Watts-Strogatz Small-world Network Research Articles

Evolutionary Cooperation in Networked Public Goods Game with Dependency Groups

Either in microlevel organizations or macrolevel societies, the individuals acquire benefits or payoffs by forming interdependency groups linked by common interests. Conducting research on the effects of interdependency groups on the evolution of cooperation could have a better understanding of the social dilemma problem. In this paper, we studied a spatial public goods game with nonlocal interdependency groups where each of participants is located in a two‐dimensional square lattice or Watts–Strogatz small‐world network with payoffs obtaining from the interactions with nearest neighbors. In terms of the enhancement factor, the effects of group density on the evolutionary cooperation can be quite different. For a low enhancement factor, the cooperation level is a nonmonotonic function with the varying density of interdependency groups in the system, which means a proper density of interdependency groups can best promote the cooperative level. For a moderate enhancement factor, a higher density of interdependency groups can always correspond to a higher cooperative level. However, if the enhancement factor is too high, a high density of interdependency groups can impede the evolutionary cooperation. We give the explanations for the different roles of group density of interdependency by using the transition probabilities of C players into D players as well as the reverse. Our findings are very helpful for the understanding of emergence cooperation as well as the cooperation regulation in the selfish individuals.

Node importance based on the weighted <i>K</i>-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 <i>K</i>, the number of infected nodes in the network is defined as the <i>K</i>-order propagation number, and the weighted sum of <i>K</i>-order propagation numbers under different values of <i>K</i> 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 <i>K</i>-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 <i>K</i>-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 <i>K</i>-shell, the weighted <i>K</i>-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.

Network heterogeneity based on <i>K</i>-order structure entropy

Structure entropy can evaluate the heterogeneity of complex networks, but traditional structure entropy has deficiencies in comprehensively reflecting the global and local network features. In this paper, we define a new structure entropy based on the number of the <i>K</i>-order neighbor nodes which refer to those nodes which a node can reach within <i>K</i> steps. It can be supposed that the more <i>K</i>-order neighbors a node has, the more important role the node plays in the network structure. Combining the formula of Shannon entropy, the <i>K</i>-order structure entropy can be defined and figured out to explain the differences among the relative importance among nodes. Meanwhile, the new structure entropy can describe the network heterogeneity from the following three aspects. The first aspect is the change tendency of structure entropy with the value of <i>K</i>. The second aspect is the structure entropy under a maximum influence scale <i>K</i>. The last aspect is the minimum value of the <i>K</i>-order structure entropy. The simulation compares the heterogeneities of five classic networks from the above three aspects, and the result shows that the heterogeneity strengthens in the from-weak-to -strong sequence:regular network, random network, WS (Watts-Strogatz) small-world network, BA (Barabási-Albert) scale-free network and star network. This conclusion is consistent with the previous theoretical research result, but hard to obtain from the traditional structure entropy. It is remarkable that the <i>K</i>-order structure entropy can better evaluate the heterogeneity of WS small-world networks and suggests that the greater small-world coefficients a network has, the stronger heterogeneity the network has. Besides, the <i>K</i>-order structure entropy can fully reflect the heterogeneity variation of star networks with network size, and reasonably explain the heterogeneity of regular networks with additional isolated nodes. It suggests that when i additional isolated nodes are added to a regular network with <i>n</i> nodes, the new network has weaker heterogeneity than the old one, but has stronger heterogeneity than the regular network with <i>n</i>+i nodes. Finally, the validity of the <i>K</i>-order structure entropy is further confirmed by simulations for the western power grid of the United States. Based on the minimum value of the <i>K</i>-order structure entropy, the heterogeneity of the western power grid is the closest to that of WS small-world networks.

