the Democratic Republic of Congo as of December 2023.MethodsWe develop a two-layer Watts-Strogatz network model. The basic reproduction number is calculated using the next-generation matrix approach. Markov chain Monte Carlo (MCMC) optimization algorithm is used to fit Mpox cases in Canada into the network model. Numerical simulations are used to assess the impact of mitigation strategies and human behavior on the final epidemic size.ResultsOur results show that the contact transmission rate of low-risk groups and susceptible humans increases when the contact transmission rate of high-risk groups and susceptible humans is controlled as the Mpox epidemic spreads. The contact transmission rate of high-risk groups after May 18, 2022, is approximately 20% lower than that before May 18, 2022. Our findings indicate a positive correlation between the basic reproduction number and the level of heterogeneity in human contacts, with the basic reproduction number estimated at 2.3475 (95% CI: 0.0749–6.9084). Reducing the average number of sexual contacts to two per week effectively reduces the reproduction number to below one.ConclusionWe need to pay attention to the re-emergence of the epidemics caused by low-risk groups when an outbreak dominated by high-risk groups is under control. Numerical simulations show that reducing the average number of sexual contacts to two per week is effective in slowing down the rapid spread of the epidemic. Our findings offer guidance for the public health authorities of the Democratic Republic of Congo in developing effective mitigation strategies.

The effect of incentive policies on the diffusion of construction and demolition waste recycling: A government perspective.

Construction and demolition (C&D) waste recycling plays a significant role in waste reduction and carbon reduction, which is critical for sustainable development. However, due to various limitations such as financial problems, C&D waste recycling industry is not well developed in developing countries. To address this problem, this study combines complex network theory and evolutionary game theory to analyse the diffusion of C&D waste recycling behaviour among enterprises under governmental incentive policies within a complex network context. The results demonstrate that the size of the network has limited effects on behaviour diffusion in Watts-Strogatz small-world network. Additionally, the study highlights the clear impact of governmental incentive probability, initial rate and connection degree on the diffusion path. By quantitatively investigating the effects of incentive tools, this study contributes to the knowledge of C&D waste management and provides valuable implications for stakeholders seeking to promote the diffusion of C&D waste recycling.

Multi-constructor CMSA for the maximum disjoint dominating sets problem

We propose the Multi-Constructor CMSA, a Construct, Merge, Solve and Adapt (CMSA) algorithm that employs multiple heuristic procedures, respectively solution constructors, for the Maximum Disjoint Dominating Sets Problem (MDDSP). At every iteration of the search procedure, the solution components built by the constructors are merged into a sub-instance, which is subsequently solved by an exact solver and then adapted to keep only beneficial solution components. In our CMSA the solution constructors are chosen at random according to their relative probabilities, which are adapted during the search, through a mechanism based on reinforcement learning. We test two variants of the new Multi-Constructor CMSA that employ, respectively, two and six solution constructors, on a new set of 3600 problem instances, encompassing random graphs, Watts–Strogatz networks and Barabási-Albert networks, generated through a Hammersley sampling procedure on the instance space. We compare our algorithm against six heuristics from the literature, as well as with the standard version of CMSA. Furthermore, we employ an integer linear programming (ILP) model that is able to achieve a good performance for small, sparse graphs. Overall, the experimental results show that all versions of CMSA outperform by a large margin the previous state of the art and that, among the variants of CMSA, the novel version that combines two constructors provides slightly better results than the other ones, more prominently on larger graphs.

