Chuansheng Shen

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

Strategy to suppress epidemic explosion in heterogeneous metapopulation networks.

We propose an efficient strategy to suppress epidemic explosion in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals and is assigned a curing rate that is proportional to kα with the node degree k and an adjustable parameter α. We perform stochastic simulations of the dynamical reaction-diffusion processes associated with the susceptible-infected-susceptible model in scale-free networks. We find that the epidemic threshold reaches a maximum when α is tuned at αopt≃1.3. This nontrivial phenomenon is robust to the change of the network size and the average degree. In addition, we carry out a mean field analysis to further validate our scheme, which also demonstrates that epidemic explosion follows different routes for α larger or less than αopt. Our work suggests that in order to efficiently suppress epidemic spreading on heterogeneous complex networks, subpopulations with higher degrees should be allocated more resources than just being linearly dependent on k.

First-order phase transition in a majority-vote model with inertia

We generalize the original majority-vote model by incorporating inertia into the microscopic dynamics of the spin flipping, where the spin-flip probability of any individual depends not only on the states of its neighbors, but also on its own state. Surprisingly, the order-disorder phase transition is changed from a usual continuous or second-order type to a discontinuous or first-order one when the inertia is above an appropriate level. A central feature of such an explosive transition is a strong hysteresis behavior as noise intensity goes forward and backward. Within the hysteresis region, a disordered phase and two symmetric ordered phases are coexisting and transition rates between these phases are numerically calculated by a rare-event sampling method. A mean-field theory is developed to analytically reveal the property of this phase transition.

Critical noise of majority-vote model on complex networks.

The majority-vote model with noise is one of the simplest nonequilibrium statistical model that has been extensively studied in the context of complex networks. However, the relationship between the critical noise where the order-disorder phase transition takes place and the topology of the underlying networks is still lacking. In this paper, we use the heterogeneous mean-field theory to derive the rate equation for governing the models dynamics that can analytically determine the critical noise f(c) in the limit of infinite network size N→∞. The result shows that f(c) depends on the ratio of 〈k〉 to 〈k(3/2)〉, where 〈k〉 and 〈k(3/2)〉 are the average degree and the 3/2 order moment of degree distribution, respectively. Furthermore, we consider the finite-size effect where the stochastic fluctuation should be involved. To the end, we derive the Langevin equation and obtain the potential of the corresponding Fokker-Planck equation. This allows us to calculate the effective critical noise f(c)(N) at which the susceptibility is maximal in finite-size networks. We find that the f(c)-f(c)(N) decays with N in a power-law way and vanishes for N→∞. All the theoretical results are confirmed by performing the extensive Monte Carlo simulations in random k-regular networks, Erdös-Rényi random networks, and scale-free networks.

Mobility and density induced amplitude death in metapopulation networks of coupled oscillators.

We investigate the effects of mobility and density on the amplitude death of coupled Landau-Stuart oscillators and Brusselators in metapopulation networks, wherein each node represents a subpopulation occupied any number of mobile individuals. By numerical simulations in scale-free topology, we find that the systems undergo phase transitions from incoherent state to amplitude death, and then to frequency synchronization with increasing the mobility rate or density of oscillators. Especially, there exists an extent of intermediate mobility rate and density that can lead to global oscillator death. Furthermore, we show that such nontrivial phenomena are robust to diverse network topologies. Our findings may invoke further efforts and attentions to explore the underlying mechanism of collective behaviors in coupled metapopulation systems.

Nucleation in scale-free networks.

We have studied nucleation dynamics of the Ising model in scale-free networks whose degree distribution follows a power law with the exponent γ by using the forward flux sampling method and focusing on how the network topology would influence the nucleation rate and pathway. For homogeneous nucleation, the new phase clusters grow from those nodes with smaller degree, while the cluster sizes follow a power-law distribution. Interestingly, we find that the nucleation rate R{Hom} decays exponentially with network size and, accordingly, the critical nucleus size increases linearly with network size, implying that homogeneous nucleation is not relevant in the thermodynamic limit. These observations are robust to the change of γ and are also present in random networks. In addition, we have also studied the dynamics of heterogeneous nucleation, wherein w impurities are initially added either to randomly selected nodes or to targeted ones with the largest degrees. We find that targeted impurities can enhance the nucleation rate R{Het} much more sharply than random ones. Moreover, ln(R{Het}/R{Hom}) scales as w{(γ-2)/(γ-1)} and w for targeted and random impurities, respectively. A simple mean-field analysis is also present to qualitatively illustrate the above simulation results.

Nucleation pathways on complex networks

Identifying nucleation pathway is important for understanding the kinetics of first-order phase transitions in natural systems. In the present work, we study nucleation pathway of the Ising model in homogeneous and heterogeneous networks using the forward flux sampling method, and find that the nucleation processes represent distinct features along pathways for different network topologies. For homogeneous networks, there always exists a dominant nucleating cluster to which relatively small clusters are attached gradually to form the critical nucleus. For heterogeneous ones, many small isolated nucleating clusters emerge at the early stage of the nucleation process, until suddenly they form the critical nucleus through a sharp merging process. Moreover, we also compare the nucleation pathways for different degree-mixing networks. By analyzing the properties of the nucleating clusters along the pathway, we show that the main reason behind the different routes is the heterogeneous character of the underlying networks.

Quenched mean-field theory for the majority-vote model on complex networks

The majority-vote (MV) model is one of the simplest nonequilibrium Ising-like model that exhibits a continuous order-disorder phase transition at a critical noise. In this paper, we present a quenched mean-field theory for the dynamics of the MV model on networks. We analytically derive the critical noise on arbitrary quenched unweighted networks, which is determined by the largest eigenvalue of a modified network adjacency matrix. By performing extensive Monte Carlo simulations on synthetic and real networks, we find that the performance of the quenched mean-field theory is superior to a heterogeneous mean-field theory proposed in a previous paper [Chen \emph{et al.}, Phys. Rev. E 91, 022816 (2015)], especially for directed networks.

How does degree heterogeneity affect nucleation on complex networks

Nucleation is an initiating process of a stable phase from a metastable phase in a first-order phase transition. Taking the Ising model as a paradigm, we investigate the dynamics of nucleation on complex networks and focus on the role played by the heterogeneity of degree distribution on nucleation rate. Using Monte Carlo simulation combined with forward flux sampling, we find that for a weak external field the nucleation rate decreases monotonically as degree heterogeneity increases. Interestingly, for a relatively strong external field the nucleation rate exhibits a nonmonotonic dependence on degree heterogeneity, in which there exists a maximal nucleation rate at an intermediate level of degree heterogeneity. Furthermore, we develop a heterogeneous mean-field theory for evaluating the free-energy barrier of nucleation. The theoretical estimations are qualitatively consistent with the simulation results. Our study suggests that degree heterogeneity plays a nontrivial role in the dynamics of phase transitions in networked Ising systems.

Mobility-enhanced signal response in metapopulation networks of coupled oscillators

We investigate the effect of mobility on the response of coupled oscillators to a subthreshold external signal in metapopulation networks, wherein each node represents a subpopulation with overdamped bistable oscillators that can randomly diffuse between nodes. With increasing mobility rate, the oscillators undergo transitions from intrawell to interwell motion, demonstrating clearly mobility-enhanced signal amplification. Moreover, the response shows nonmonotonic dependence on the mobility rate, i.e., a maximal gain occurs at a moderate level of mobility. This interesting phenomenon is robust against variations in the overall density, network size, as well as network topology. In addition, a simple mean-field analysis is carried out to qualitatively illustrate the simulation results.