Abstract
In this work, we investigate a model of an adaptive networked dynamical system, where the coupling strengths among phase oscillators coevolve with the phase states. It is shown that in this model the oscillators can spontaneously differentiate into two dynamical groups after a long time evolution. Within each group, the oscillators have similar phases, while oscillators in different groups have approximately opposite phases. The network gradually converts from the initial random structure with a uniform distribution of connection strengths into a modular structure that is characterized by strong intra-connections and weak inter-connections. Furthermore, the connection strengths follow a power-law distribution, which is a natural consequence of the coevolution of the network and the dynamics. Interestingly, it is found that if the inter-connections are weaker than a certain threshold, the two dynamical groups will almost decouple and evolve independently. These results are helpful in further understanding the empirical observations in many social and biological networks.
GENERAL SCIENTIFIC SUMMARY Introduction and background. Modularity is common in many social and biological networked systems, and is generally believed to correspond to certain functional groups. Usually, the distribution of connection strengths follows a power law. In the past decade, attention has mainly been paid to the topological structure of the networks, and the dynamics on the networks, respectively. Nevertheless, in various realistic systems, in principle the network topology and dynamics are strongly dependent on each other. Thus any network structures and dynamical patterns that emerged are actually the results of the co-evolution of the network dynamics and topology. Recently, attention has been paid to the adaptive co-evolutionary networks. However, so far, how the dynamical groups are generated during the co-evolution of network structure and dynamics has not been investigated from the point of view of complex networks. Motivated by this idea, we present a toy adaptive network model consisting of phase oscillators, where the connection strengths among oscillators are coupled with the dynamical states.
Main results. By adopting the simple evolution rule, the dynamical groups can be spontaneously formed in our model, i.e. in-phase and anti-phase synchronized states simultaneously exist in our system. Simultaneously, the connection strengths in the network can self-organize into a power law distribution from the initial random distribution. In addition, communities, which correspond to the dynamical groups in our model, can also be spontaneously formed.
Wider implications. This investigation may be helpful in further understanding the behaviors of many real networked dynamical systems, such as the evolution of social networks and biological networks.
Figure. Characterization of the formation of the dynamical groups and the modular structure of the network. (a) The evolution of the oscillator states. (b) The evolution of the order parameters, where F keeps increasing, but R always maintains very small values, indicating that the dynamical groups have formed. (c) The evolution of the average link weight, where the average weight of the inter links (<winter>) decreases all the time, while the intra link weight (<wintra>) keeps increasing. (d) The distribution of the connections strengths for the network at t = 3000, 9000 and 15 000. The longer the time t, the more obvious the power law distribution is. (e) The snapshot of weight matrix wmk at t = 3000, where modular structure occurs simultaneously with the formation of dynamical groups.