Effect of inhibitory spike-timing-dependent plasticity on fast sparsely synchronized rhythms in a small-world neuronal network

We consider the Watts–Strogatz small-world network (SWN) consisting of inhibitory fast spiking Izhikevich interneurons. This inhibitory neuronal population has adaptive dynamic synaptic strengths governed by the inhibitory spike-timing-dependent plasticity (iSTDP). In previous works without iSTDP, fast sparsely synchronized rhythms, associated with diverse cognitive functions, were found to appear in a range of large noise intensities for fixed strong synaptic inhibition strengths. Here, we investigate the effect of iSTDP on fast sparse synchronization (FSS) by varying the noise intensity D. We employ an asymmetric anti-Hebbian time window for the iSTDP update rule [which is in contrast to the Hebbian time window for the excitatory STDP (eSTDP)]. Depending on values of D, population-averaged values of saturated synaptic inhibition strengths are potentiated [long-term potentiation (LTP)] or depressed [long-term depression (LTD)] in comparison with the initial mean value, and dispersions from the mean values of LTP/LTD are much increased when compared with the initial dispersion, independently of D. In most cases of LTD where the effect of mean LTD is dominant in comparison with the effect of dispersion, good synchronization (with higher spiking measure) is found to get better via LTD, while bad synchronization (with lower spiking measure) is found to get worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to that in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD). Emergences of LTD and LTP of synaptic inhibition strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, we also investigate the effects of network architecture on FSS by changing the rewiring probability p of the SWN in the presence of iSTDP.

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.

Stochastic multiresonance in coupled excitable FHN neurons.

In this paper, effects of noise on Watts-Strogatz small-world neuronal networks, which are stimulated by a subthreshold signal, have been investigated. With the numerical simulations, it is surprisingly found that there exist several optimal noise intensities at which the subthreshold signal can be detected efficiently. This indicates the occurrence of stochastic multiresonance in the studied neuronal networks. Moreover, it is revealed that the occurrence of stochastic multiresonance has close relationship with the period of subthreshold signal Te and the noise-induced mean period of the neuronal networks T0. In detail, we find that noise could induce the neuronal networks to generate stochastic resonance for M times if Te is not very large and falls into the interval (M×T0,(M+1)×T0) with M being a positive integer. In real neuronal system, subthreshold signal detection is very meaningful. Thus, the obtained results in this paper could give some important implications on detecting subthreshold signal and propagating neuronal information in neuronal systems.

Directed and undirected network evolution from Euler–Lagrange dynamics

In this paper, we investigate both undirected and directed network evolution using the Euler–Lagrange equation. We use the Euler–Lagrange equation to develop a variational principle based on the von Neumann entropy for time-varying network structure. Commencing from recent work to approximate the von Neumann entropy using simple degree statistics, the changes in entropy between different time epochs are determined by correlations in the degree difference in the edge connections. Our Euler–Lagrange equation minimises the change in entropy and allows to develop a dynamic model to simulate the changes of node degree with time. We first explore the effect of network dynamics on the three widely studied complex network models, namely (a) Erdős-Rényi random graphs, (b) Watts-Strogatz small-world networks, and (c) Barabási-Albert scale-free networks. Our model effectively captures both undirected and directed structural transitions in the dynamic network models. We apply our model to a network time sequence representing the evolution of stock prices on the New York Stock Exchange(NYSE) and sequences of Drosophila gene regulatory networks containing different developmental phases of the organism from embryo to adult. Here we use the model to differentiate between periods of stable and unstable stock price trading and to detect periods of anomalous network evolution. Our experiments show that the presented model not only provides an accurate simulation of the degree statistics in time-varying networks but also captures the topological variations taking place when the structure of a network changes violently.

Research on edges immunization strategy for complex network based on SIS-CA model

This paper proposes a targeted edges immunization strategy, which aims to immune a certain number of the most important edges from the perspective of protecting transmission path. To better simulate the dynamic changes of risks propagation in real complex networks, an SIS-CA (Susceptible-Infected-Susceptible Cellular Automation) propagation model is developed, which combines SIS (Susceptible-Infected-Susceptible) model with the cellular automata (CA) method and takes nonuniformity and adjustment time into consideration. Based on SIS-CA propagation model, the feasibility of targeted edges immunization strategy was analysed in comparison with other targeted immunization strategy in the WS (Watts-Strogatz) small-world network model, BA (Barabasi-Albert) scale-free network model, and an aircraft manufacturing enterprise collaboration network (MECN) known to have the features of complex networks. The test results show that compared with conventional targeted immunization strategies, the proposed targeted edges immunization strategy obtains a better immunization efficacy via protecting fewer key edges.