Using deterministic self-avoiding walks as a small-world metric on Watts–Strogatz networks

Using deterministic self-avoiding walks as a small-world metric on Watts–Strogatz networks

A Scalable Distributed Dynamical Systems Approach to Learn the Strongly Connected Components and Diameter of Networks

Finding strongly connected components (SCCs) and the diameter of a directed network play a key role in a variety of discrete optimization problems, and subsequently, machine learning and control theory problems. On the one hand, SCCs are used in solving the 2-satisfiability problem, which has applications in clustering, scheduling, and visualization. On the other hand, the diameter has applications in network learning and discovery problems enabling efficient internet routing and searches, as well as identifying faults in the power grid. In this paper, we leverage consensus-based principles to find the SCCs in a scalable and distributed fashion with a computational complexity of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(Dd_{\rm{in-degree}}^{\max })$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D$</tex-math></inline-formula> is the (finite) diameter of the network and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d_{\rm{in-degree}}^{\max }$</tex-math></inline-formula> is the maximum in-degree of the network. Additionally, we prove that our algorithm terminates in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$D+1$</tex-math></inline-formula> iterations, which allows us to retrieve the diameter of the network. We illustrate the performance of our algorithm on several random networks, including Erdö-Rényi, Barabási-Albert, and Watts-Strogatz networks.

A machine-learning procedure to detect network attacks

Abstract The goal of this note is to assess whether simple machine-learning algorithms can be used to determine whether and how a given network has been attacked. The procedure is based on the k-Nearest Neighbour and the Random Forest classification schemes, using both intact and attacked Erdős–Rényi, Barabasi–Albert and Watts–Strogatz networks to train the algorithm. The types of attacks we consider here are random failures and maximum-degree or maximum-betweenness node deletion. Each network is characterized by a list of four metrics, namely the normalized reciprocal maximum degree, the global clustering coefficient, the normalized average path length and the degree assortativity: a statistical analysis shows that this list of graph metrics is indeed significantly different in intact or damaged networks. We test the procedure by choosing both artificial and real networks, performing the attacks and applying the classification algorithms to the resulting graphs: the procedure discussed here turns out to be able to distinguish between intact networks and those attacked by the maximum-degree of maximum-betweenness deletions, but cannot detect random failures. Our results suggest that this approach may provide a basis for the analysis and detection of network attacks.

Spatial distribution of heterogeneity as a modulator of collective dynamics in pancreatic beta-cell networks and beyond

We study the impact of spatial distribution of heterogeneity on collective dynamics in gap-junction coupled beta-cell networks comprised on cells from two populations that differ in their intrinsic excitability. Initially, these populations are uniformly and randomly distributed throughout the networks. We develop and apply an iterative algorithm for perturbing the arrangement of the network such that cells from the same population are increasingly likely to be adjacent to one another. We find that the global input strength, or network drive, necessary to transition the network from a state of quiescence to a state of synchronised and oscillatory activity decreases as network sortedness increases. Moreover, for weak coupling, we find that regimes of partial synchronisation and wave propagation arise, which depend both on network drive and network sortedness. We then demonstrate the utility of this algorithm for studying the distribution of heterogeneity in general networks, for which we use Watts-Strogatz networks as a case study. This work highlights the importance of heterogeneity in node dynamics in establishing collective rhythms in complex, excitable networks and has implications for a wide range of real-world systems that exhibit such heterogeneity.

Quantum transport efficiency in noisy random-removal and small-world networks

We report the results of an in-depth study of the role of graph topology on quantum transport efficiency in random removal and Watts–Strogatz networks. By using four different environmental models—noiseless, driven by classical random telegraph noise (RTN), thermal quantum bath, and bath + RTN—we compare the role of the environment and of the change in network topology in determining the quantum transport efficiency. We find that small and specific changes in network topology is more effective in causing large change in efficiency compared to that achievable by environmental manipulations for both network classes. Furthermore, we have found that noise dependence of transport efficiency in Watts–Strogatz networks can be categorized into six classes. In general, our results highlight the interplay that network topology and environment models play in quantum transport, and pave the way for transport studies for networks of increasing size and complexity—when going beyond so far often used few-site transport systems.

First encounters on Watts-Strogatz networks and Barabási-Albert networks.

