Separation And Integration With The Change Of Connection Structure

In this paper,using the model of phase oscillator,we study the effect of the connection structure of coupled oscillator system on the collective behavior of oscillator which are named as segregation and integration.Synchronization is a classical dynamics problem in study of the coupled oscillator system,and is used to understand various synchronization behaviors in natural phenomena.Segregation and integration are based on the synchronization and are collective behaviors proposed according to natural complex systems,such as neural systems.We use computer simulation and analytical analysis to study how the segregation and integration change with deleting nodes or connections.In this study,the oscillator model is used to give a theoretical explanation of the change of separation and integration.The research were as follows:First,we simulate the effect of network structure on parameters measuring the complexity of networks.We use two fully connected networks as modules to construct a modular network by removing connections,e.g.,connections within modules are rewired into inter-modular connections.In the processes of rewiring,we compute various measurements of the network structure properties and the collective behavior of oscillator,including the network modularity,the Pearson correlation coefficient of oscillators,the degree of synchronization,the segregation index,and the Shannon entropy.We use two methods to calculate the segregation of the network,and our calculations show that the results of the two methods are the same.Moreover,our simulation confirms that in the vicinity of value of rewiring probability exhibiting segregation and integration(S-I)balance,the Pearson correlation coefficient of the network is smallest,and the network has the largest complexity,that is,the Shannon entropy is the largest.Finally,our research shows that the measurements of various complexity and the S-I balance proposed in literatures are proper for the coupled oscillator model.Second,we simulate the effect of the degeneration of network structure on segregation and integration in the S-I balance network.The process of degeneration of the network is simulated by randomly removing nodes or links from the network.These structural changes simulate with brain aging.The monotonic change of segregation and integration with the aging of the brain was observed,but the dynamical mechanism is lack.By changing the network structure,we show that the collective behavior of the S-I balance network will change in the way as the same as the normal aging process,that is,segregation decreases and integration increases.The behavior of segregation depends on the network structure.When the reconnection ratio is very small,the network is isolated modules,segregation and integration are not affected by the change of the network.When the reconnection ratio is very large,the network is closer to a random network,segregation and integration does not change after removing nodes or links.The behavior only appears in the vicinity of S-I balance.Third,we give an explanation of the mechanism of the change of segregation and integration,and reveal the role of the modular network structure in the change.In the process of removing nodes or links,we compute the variation of the degree of each node including thechange amount of intra-modular and inter-modular degree.The results show that all nodes lose intra-modular connections,however a part of nodes do not lose inter-modular connections.The distribution of the change of inter-modular connection degrees is asymmetric which improves the integration of the network.Then we analyzed the region where the network exhibits asymmetry.We show that nearby the S-I balance point,the change in the inter-modular degree exhibits an obvious asymmetric.We also investigate the distribution of variation in the degree of node in real brain networks,which consists with the results of the model.When some nodes or links are removed from the biological realistic network,the distribution of variation in the inter-modular degree of node is asymmetric.