The Consensus Problem On Complex Network And The Application Of The Related Model In Brain Science

The graph-based model provides an important method to study complex system dynamics.The dynamics of many complex systems can be modeled using the dy-namical system on the graph.The average agreement on the graph is a dynamic system widely studied,which has many important applications in the field of cybernetics.This thesis studies the consensus problem of the average protocol on the graph.The purpose of the consensus problem on the graph is to obtain a sufficient condition that is mild and easy to verify with the structure of the graph,so that the dynamic system defined by the average protocol can reach consensus as time grows to infinity.In this thesis,consen-sus reaching can be equivalent to the weak ergodicity of the Markov process,and the ergodicity has studied by estimating the Dobrushin coefficient of the Markov chain.We are developing a method to deal with the consensus problem,the static and the time-varying network can be dealt with an unified viewpoint,and derive the necessary and sufficient conditions of the consensus reaching.Moreover,a more milder suffi-cient conditions for(almost sure)consensus are presented for the case of time-varying(and stochastic)networks.The thesis also studies the average protocol with random noise on two-dimensional grid and shortcut model(small-world model).The purpose of this model is to study the behavior of human brain in resting-state.When the brain is in a resting-state,the measure of activity of each voxel in the brain can be measured by fMRI scanning.Con-sidering the large number of neurons have connected in the brain,thus they construct a complex graph.It is shown that the neurons of the brain have long-distance con-nection in many experiments.And the graph of neurons connection is a small-world network.We have used the average agreement with noise on the small-world network to describe the activities of the brain in the resting-state,and construct the statistics in-cluding the spatial correlation function and the spatial entropy to reflect the difference of the underlying nerve connections between different persons.From this viewpoint,we analyze the fMRI data of schizophrenic patients,and obtained the difference be-tween the brain activity data of normal persons and the patients,and obtained some biological significance.