Stochastic spike synchronization in a small-world neural network with spike-timing-dependent plasticity

We consider the Watts–Strogatz small-world network (SWN) consisting of subthreshold neurons which exhibit noise-induced spikings. This neuronal network has adaptive dynamic synaptic strengths governed by the spike-timing-dependent plasticity (STDP). In previous works without STDP, stochastic spike synchronization (SSS) between noise-induced spikings of subthreshold neurons was found to occur in a range of intermediate noise intensities. Here, we investigate the effect of additive STDP on the SSS by varying the noise intensity. Occurrence of a “Matthew” effect in synaptic plasticity is found due to a positive feedback process. As a result, good synchronization gets better via long-term potentiation of synaptic strengths, while bad synchronization gets worse via long-term depression. Emergences of long-term potentiation and long-term depression of synaptic strengths are intensively investigated via microscopic studies based on the pair-correlations between the pre- and the post-synaptic IISRs (instantaneous individual spike rates) as well as the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, the effects of multiplicative STDP (which depends on states) on the SSS are studied and discussed in comparison with the case of additive STDP (independent of states). These effects of STDP on the SSS in the SWN are also compared with those in the regular lattice and the random graph.

An improved synchronization likelihood method for quantifying neuronal synchrony

Indirect quantification of the synchronization between two dynamical systems from measured experimental data has gained much attention in recent years, especially in the computational neuroscience community where the exact model of the neuronal dynamics is unknown. In this regard, one of the most promising methods for quantifying the interrelationship between nonlinear non-stationary systems is known as Synchronization Likelihood (SL), which is based on the likelihood of the auto-recurrence of embedding vectors (similar patterns) in multiple dynamical systems. However, synchronization likelihood method uses the Euclidean distance to determine the similarity of two patterns, which is known to be sensitive to outliers. In this study, we propose a discrete synchronization likelihood (DSL) method to overcome this limitation by using the Manhattan distance in the discrete domain (l1 norm on discretized signals) to identify the auto-recurrence of embedding vectors. The proposed method was tested using unidirectional and bidirectional identical/non-identical coupled Hénon Maps, a Watts-Strogatz small-world network with nonlinearly coupled nodes based on Kuramoto model and the real-world ADHD-200 fMRI benchmark dataset. According to the results, the proposed method shows comparable and in some cases better performance than the conventional SL method, especially when the underlying highly connected coupled dynamical system goes through subtle changes in the bivariate case or sudden shifts in the multivariate case.

P.1.j.001 - Hormone and menstrual cycle effects on sleep spindle parameters in oral contraceptive users and controls

We consider the Watts–Strogatz small-world network (SWN) consisting of inhibitory fast spiking Izhikevich interneurons. This inhibitory neuronal population has adaptive dynamic synaptic strengths governed by the inhibitory spike-timing-dependent plasticity (iSTDP). In previous works without iSTDP, fast sparsely synchronized rhythms, associated with diverse cognitive functions, were found to appear in a range of large noise intensities for fixed strong synaptic inhibition strengths. Here, we investigate the effect of iSTDP on fast sparse synchronization (FSS) by varying the noise intensity D. We employ an asymmetric anti-Hebbian time window for the iSTDP update rule [which is in contrast to the Hebbian time window for the excitatory STDP (eSTDP)]. Depending on values of D, population-averaged values of saturated synaptic inhibition strengths are potentiated [long-term potentiation (LTP)] or depressed [long-term depression (LTD)] in comparison with the initial mean value, and dispersions from the mean values of LTP/LTD are much increased when compared with the initial dispersion, independently of D. In most cases of LTD where the effect of mean LTD is dominant in comparison with the effect of dispersion, good synchronization (with higher spiking measure) is found to get better via LTD, while bad synchronization (with lower spiking measure) is found to get worse via LTP. This kind of Matthew effect in inhibitory synaptic plasticity is in contrast to that in excitatory synaptic plasticity where good (bad) synchronization gets better (worse) via LTP (LTD). Emergences of LTD and LTP of synaptic inhibition strengths are intensively investigated via a microscopic method based on the distributions of time delays between the pre- and the post-synaptic spike times. Furthermore, we also investigate the effects of network architecture on FSS by changing the rewiring probability p of the SWN in the presence of iSTDP.

