The Kuramoto model in complex networks

Rodrigues, Francisco A.; Peron, Thomas; Ji, Peng; Kurths, Jürgen

doi:10.1016/j.physrep.2015.10.008

Cited by 810 publications

(681 citation statements)

References 430 publications

(1,095 reference statements)

Supporting

Mentioning

620

Contrasting

Unclassified

Order By: Relevance

“…Phase transitions in ensembles of complex networked systems have been a subject of intense research in statistical physics and many other disciplines [1]. These results are of fundamental importance for understanding various dynamical processes in real world, such as percolation [2,3], epidemic spreading [4], synchronization [5,6], and collective phenomena in social networks [7].…”

mentioning

confidence: 99%

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

et al. 2017

Self Cite

View full text Add to dashboard Cite

We generalize the original majority-vote model by incorporating an 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 type to a discontinuous or an explosive 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. Recently, explosive or discontinuous transitions in complex networks have received growing attention since the discovery of an abrupt percolation transition in random networks [8,9] and scale-free networks [10,11]. Later studies affirmed that this transition is actually continuous but with an unusual finite size scaling [12][13][14], yet many related models show truly discontinuous and anomalous transitions (cf.[15] for a recent review). Striking different from continuous phase transitions, in an explosive transition an infinitesimal increase of the control parameter can give rise to a considerable macroscopic effect. Subsequently, an explosive phenomenon was found in the dynamics of cascading failures in interdependent networks [16][17][18], in contrast to the secondorder continuous phase transition found in isolated networks. More recently, such explosive phase transitions have been reported in various systems, such as explosive synchronization due to a positive correlation between the degrees of nodes and the natural frequencies of the oscillators [19][20][21] [25][26][27].In this paper we report an explosive order-disorder phase transition in a generalized majority-vote (MV) model by incorporating the effect of individuals' inertia (called inertial MV model ). The MV model is one of the simplest nonequilibrium generalizations of the Ising model that displays a continuous order-disorder phase transition at a critical value of noise [28]. It has been extensively studied in the context of complex networks, including random graphs [29,30], small world networks [31][32][33], and scale-free networks [34,35]. However, the continuous nature of the order-disorder phase transition is not affected by the topology of the underlying networks [36]. In our model, we have included a substantial change to make it more realistic, namely the state update of each node depends not only on the states of its neighboring nodes, but also on its own state. In fact, in a social or biological context individuals have a tendency for beliefs to endure once formed. In a recent experimental study, behavioral inertia was found to be essential for collective turning of starling flocks [37]. We refer this m...

show abstract

mentioning

confidence: 99%

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

et al. 2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…For reviews, see Refs. [3][4][5][6][7][8][9].The governing equations for the Kuramoto model arėfor i = 1, . .…”

mentioning

confidence: 99%

Kuramoto model with uniformly spaced frequencies: Finite-Nasymptotics of the locking threshold

Ottino-Loffler

Strogatz

2016

Phys. Rev. E

View full text Add to dashboard Cite

We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, N , is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite N , the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite N have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when N 1. The leading correction to the infinite-N result scales like either N −3/2 or N −1 , depending on whether the frequencies are evenly spaced according to a midpoint rule or an endpoint rule. These scaling laws agree with numerical results obtained by Pazó [Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics. INTRODUCTIONIn 1975, Kuramoto proposed an elegant model of coupled nonlinear oscillators, now known as the Kuramoto model [1,2]. Since then the model has been applied to a wide range of physical, biological, chemical, social, and technological systems, and its analysis has stimulated theoretical work in nonlinear dynamics, statistical physics, network science, control theory, and pure mathematics. For reviews, see Refs. [3][4][5][6][7][8][9].The governing equations for the Kuramoto model arėfor i = 1, . . . , N , where θ i is the phase of oscillator i and ω i is its natural frequency. Inspired by Winfree's work on self-synchronizing systems of biological oscillators [10], Kuramoto restricted attention to attractive coupling, K > 0, and assumed that the ω j were randomly distributed across the population according to a prescribed probability distribution g(ω), which he took to be unimodal and symmetric about its mean. Without loss of generality the mean frequency can be set to 0 by going into a rotating frame, and the coupling K can be set to K = 1 by rescaling time. Both of these normalizations will be assumed in what follows. One adjustable parameter remains: the characteristic width γ of the frequency distribution. When γ is sufficiently large, numerical simulations show that the oscillators behave incoherently and run at their natural frequencies. At the other extreme, when γ = 0 the oscillators have identical frequencies and approach a perfectly synchronized solution with θ i (t) = θ j (t) for all i, j and t. Kuramoto's achievement was to analyze the dynamics of the model in between these two extremes. He solved the model in the continuum limit N → ∞ and obtained a number of beautiful results [1,2], opening up a fruitful line of research for many subsequent studies (reviewed in [3][4][5][6]). In particular, he showed that as γ is decreased from large values, a bifurcation takes place at a critical value γ c . This bifurcation gives rise to a branch of partially synchronized states: the ...

show abstract

“…One of the simple paradigm to understand the synchronization phenomenon is the Kuramoto model [25,26], which has been applied to study the synchronization patterns and to identify community in complex networks. Consider a network of N vertices joined in pairs by E links, which represent the interaction between vertices, for example, the acquaintance or collaborations between individual in social networks.…”

Section: Methodology and Analysismentioning

confidence: 99%

“…Note that vertices are represented by their corresponding oscillators in the following part of our paper. Many previous literatures show that there exists a critical coupling λ c , above which synchronization emerges spontaneously [25,26]. We here realize the Kuramoto model on complex networks using the Runge-Kutta methods.…”

Section: Methodology and Analysismentioning

confidence: 99%

A dynamical approach to identify vertices' centrality in complex networks

et al. 2017

View full text Add to dashboard Cite

In this paper, we proposed a dynamical approach to assess vertices' centrality according to the synchronization process of the Kuramoto model. In our approach, the vertices' dynamical centrality is calculated based on the Difference of vertices' Synchronization Abilities (DSA), which are different from traditional centrality measurements that are related to the topological properties. Through applying our approach to complex networks with a clear community structure, we have calculated all vertices' dynamical centrality and found that vertices at the end of weak links have higher dynamical centrality. Meanwhile, we analyzed the robustness and efficiency of our dynamical approach through testing the probabilities that some known vital vertices were recognized. Finally, we applied our dynamical approach to identify community due to its satisfactory performance in assessing overlapping vertices. Our present work provides a new perspective and tools to understand the crucial role of heterogeneity in revealing the interplay between the dynamics and structure of complex networks.

show abstract

The Kuramoto model in complex networks

Cited by 810 publications

References 430 publications

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

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

Kuramoto model with uniformly spaced frequencies: Finite-Nasymptotics of the locking threshold

A dynamical approach to identify vertices' centrality in complex networks

Contact Info

Product

Resources

About