Watts-Strogatz Network Research Articles

Neuronal avalanches in Watts-Strogatz networks of stochastic spiking neurons.

Networks of stochastic leaky integrate-and-fire neurons, both at the mean-field level and in square lattices, present a continuous absorbing phase transition with power-law neuronal avalanches at the critical point. Here we complement these results showing that small-world Watts-Strogatz networks have mean-field critical exponents for any rewiring probability p>0. For the ring (p=0), the exponents are the same from the dimension d=1 of the directed-percolation class. In the model, firings are stochastic and occur in discrete time steps, based on a sigmoidal firing probability function. Each neuron has a membrane potential that integrates the signals received from its neighbors. The membrane potentials are subject to a leakage parameter. We study topologies with a varied number of neuron connections and different values of the leakage parameter. Results indicate that the dynamic range is larger for p=0. We also study a homeostatic synaptic depression mechanism to self-organize the network towards the critical region. These stochastic oscillations are characteristic of the so-called self-organized quasicriticality.

Cascading failures in coupled map lattices with sustained attack

Many realistic networked systems may face sustained attack over a sequence of time. In this paper, we propose a cascading failure model in coupled map lattices (CMLs) with sustained attack, where a number of crucial nodes will be attacked one by one at each time step. The effects of four attacking strategies: high-degree strategy (HDS), high-clustering coefficient strategy, high-closeness strategy and high-betweenness strategy are compared. This shows that the performance of HDS is better than that of other attacking strategies when the value of the outside attack is small. The effectiveness of HDS on Watts–Strogatz (WS) small-world networks and two real-world networks is extensively investigated. The results indicate that increasing the value of the rewiring probability of WS networks can make the network more robust to resist sustained attack. The sparser the network structure is, the faster the diffusion velocity of cascading failures is. A smaller coupling strength in CMLs can restrain the diffusion velocity of cascades. We also extend our CML-based model with sustained attack from one node strategy (ONS) to two nodes strategy (TNS) and compare the effect of ONS and TNS with different values of the outside attack. It is found that ONS outperforms TNS for small values of the outside attack, but TNS would be a better choice when the outside attack and the time step are all large. Our work will provide new insights into the problem of protecting complex networks against cascading failures with respect to malicious attack.

Path to fixation of evolutionary processes in graph-structured populations

We study the spreading of a single mutant in graph-structured populations with a birth-death update rule. We use a mean-field approach and a Markov chain dynamics to investigate the effect of network topology on the path to fixation. We obtain approximate analytical formulas for average time versus the number of mutants in the fixation process starting with a single mutant for several network structures, namely, cycle, complete graph, two- and three-dimensional lattices, random graph, regular graph, Watts–Strogatz network, and Barabasi–Albert network. In the case of the cycle and complete graph, the results are accurate and in line with the results obtained by other methods. In the case of two- and three-dimensional lattice structures, some efforts are made in other studies to provide an analytical justification for simulation results of the evolutionary process, but they can explain just the onset of the fixation process, not the whole process. The results of the analytical approach of the present paper are well fitted to the simulation results throughout the whole fixation process. Moreover, we analyze the dynamics of evolution for a number of complex structures, and in all cases, we obtain analytical results which are in good agreement with simulations. Our results may shed some light on the process of fixation during the whole path to fixation.

Effect of chaotic agent dynamics on coevolution of cooperation and synchronization.

The effect of chaotic dynamical states of agents on the coevolution of cooperation and synchronization in a structured population of the agents remains unexplored. With a view to gaining insights into this problem, we construct a coupled map lattice of the paradigmatic chaotic logistic map by adopting the Watts-Strogatz network algorithm. The map models the agent's chaotic state dynamics. In the model, an agent benefits by synchronizing with its neighbors, and in the process of doing so, it pays a cost. The agents update their strategies (cooperation or defection) by using either a stochastic or a deterministic rule in an attempt to fetch themselves higher payoffs than what they already have. Among some other interesting results, we find that beyond a critical coupling strength, which increases with the rewiring probability parameter of the Watts-Strogatz model, the coupled map lattice is spatiotemporally synchronized regardless of the rewiring probability. Moreover, we observe that the population does not desynchronize completely-and hence, a finite level of cooperation is sustained-even when the average degree of the coupled map lattice is very high. These results are at odds with how a population of the non-chaotic Kuramoto oscillators as agents would behave. Our model also brings forth the possibility of the emergence of cooperation through synchronization onto a dynamical state that is a periodic orbit attractor.