Impact of Partial Time Delay on Temporal Dynamics of Watts–Strogatz Small-World Neuronal Networks

In this paper, we mainly discuss effects of partial time delay on temporal dynamics of Watts–Strogatz (WS) small-world neuronal networks by controlling two parameters. One is the time delay [Formula: see text] and the other is the probability of partial time delay [Formula: see text]. Temporal dynamics of WS small-world neuronal networks are discussed with the aid of temporal coherence and mean firing rate. With the obtained simulation results, it is revealed that for small time delay [Formula: see text], the probability [Formula: see text] could weaken temporal coherence and increase mean firing rate of neuronal networks, which indicates that it could improve neuronal firings of the neuronal networks while destroying firing regularity. For large time delay [Formula: see text], temporal coherence and mean firing rate do not have great changes with respect to [Formula: see text]. Time delay [Formula: see text] always has great influence on both temporal coherence and mean firing rate no matter what is the value of [Formula: see text]. Moreover, with the analysis of spike trains and histograms of interspike intervals of neurons inside neuronal networks, it is found that the effects of partial time delays on temporal coherence and mean firing rate could be the result of locking between the period of neuronal firing activities and the value of time delay [Formula: see text]. In brief, partial time delay could have great influence on temporal dynamics of the neuronal networks.

Effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks.

In this paper, we study effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks. Our focus is on the impact of two parameters, namely the time delay τ and the probability of partial time delay pdelay, whereby the latter determines the probability with which a connection between two neurons is delayed. Our research reveals that partial time delays significantly affect phase synchronization in this system. In particular, partial time delays can either enhance or decrease phase synchronization and induce synchronization transitions with changes in the mean firing rate of neurons, as well as induce switching between synchronized neurons with period-1 firing to synchronized neurons with period-2 firing. Moreover, in comparison to a neuronal network where all connections are delayed, we show that small partial time delay probabilities have especially different influences on phase synchronization of neuronal networks.

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.

Effects of hybrid synapses and partial time delay on stochastic resonance in a small-world neuronal network

In real neuronal systems, information transition delay is an inevitable factor. However, between some neurons, neuronal information is transmitted instantaneously or the time delay is too small and can be neglected. Thus, differing from the conventional studies where all connections are considered to be delayed, here we mainly focus on the effect of partial time delay on stochastic resonance in a Watts-Strogatz small-world neuronal network. Meanwhile, in the same neuronal network, the electrical and chemical synapses usually coexist. Thus, effects of hybrid synapses are also considered. Firstly, in the absence of time delay, noise could induce stochastic resonance when the neuronal network contains much more excitatory synapses than inhibitory ones; while it cannot induce stochastic resonance vise verse. Interestingly, it is further revealed that when the ratio of excitatory synapse to inhibitory synapse is approximately 4:1, noise-induced stochastic resonance is more robust. Thus, to discuss the effects of other factors on noise-induced stochastic resonance, we set this ratio to be 4:1. In the absence of time delay, we also consider effects of chemical synapses with a ratio of excitatory synapse to inhibitory synapse of 4:1 on the noise-induced stochastic resonance. The obtained results show that the noise could always induce stochastic resonance no matter how the probability of chemical synapses varies. And the optimal noise intensity increases linearly with the probability of chemical synapses increasing. For partial time delay, it is surprisingly found that the stochastic resonance could appear multiple times with the variation of the time delay being just for small partial time delay probability. Moreover, chemical synapse is found to facilitate this effect of partial time delay. Finally, by analyzing the joint effects of partial time delay and noise intensity, it is found that the larger the time delay and the partial time delay probability are, the wider the optimal noise region corresponding to large response amplitude is.

Вычисление областей устойчивости дискретных моделей больших нейронных сетей типа small world

The article is devoted to description of discrete models of small world networks with a large number of neurons with a certain parameter p varying from 0 to 1. For p=0 have model, regular neural networks, which is a ring network in which each neuron interacts with several neighbors on the ring. In the case p=1 have a model with randomly distributed connections. When the values of $p$ not exceeding 0.1 have the Watts-Strogatz small world network. Such a neural network can be models of different neural structures in living organisms, for example, the hippocampus of the mammalian brain. This paper examines the dynamics of change areas of stability of such neural networks when 0\leq p \leq 0.1. Numerical experiments show an increase in sustainability in the transition from a regular network to small world.

