Phase Transition And Critical Phenomena In Complex Systems

As the experimental techniques keep developing,the details of every agent's behavior in complex systems can be recorded,and a lot of scale-free phenomena are observed among these data,which indicates that some real systems stay on critical states.Since structure and dynamics of complex systems are complex,the critical phenomena of complex systems not reveal novel universality class but also present non-equilibrium character.The continuous phase transition theory plays a fundamental role in studying critical phenomena of complex systems.Meanwhile,a series of new models with unique critical phenomena are proposed,which gives people a new angle to study the phase transition and critical phenomena.In this thesis,we introduce our works from two aspects,percolation and critical transition induced by dynamics.In chapter two,we make a brief introduction about the background knowledge,including phase transition and critical phenomena in physical systems,critical phenomena in complex systems,structural feature of complex systems,and complex networks.We also state our research motivation and the structure of this thesis.Clique is a complete subgraph which plays a significant role in dividing community structure of complex networks and dynamical stability.It is widely accepted that the critical exponents only depend on the space dimension and order parameter dimension.However,it has been observed that the clique percolation on ER graph belongs to different universality class for different types of clique connection.To verify the hypothesis about universality class,in the chapter two,we study clique percolation on 2 dimensional Moore lattice.We find the different type of clique connections we choose will not affect the critical exponents of 2-dimensional clique percolation,and the analytical analysis support this conclusion.We also propose an analytical clique percolation model,and we prove that this infinite dimension systems have same critical exponent ? with bond percolation model.It also states that the unique critical phenomena are due to the special structure of ER graph.In the clique percolation study,we also develop a numerical method to estimate the critical exponent ?,and it can be easily used for different percolation model on complex networks.Basing on this method,we study the explosive percolation on lattice and k-core percolation on ER graph based on this method,and obtain the same critical exponents with references.We also find that the discontinuity in k-core percolation is induced by the largest characteristic cluster merging,and the critical phenomena comes from the second characteristic merging.In the last part of chapter two,we study the fluctuation correlation in percolation model.We find largest eigenvalues and the eigenvalue distribution of fluctuation correlation matrix show the characters of critical phenomena.The numerical results of serval systems with different dimensions indicate there is a simple scaling relation in fluctuation correlation spectrum.We also notice that the fluctuation correlation function can be rescaled into the similar form in collective motion system.In chapter three,we study the flux fluctuation in complex systems,cascading failure in supply-demand systems and dynamics mode transition in minority game.In real transition process,the flux fluctuation usually falls into two scaling form,and a following theory explains the phenomena successfully.However,a striking derivation is noticed in the numerical experiments of small systems.In chapter three,we propose a revised flux-fluctuation theory.It can accurately describe the flux-fluctuation of small systems in which the external drive fluctuates weakly,and for strong fluctuating drive,the revised theory is approximate to original theory.We also develop this theory,and derive a region flux-fluctuation theory which can describe the flux-fluctuation in a local region.The sharing flux between two nodes,defined as the number of data packet passing through these two nodes simultaneously,can be accurately estimated by the region flux-fluctuation theory.Supply-demand system is a common complex transport system.We study the cascading transition in a simplified model of supply-demand system.We find that the cascading failure spreads along two layers of the network.It is also found that there is a critical degree of redundancy.If the system redundancy is smaller than the critical redundancy,single edge failure can easily cause a global cascading,and the number of surviving nodes has a ceiling for global cascading.If the system's redundancy is larger than critical redundancy,a critical layer structure will appear in the system.With the initial failure closing to critical layer,the cascading duration increases as power law function.When the redundancy is approaching critical point,the critical behaviors can be observed in the number of fail nodes and duration time.We explain these abundant critical phenomena by percolation model on interdependent network.We study the critical transition of dynamics mode change in minority game.The orbit with period three are observed in the minority game which contains multi-resources.As the parameter approaching critical point,the lifetime of period three orbit is decreasing exponentially for the system constituted by finite number of agents.The exponential factor increases as power-law function of the number of agents.We also derive the lifetime of period three orbit,the basin of attraction and the feasible region of the orbit with higher period.We give the overall summary and propose a researching plan in chapter four.