Age separation dramatically reduces COVID-19 mortality rate in a computational model of a large population.

COVID-19 pandemic has caused a global lockdown in many countries throughout the world. Faced with a new reality, and until a vaccine or efficient treatment is found, humanity must figure out ways to keep the economy going, on one hand, while keeping the population safe, on the other hand, especially those that are susceptible to this virus. Here, we use a Watts–Strogatz network simulation, with parameters that were drawn from what is already known about the virus, to explore five different scenarios of partial lockdown release in two geographical locations with different age distributions. We find that separating age groups by reducing interactions between them protects the general population and reduces mortality rates. Furthermore, the addition of new connections within the same age group to compensate for the lost connections outside the age group still has a strong beneficial influence and reduces the total death toll by about 62%. While complete isolation from society may be the most protective scenario for the elderly population, it would have an emotional and possibly cognitive impact that might outweigh its benefit. Therefore, we propose creating age-related social recommendations or even restrictions, thereby allowing social connections while still offering strong protection for the older population.

A weighted networked SIRS epidemic model

Population mobility in an SIRS epidemic model is considered via graph Laplacian diffusion. We show the existence and uniqueness of solutions to the SIRS model defined on weighed graph. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1. By constructing Lyapunov function, we show that the endemic equilibrium is globally asymptotically stable for the model with the same diffusion rates if the basic reproduction number is greater than 1. With numerical simulations, we apply our generalized weighed graph to Watts-Strogatz network, where the degree of node is illustrated to determine the peak number of infectious population. It also indicates that the network has an impact on small-time behavior of epidemic transmission.

Bullshit in a network structure: the two-sided influence of self-generated signals.

In today’s social network age, information flowing in networks does not derive solely from external sources; people in the network also independently generate signals. These self-generated signals may not be deliberate lies, but they may not bear any relationship with the truth, either. Following the philosopher Harry G. Frankfurt, we refer to such self-generated signals as bullshit. We present an information diffusion model that allows nodes which hold no value to spread information, capturing the diffusion of bullshit information. The presence of self-generated signals (i.e., bullshit) increases the amount of information available for transmission in the network. However, participants in the spread process respond to the existence of such self-generated information by receiving data from internal sources with caution. These two contradictory forces—the increase in information transmission on the one hand, and in suspicion on the other—result in a two-sided effect of bullshit on the total spread time. We first take a numerical approach, simulating our model on Watts–Strogatz networks and building a decision tree to characterize the effects of bullshit given different network structures. We find that increasing the rate of self-generated information may have either a monotonic or non-monotonic effect on the rumor spread time, depending on the network structure and rate of non-self-generated internal communications. Then, taking an analytical approach, we analyze the spread behavior for cliques, and identify the conditions for monotonic behavior in a 2-clique network.

Performance Comparison of Nonlinear Kalman Filters in Epidemic Tracking on Networks

The problem of tracking epidemic spreading on networks is relevant to the control of morbidity. The transmission dynamics of an epidemic can be described by a simple compartmental model. Specifically, the estimation of epidemic spreading on networks can be achieved by a nonlinear Kalman filter, which is a tool for state estimation of nonlinear systems. In this article, epidemic spreading on networks is described by compartmental models, such as susceptible–infected–susceptible, susceptible–infected–recovered, and susceptible–infected–recovered–susceptible models. The dynamics of epidemic spreading on homogeneous networks, including Erdos–Renyi network, Watts–Strogatz network, and Newman–Watts network, are estimated by several nonlinear Kalman filters, including extended Kalman filter, unscented Kalman filter, and third-degree and fifth-degree cubature Kalman filters. The performance comparison in terms of accuracy and stability forms a guideline of utilizing nonlinear Kalman filters for tracking epidemic spreading. The theoretical analysis has been validated through numerical experiments.

