Research On Some Problems Of Self-organized Criticality And Complex Networks

In this paper, we studied several problem aboutt self-organized criticality and complex networks in order to improve the understanding about them. These problems are the universality class of the Abelian stochastic sandpile model and directed sandpile model, sandpile avalanche dynamics in the directd complex netwroks,and the structure and function of complex transport networks.Firstly, we studied only self-organized criticality, that is the universality class of the Abelian stochastic sandpile model and directed sandpile model, and performed as followes. 1) We introduced a Abelian stochastic sandpile model where the number of grain transported is random, and studied it by the finite size scaling law and moment analysis. 2) According the local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site when all its neighbors topple for once, we introduced two Abelian quenched random directed sandpile models. By the finite size scaling law and moment analysis,we studied comparatively the two models, Abelian deterministic directed sandpile model and Abelian stochastic directed sandpile model. 3) By the finite size scaling law and moment analysis ,we studied comparatively the Abelian stochastic directed sandpile model and non-Abelian stochastic directed sandpile model. 4) we introduced a non-Abelian deterministic directed sandpile model, and by the finite size scaling law and moment analysis,we studied comparatively the Abelian deterministic directed sandpile model and non-Abelian deterministic directed sandpile model.The results show that: 1) Our Abelian stochastic sandpile model has the same critical exponents as Manna and Oslo models, which lead us to conjecture that all Abelian stochastic sandpile models with only different random toppling rules belong to the same universality. 2)For the Abelian deterministic directed sandpile model and Abelian stochastic directed sandpile model, the numerical values of critical exponents are consistent completely with the exact solutions, which verifies that the moment method is rather reliable and accurate. 3) The Abelian quenched random directed sandpile model with the local flow balance has the same critical exponents with Abelian deterministic directed sandpile model, and the Abelian quenched random directed sandpile model without the local flow balance has the same critical exponents with Abelian stichastic directed sandpile model. These results show that the presence or absence of the local flow balance determines the universality class of the Abelian directed sandpile model. 4) The corresponding critical exponents differ for Abelian stochastic directed sandpile model and non-Abelian stochastic directed sandpile model, which shows that the Abelian symmetry breaking changes universality class for the avalanches. This result is contrary with recently performed investigations in [68]. 5) The corresponding critical exponents differ for Abelian deterministic directed sandpile model and non-Abelian deterministic directed sandpile model, and there exist the long-range spatial correlations within the metastable states in the non-Abelian model. These results show that the Abelian symmetry breaking change both the universality classes for both avalanches and spatial structure within the metastable. 6)The physical origin of the different critical behavior has to be ascribed to the presence of the different avalanche structure in models.We think the self-organized criticality must connect with complex networks. Since the sandpile avalanche dynamics on undirected small-world network has been studied, so that we studied the sandpile avalanche dynamics on directed small-world network and compared it with the corresponding dynamics on undirected small-world network. The results show that: 1) The avalanche size and duration distribution follow a power law for all rewiring probability p . 2) For each system size, there exists a very small critical probability pc (L). For p < pc(L), the values of critical exponents are sensitive to p , however, for p≥pc(L) they do not depend on p and close to mean-field solution in Euclidean space. 3) The critical probability pc (L) decrease with increasing system size. By the scaling analysis, we find p c(∞)→0 if L→∞. 4) Under the condition of L→∞, for all p >0, the scaling behavior of the system is consistent with the mean-field prediction, which indicate that the direction changes the self-organized critical behavior on small-world network.We also studied the structure of complex transport networks which must depend on the structure of the basic networks and the dynamical factor. So, we introduce a complex transport network model with a dynamical parameterαbased on ranking choice strategy. Varyingαfrom 0 to∞allows a smooth transition between thr case of random diffusion and that of gradient transport. Based on the substrate of random and scale-free networks, we studied the effect the distribution of the scaling field and parameterαon the distribution of degree. The results show that: 1) The in-degree distribution of the transport network does not depend on the distribution of the scaling field. 2) For scale-free network, the in-degree distribution is a power-law function, the power exponentγdoes not dependαand consistent with the power exponents of the substrate. 3) For random network, the in-degree distribution is a power-law function for allα>0, but the power exponentγdoes dependα. By the scaling anasisly, we obtain the relationγ=1+1/a, which indicates that complex transport network model based on ranking preferential choice can show the power-law degree distribution with all power exponents.We also studied the function of complex transport networks which must also depend on the structure of the basic networks and the dynamical factor. So, we introduce a weighted complex transport network model with a dynamical parameterαbased on the distribution of potential difference, where the parameterαcharacterizing the effect of local jamming. Based on the substrate of random and scale-free networks, we studied the effect of the average connectivity k of the network and parameterαon jamming. In addition, the jamming effect of the two networks is compared. The results show that: 1) k andαplay an important pole for the jamming. It is found that for both two networks, for sufficiently large k , above a critical valueαc, the value of J increases with k , while forα<αc the opposite occurs. The value ofαcdepend on the network structure. 2) By comparing the jamming effect of the two networks, we found a more complicated crossover phenomenon, which is described by two parametersαand ?k ?. The phenomenon shows that scale-free networks have a higher level of jamming than random networks for a very wide region ofαand ?k ? value, and only when ?k ? andαare both sufficiently large, scale-free networks have a lower level of jamming.