The Watts-Strogatz networks are important models that interpolate between regular lattices and random graphs, and Barabási-Albert networks are famous models that explain the origin of the scale-free networks. Here, we consider the first encounters between two particles (e.g., prey A and predator B) embedded in the Watts-Strogatz networks and the Barabási-Albert networks. We address numerically the mean first-encounter time (MFET) while the two particles are moving and the mean first-passage time (MFPT) while the prey is fixed, aiming at uncovering the impact of the prey's motion on the encounter time, and the conditions where the motion of the prey would accelerate (or slow) the encounter between the two particles. Different initial conditions are considered. In the case where the two particles start independently from sites that are selected randomly from the stationary distribution, on the Barabási-Albert networks, the MFET is far less than the MFPT, and the impact of prey's motion on the encounter time is enormous, whereas, on the Watts-Strogatz networks (including Erdős-Rényi random networks), the MFET is about 0.5-1 times the MFPT, and the impact of prey's motion on the encounter time is relatively small. We also consider the case where prey A starts from a fixed site and the predator starts from a randomly drawn site and present the conditions where the motion of the prey would accelerate (or slow) the encounter between the two particles. The relation between the MFET (or MFPT) and the average path length is also discussed.

A new centrality measure based on neighbor loop structure for network dismantling

Nearly all real-world networks are complex networks and usually are in danger of collapse. Therefore, it is crucial to exploit and understand the mechanisms of network attacks and provide better protection for network functionalities. Network dismantling aims to find the smallest set of nodes such that after their removal the network is broken into connected components of sub-extensive size. To overcome the limitations and drawbacks of existing network dismantling methods, this paper focuses on network dismantling problem and proposes a neighbor-loop structure based centrality metric, NL, which achieves a balance between computational efficiency and evaluation accuracy. In addition, we design a novel method combining NL-based nodes-removing, greedy tree-breaking and reinsertion. Moreover, we compare five baseline methods with our algorithm on ten widely used real-world networks and three types of model networks including Erdös-Rényi random networks, Watts-Strogatz small-world networks and Barabási-Albert scale-free networks with different network generation parameters. Experimental results demonstrate that our proposed method outperforms most peer methods by obtaining a minimal set of targeted attack nodes. Furthermore, the insights gained from this study may be of assistance to future practical research into real-world networks.

Synchronization of frustrated phase oscillators in the small-world networks

We numerically study the synchronization of an identical population of Kuramoto–Sakaguchi phase oscillators in Watts–Strogatz networks. We find that, unlike random networks, phase shift could enhance the synchronization in small-world networks. We also observe abrupt phase transition with hysteresis at some values of phase shifts in small-world networks, signs of an explosive phase transition. Moreover, we report the emergence of Chimera states at some values of phase shift close to the transition points, which consist of spatially coexisting synchronized and desynchronized domains.

Asymptotic and transient dynamics of SEIR epidemic models on weighted networks

We study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion, that is, on a weighted network. First, we establish the existence and uniqueness of solutions to the SEIR model defined on a weighed graph. Then by constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally, we apply our generalized weighed graph to Watts–Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine peak numbers of the infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behaviour of the epidemic transmission.

Phase synchronization and measure of criticality in a network of neural mass models

Synchronization has an important role in neural networks dynamics that is mostly accompanied by cognitive activities such as memory, learning, and perception. These activities arise from collective neural behaviors and are not totally understood yet. This paper aims to investigate a cortical model from this perspective. Historically, epilepsy has been regarded as a functional brain disorder associated with excessive synchronization of large neural populations. Epilepsy is believed to arise as a result of complex interactions between neural networks characterized by dynamic synchronization. In this paper, we investigated a network of neural populations in a way the dynamics of each node corresponded to the Jansen–Rit neural mass model. First, we study a one-column Jansen–Rit neural mass model for four different input levels. Then, we considered a Watts–Strogatz network of Jansen–Rit oscillators. We observed an epileptic activity in the weak input level. The network is considered to change various parameters. The detailed results including the mean time series, phase spaces, and power spectrum revealed a wide range of different behaviors such as epilepsy, healthy, and a transition between synchrony and asynchrony states. In some points of coupling coefficients, there is an abrupt change in the order parameters. Since the critical state is a dynamic candidate for healthy brains, we considered some measures of criticality and investigated them at these points. According to our study, some markers of criticality can occur at these points, while others may not. This occurrence is a result of the nature of the specific order parameter selected to observe these markers. In fact, The definition of a proper order parameter is key and must be defined properly. Our view is that the critical points exhibit clear characteristics and invariance of scale, instead of some types of markers. As a result, these phase transition points are not critical as they show no evidence of scaling invariance.