Rumor spreading model considering rumor credibility, correlation and crowd classification based on personality

The study of rumor spreading or rumor controlling is important and necessary because rumors can cause serious negative effects on society. The process of rumor spreading is influenced by many factors. In this paper, we suggest that people with different personalities will behave differently after hearing rumors. Thus, we divide the population into two types: radical people and steady people. Furthermore, we suggest that the credibility of rumors and the correlation between rumors and people’s lives are important factors that will influence the spread of rumors. Based on these considerations, we propose the SEIsIrR model. We establish differential equations to describe the dynamics of the rumor spreading process in homogeneous and heterogeneous networks. Using the Jacobian matrix and next generation matrix, we obtain the spreading threshold of the SEIsIrR model and discuss the relationship of the spreading threshold between homogeneous networks and heterogeneous networks. We employ a real rumor dataset obtained from Twitter to verify the SEIsIrR model and perform numerical simulations in Watts-Strogatz (WS) networks and Barabasi-Albert (BA) networks to verify the obtained spreading thresholds and discuss the impacts of these factors on the rumor spreading process and the differences in the rumor spreading processes between WS networks and BA networks. The simulation results show that these factors influence the speed and range of rumor spreading.

Small-worldness favours network inference in synthetic neural networks

A main goal in the analysis of a complex system is to infer its underlying network structure from time-series observations of its behaviour. The inference process is often done by using bi-variate similarity measures, such as the cross-correlation (CC) or mutual information (MI), however, the main factors favouring or hindering its success are still puzzling. Here, we use synthetic neuron models in order to reveal the main topological properties that frustrate or facilitate inferring the underlying network from CC measurements. Specifically, we use pulse-coupled Izhikevich neurons connected as in the Caenorhabditis elegans neural networks as well as in networks with similar randomness and small-worldness. We analyse the effectiveness and robustness of the inference process under different observations and collective dynamics, contrasting the results obtained from using membrane potentials and inter-spike interval time-series. We find that overall, small-worldness favours network inference and degree heterogeneity hinders it. In particular, success rates in C. elegans networks – that combine small-world properties with degree heterogeneity – are closer to success rates in Erdös-Rényi network models rather than those in Watts-Strogatz network models. These results are relevant to understand better the relationship between topological properties and function in different neural networks.

Local acceptance and emergence of consensus in a heterogeneous small-world network of agents with and without memory

We formulate a spatiotemporal model of social interaction that explores the role of local acceptance or influence in consensus-building. It is applied to a population of interacting agents that forms a small-world Watts–Strogatz network to determine the level of consensus that emerges as a function of the local acceptance parameter α(0≤α≤1), agent memory capacity, and interaction scale. Parameter α sets the probability that an agent copies the majority state of its neighbors while (1−α) is the corresponding likelihood that it conforms with the minority state. An agent with memory may retain its present state regardless of the local state of its neighborhood. We track the emergence of global agreement in a heterogeneous population of agents with and without memory, using density metrics that correspond to three different interaction scales — entire population, neighborhoods, and pairs of agents. We have found that for a given α and population heterogeneity, the degree of agreement would depend strongly on the chosen interaction scale and network connectivity. Unlike its disordered counterpart, a homogeneous small-world network with memory, is unable to reach a simple majority of 51% over a wide range of α < 0.83. The degree of agreement then peaks towards 80% at α=0.95 before decreasing rapidly back to 50% as α→1.0. Memory prevents a heterogeneous small-world network from achieving full agreement at α=1.0, even when the agents with memory is a weak minority. All density profiles are asymmetric about α=0.5, indicating the uneven impact of copying the majority or minority state in consensus building. They are also distinct and difficult to extrapolate from each other. The model provides an adaptive platform for exploring the spatiotemporal performance of existing collective decision-making strategies either separately or in multiple combination of each other. It can be deployed to probe the possible triggers of strategy modification in adaptive social networks. We have identified its presence in animal groups that were observed while in pursuit of predefined tasks such as group emergency evacuation.

Small-world networks and synchronisation in an agent-based model of civil violence

ABSTRACTThe rapid evolution and current ubiquity of social media as a form of communication calls for a revision of many models of collective behaviour. In this paper, we modify a classic agent-based model of civil violence by Epstein (2002) consisting of citizen and law-enforcement agents by integrating a Watts-Strogatz small-world network (SWN). The SWN simulates non-local connections between citizens, enabling influence by both local and distant neighbours and providing an analogue to social media. The objective was to examine the influence of non-local connections on civil violence dynamics for varied law-enforcement concentration and network density. For lower law-enforcement concentrations, the SWN influence leads to more frequent large-scale violent outbursts, while for higher law-enforcement concentrations, outcomes depended most strongly on the number of local neighbours. The long-range coupling across the lattice due to the SWN provides a new mechanism for non-trivial dynamics and leads to a synchronisation effect.

