Inhomogeneous Percolation On Bethe Lattices And Multiplex Networks

By the theory of discrete-dynamic system,generalised recursive approach,generating function approach,sensitivity analysis,fixed-point iteration approach and numerical simulation,this thesis focuses on the inhomogeneous percolation and some applications on several lattices or networks,including the classical Bethe lattice,the irregular Bethe lattice,multilayer finite fixed network and multilayer giant random network.During percolation process,the formation of percolation clusters with distinct sizes,occurring of spanning cluster(percolating),critical occupation probability,mean size of percolation cluster,percolating probability,and the critical phenomenon including scaling invariance,fractal characteristic and critical exponents are principal quantities and phenomena.In this paper,percolation phase transition and critical phenomena is displayed by critical occupation probability,average cluster size,percolating probability and part application.The thesis can be mainly divided into the following four sections.Firstly inhomogeneous site percolation on Bethe Lattices with two occupation probabilities is investigated,and then the result is extended to percolation with 8)occupation probabilities.The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability with given occupation probability ,the critical occupation probability,and the average cluster size.Moreover,using the above theory,the diffusion behaviour of an infectious disease(SARS)is discussed in detail and specific disease-control strategies is presented in consideration of groups with different infection probabilities.Then an inhomogeneous site percolation on an irregular Bethe lattice,for considering that percolation often occurs on irregular grids or lattices with variable site neighbours in real-world problems.The explicit expression for clustersize distribution of this percolation is derived based on probability theory.Moreover,the exact formulas for critical occupation probability,mean cluster size,and percolation probability are obtained using generating function method and generalised recursive approach.In addition,sensitivity analysis and numerical simulation are given to deepen and illustrate the results.Next a kind of high-order inhomogeneous bond percolation on multilayer infinite random network are focused on.By using the generalized recursive algorithm,fixed point iterative algorithm,the theory of fixed point,stability analysis,and bifurcation analysis methods of discrete-time dynamic system,the percolation transition(percolating)critical condition,and percolating probability,are obtained.In particular,percolation analysis and sensitivity analysis also showed that percolation transition(or percolating)is closely related to the mean degree of sites at every layer and the correlation coefficient between distinct layers.With the increasing of average degree,the reliability of the network will increase,the supercritical region will expand,and the percolating probability will increase;and with the increasing of correlation coefficient from-1 to 1,the convexity of the critical surface of percolating will change firstly from the downhill convex to linear,and then to uphill convex.Last a kind of generalization of high-order inhomogeneous bond percolation is studied on multilayer finite fixed network.The inhomogeneous percolation means that edges in each layer are occupied or removed with distinct probabilities,independently and randomly.Firstly,an analytical approach to simplify the inhomogeneous percolation on multilayer finite fixed network is presented,by decomposing the layers into disjoint ones,fusing inhomogeneous percolation process and combining weighted layers together.Then employing generalised recursive approach,discrete-time dynamical system analysis and fixed-point iteration method,we derive the percolating probability and the exact analytical results for critical occupation probability,demonstrate the percolation transition and critical phenomenon of inhomogeneous percolation on these two networks.It is found that,for inhomogeneous percolation on multilayer finite fixed network,the conjoining of the network speeds up the percolating process.