Synchronization in multiplex models of neuron-glial systems: Small-world topology and inhibitory coupling.

In this work, we investigate the impact of mixed coupling on synchronization in a multiplex oscillatory network. The network mimics the neural-glial systems by incorporating interacting slow ("glial") and fast ("neural") oscillatory layers. Connections between the "glial" elements form a regular periodic structure, in which each element is connected to the eight other neighbor elements, whereas connections among "neural" elements are represented by Watts-Strogatz networks (from regular and small-world to random Erdös-Rényi graph) with a matching mean node degree. We find that the random rewiring toward small-world topology readily yields the dynamics close to that exhibited for a completely random graph, in particular, leading to coarse-graining of dynamics, suppressing multi-stability of synchronization regimes, and the onset of Kuramoto-type synchrony in both layers. The duration of transient dynamics in the system measured by relaxation times is minimized with the increase of random connections in the neural layer, remaining substantial only close to synchronization-desynchronization transitions. "Inhibitory" interactions in the "neural" subnetwork layer undermine synchronization; however, the strong coupling with the "glial" layer overcomes this effect.

Detection of functional communities in networks of randomly coupled oscillators using the dynamic-mode decomposition.

Dynamic-mode decomposition (DMD) is a versatile framework for model-free analysis of time series that are generated by dynamical systems. We develop a DMD-based algorithm to investigate the formation of functional communities in networks of coupled, heterogeneous Kuramoto oscillators. In these functional communities, the oscillators in a network have similar dynamics. We consider two common random-graph models (Watts-Strogatz networks and Barabási-Albert networks) with different amounts of heterogeneities among the oscillators. In our computations, we find that membership in a functional community reflects the extent to which there is establishment and sustainment of locking between oscillators. We construct forest graphs that illustrate the complex ways in which the heterogeneous oscillators associate and disassociate with each other.

Complex networks of interacting stochastic tipping elements: Cooperativity of phase separation in the large-system limit.

Tipping elements in the Earth system have received increased scientific attention over recent years due to their nonlinear behavior and the risks of abrupt state changes. While being stable over a large range of parameters, a tipping element undergoes a drastic shift in its state upon an additional small parameter change when close to its tipping point. Recently, the focus of research broadened towards emergent behavior in networks of tipping elements, like global tipping cascades triggered by local perturbations. Here, we analyze the response to the perturbation of a single node in a system that initially resides in an unstable equilibrium. The evolution is described in terms of coupled nonlinear equations for the cumulants of the distribution of the elements. We show that drift terms acting on individual elements and offsets in the coupling strength are subdominant in the limit of large networks, and we derive an analytical prediction for the evolution of the expectation (i.e., the first cumulant). It behaves like a single aggregated tipping element characterized by a dimensionless parameter that accounts for the network size, its overall connectivity, and the average coupling strength. The resulting predictions are in excellent agreement with numerical data for Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks of different size and with different coupling parameters.

Feedback driven message spreading on network

This paper focuses on the role of feedback mechanism on message propagation. In this paper, we study the effect of feedback on message propagation in terms of both the motivation of the communicator to propagate the message and the trust of the message recipient in the communicator, and we design a decay mechanism of motivation based on Newton's cooling law. Based on these considerations, we propose a model of message propagation based on the feedback mechanism. We verify the effect of feedback on message propagation by performing numerical simulations in the Watts-Strogatz (WS) and Barabasi-Albert (BA) networks. The simulation results show that the feedback mechanism leads to faster and more persistent message propagation and that the degree of influence on message propagation varies across different network structures. The simulation also results show that the feedback mechanism changes the structure of the social network and makes the nodes between the networks more closely connected.