A simple and generic paradigm for creating complex networks using the strategy of vertex selecting-and-pairing

In many networks of scientific interest we know that the link between any pair of vertices conforms to a specific probability, such as the link probability in the Barab\'asi-Albert scale-free networks. Here we demonstrate how the distributions of link probabilities can be utilized to generate various complex networks simply and effectively. We focus in particular on the problem of complex network generation and develop a straightforward paradigm by using the strategy of vertex selecting-and-pairing to create complex networks more generic than other relevant approaches. Crucially, our paradigm is capable of generating various complex networks with varied degree distributions by using different probabilities for selecting vertices, while in contrast other relevant approaches can only be used to generate a specific type of complex networks. We demonstrate our paradigm on four synthetic Barab\'asi-Albert scale-free networks, four synthetic Watts-Strogatz small-world networks, and on a real email network with known degree distributions.

Cascading failures with coupled map lattices on Watts–Strogatz networks

In this paper, we present a cascading failure model based on coupled map lattices, in which a given number of crucial nodes are perturbed at the same time. By comparing the effect of five attacking strategies: random strategy, high-degree strategy, high-clustering coefficient strategy, high-betweenness strategy and high-closeness strategy (HCS) on Watts–Strogatz (WS) small-world networks, we found that HCS outperforms the other attacking strategies with different number of perturbed nodes. The efficiency of HCS on WS small-world networks is explored extensively. The results show that increasing the rewiring probability can reduce the range of cascading failures, but it will enhance the propagation speed of cascades. Furthermore, a more compact network structure will lead to a faster cascading failure propagation speed, and the propagation speed of cascades is inversely proportional to the characteristic path length of WS small-world networks. Our work will highlight a better understanding of cascading failures on complex networks.

Node importance based on the weighted &lt;i&gt;K&lt;/i&gt;-order propagation number algorithm

The measurement of node importance is significant for analyzing a network structure. To comprehensively reflect the global and local network features, in this paper we abstract the propagating process of epidemic diseases based on the network topology structure, and then respectively sets each node as an infection source. After a certain dissemination time &lt;i&gt;K&lt;/i&gt;, the number of infected nodes in the network is defined as the &lt;i&gt;K&lt;/i&gt;-order propagation number, and the weighted sum of &lt;i&gt;K&lt;/i&gt;-order propagation numbers under different values of &lt;i&gt;K&lt;/i&gt; is taken as the important index of nodes. The simulation experiments of Watts-Strogatz small-world networks and a dolphin network show that the weighted &lt;i&gt;K&lt;/i&gt;-order propagation number algorithm is more effective than the traditional method in evaluating the importance of nodes. For the Watts-Strogatz small-world networks, it can reflect the influence of long-distance connections on information transmission in detail. For the dolphin network, the weighted &lt;i&gt;K&lt;/i&gt;-order propagation number algorithm significantly raises the profile of those nodes which play a key role in the information communication between different dolphin communities. In addition, in this paper we use a deliberate attacking method to analyze the western power grid of the United States, the road transportation network of the Chicago region, the co-authorship network in network science and the axonal tracts’ network between neurons of mouse. According to the specific order of the node importance from high to low, network nodes are attacked in turn, that is, all edges of the attacked nodes are removed. The analysis results of network parameters such as the network efficiency and the node number of the maximum sub-graph changing with the attacking times show that comparing with other evaluation indices of node importance such as degree, Ren method, Chen method, eigenvector method, Katz index, PageRank, CI method and &lt;i&gt;K&lt;/i&gt;-shell, the weighted &lt;i&gt;K&lt;/i&gt;-order propagation number algorithm focuses much on destroying the major structure, and all of the above four networks will break down if only a small number of important nodes are attacked. For example, after attacking only 100 nodes, the network efficiency of the western power grid of the United States is down by 90%, and after attacking 200 nodes, the network scale of the maximum sub-graph is nearly 3% of the original network.

A Stochastic Information Diffusion Model in Complex Social Networks

The number of potential participants in information diffusion is dynamically changing, and the topological structure of a real social network is affected by environment noise. In this paper, a stochastic information diffusion model considering both population perturbation and connectivity variation has been proposed. For the model on homogeneous network, we show the existence and uniqueness of the global positive solution and give a sufficient condition of extinction of information. A series of numerical simulations on Watts-Strogatz (WS) network, Barabasi-Albert (BA) network and real-world Facebook network have been conducted to verify the theoretical analysis and evaluate the sensitivity of the proposed model to relevant parameters. We find that both population perturbation and connectivity variation have great impacts on information diffusion process. In general, the connectivity variation noise promotes the spread of information, and the population perturbation noise corresponding to I-infected individuals inhibit the spread of information. In the case of large noise intensity, the population perturbation noise corresponding to I-infected individuals plays a decisive role compared with the connectivity variation noise.