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.

Stochastic multi-resonance induced by partial time delay in a Watts-Strogatz small-world neuronal network

In a neuronal system, propagation speed of neuronal information is mainly determined by the length, the diameter, and the kind of the axons between the neurons. Thus, some communications between neurons are not instantaneous, and others are instantaneous or with some negligible delay. In the past years, effects of time delay on neuronal dynamics, such as synchronization, stochastic resonance, firing regularity, etc., have been investigated. For stochastic resonance, it has been reported recently that stochastic multi-resonance in a neuronal system can be induced by time delay. However, in these studies, time delay has been introduced to every connection of the neuronal system. As mentioned in the beginning, in a real neuronal system, communication between some neurons can be instantaneous or with some negligible delays. Thus, considering the effect of partial time delay (time delay is called as partial time delay if only part of connections are delayed) on neuronal dynamics could be more meaningful.In this paper, we focus on discussing effect of partial time delay on response amplitude of a Watts-Strogatz neuronal network which is locally modeled by Rulkov map. With the numerically obtained results, we can see that partial time delay can induce a stochastic multi-resonance which is indicated by the multi-peak characteristics in the variation of response amplitude with partial time delay. Namely, partial time delay could also induce stochastic multi-resonance in a neuronal system. Moreover, we also find that optimal response amplitude can be reached in much wider range of the partial time delay value when delayed connections are less (i.e., the partial time delay probability is small). This is different from the case in which all connections are delayed, where response amplitude become optimal only when time delay is nearly the multiples of external signal's period. But the range of the partial time delay value becomes narrower and narrower with the increasing of the partial time delay probability and when the partial time delay probability is large enough, response amplitude becomes optimal only when time delay is nearly the multiples of external signal period. It is similar to the case where all connections are delayed. Furthermore, effects of random rewiring probability and total link number in the neuronal network on partial time delay induced stochastic multi-resonance are also studied. It is found that partial time delay induced stochastic multi-resonance is robust to random rewiring probability but not robust to total link number. Stochastic resonance is a very important nonlinear phenomenon in neuroscience, thus, our obtained results could have some implications in this field.

Synchronization properties of heterogeneous neuronal networks with mixed excitability type.

We study the synchronization of neuronal networks with dynamical heterogeneity, showing that network structures with the same propensity for synchronization (as quantified by master stability function analysis) may develop dramatically different synchronization properties when heterogeneity is introduced with respect to neuronal excitability type. Specifically, we investigate networks composed of neurons with different types of phase response curves (PRCs), which characterize how oscillating neurons respond to excitatory perturbations. Neurons exhibiting type 1 PRC respond exclusively with phase advances, while neurons exhibiting type 2 PRC respond with either phase delays or phase advances, depending on when the perturbation occurs. We find that Watts-Strogatz small world networks transition to synchronization gradually as the proportion of type 2 neurons increases, whereas scale-free networks may transition gradually or rapidly, depending upon local correlations between node degree and excitability type. Random placement of type 2 neurons results in gradual transition to synchronization, whereas placement of type 2 neurons as hubs leads to a much more rapid transition, showing that type 2 hub cells easily "hijack" neuronal networks to synchronization. These results underscore the fact that the degree of synchronization observed in neuronal networks is determined by a complex interplay between network structure and the dynamical properties of individual neurons, indicating that efforts to recover structural connectivity from dynamical correlations must in general take both factors into account.

Risk and Resilience Analysis of Complex Network Systems Considering Cascading Failure and Recovery Strategy Based on Coupled Map Lattices

Risk and resilience are important and challenging issues in complex network systems since a single failure may trigger a whole collapse of the systems due to cascading effect. New theories, models, and methods are urgently demanded to deal with this challenge. In this paper, a coupled map lattices (CML) based approach is adopted to analyze the risk of cascading process in Watts-Strogatz (WS) small-world network and Barabási and Albert (BA) scale-free network, respectively. Then, to achieve an effective and robust system and provide guidance in countering the cascading failure, a modified CML model with recovery strategy factor is proposed. Numerical simulations are put forward based on small-world CML and scale-free CML. The simulation results reveal that appropriate recovery strategies would significantly improve the resilience of networks.