Modeling rumor propagation and refutation with time effect in online social networks

Rumor propagation and refutation form an important issue for spreading dynamics in online social networks (OSNs). In this paper, we introduce a novel two-stage rumor propagation and refutation model with time effect for OSNs. The dynamical mechanism of rumor propagation and refutation with time effect is investigated deeply. Then a two-stage model and the corresponding mean-field equations in both homogeneous and heterogeneous networks are obtained. Monte Carlo simulations are conducted to characterize the dynamics of rumor propagation and refutation in both Watts–Strogatz network and Barabási–Albert network. The results show that heterogeneous networks yield the most effective rumor and anti-rumor spreading. Besides, the sooner authority releases anti-rumor and the more attractive anti-rumor is, the less rumor influence is. What’s more, these findings suggest that individuals’ ability to control themselves and identify rumor accurately should be improved to reduce negative impact of rumor effectively. The results are helpful to understand better the mechanism of rumor propagation and refutation in OSNs.

Efficient weighting strategy for enhancing synchronizability of complex networks

Networks with high propensity to synchronization are desired in many applications ranging from biology to engineering. In general, there are two ways to enhance the synchronizability of a network: link rewiring and/or link weighting. In this paper, we propose a new link weighting strategy based on the concept of the neighborhood subgroup. The neighborhood subgroup of a node i through node j in a network, i.e. [Formula: see text], means that node u belongs to [Formula: see text] if node u belongs to the first-order neighbors of j (not include i). Our proposed weighting schema used the local and global structural properties of the networks such as the node degree, betweenness centrality and closeness centrality measures. We applied the method on scale-free and Watts–Strogatz networks of different structural properties and show the good performance of the proposed weighting scheme. Furthermore, as model networks cannot capture all essential features of real-world complex networks, we considered a number of undirected and unweighted real-world networks. To the best of our knowledge, the proposed weighting strategy outperformed the previously published weighting methods by enhancing the synchronizability of these real-world networks.

Impact of network randomness on multiple opinion dynamics

People often face the challenge of choosing among different options with similar attractiveness. To study the distribution of preferences that emerge in such situations, a useful approach is to simulate opinion dynamics on top of complex networks, composed by nodes (individuals) and their connections (edges), where the state of each node can be one amongst several opinions including the undecided state. We analyze two different dynamics: the one proposed by Travieso and Fontoura (TF) and the plurality rule (PR), which are paradigmatic of outflow and inflow dynamics, respectively. We are specially interested in the impact of the network randomness on the final distribution of opinions. For that purpose, we consider Watts–Strogatz networks, which possess the small-world property, and where randomness is controlled by a probability p of adding random shortcuts to an initially regular network. Depending on the value of p, the average connectivity 〈k〉, and the initial conditions, the final distribution can be basically (i) consensus, (ii) coexistence of different options, or (iii) predominance of indecision. We find that, in both dynamics, the predominance of a winning opinion is favored by increasing the number of reconnections (shortcuts), promoting consensus. In contrast to the TF case, in the PR dynamics, a fraction of undecided nodes can persist in the final state. In such cases, a maximum number of undecided nodes occurs within the small-world range of p, due to ties in the decision group.

Network and equation-based models in epidemiology

Network and equation-based (EB) models are two prominent methods used in the study of epidemics. While EB models use a global approach to model aggregate population, network models focus on the behavior of individuals in the population. The two approaches have been used in several areas of research, including finance, computer science, social science and epidemiology. In this study, epidemiology is used to contrast EB models with network models. The methods are based on the assumptions and properties of compartmental models. In EB models we solve a system of ordinary differential equations and in network models we simulate the spread of epidemics on contact networks using bond percolation. We examine the impact of network structures on the spread of infection by considering various networks, including Poisson, Erdős Rényi, Scale-free, and Watts–Strogatz small-world networks, and discuss how control measures can make use of the network structures. In addition, we simulate EB assumptions on Watts–Strogatz networks to determine when the results are similar to that of EB models. As a case study, we use data from the 1918 Spanish flu pandemic and that from measles outbreak to validate our results.