Research On The Modeling And Vulnerability Of Fractal Complex Networks For Brain Network Analysis

The brain networks not only have obvious small-world characteristics,but also have fractal,high modularity and extremely low vulnerability.The traditional small-world models did not have a highly global modular and fractal structure,which indicated that the small-world features were not sufficient to describe the topological features of the brain networks.And the relationship between fractal characteristics and network vulnerability has not yet been clear.In response to these problems,previous research believed that the addition of long-distance and long-range edges made small-world features appear in fractal networks with large-world features.At the same time,most of the network vulnerability was analyzed based on the degree exponent.This article mainly drawed on the emergence characteristics of complex systems,that was,changes of local feature made the system emerge with some non-superimposed overall characteristics,and analyzed the change law of the overall characteristics of complex networks from the perspective of local connections.This subject studied fractal complex network modeling and vulnerability based on the structural characteristics of nodes in a local area.The main research content was the coexistence of fractal and small-world features and the impact of fractal on network vulnerability.Based on the above theoretical research results,the complex brain networks generated by functional magnetic resonance imaging experiments of normal subjects under dual-task conditions were used as an example to understand the small world,fractal,and vulnerability characteristics of the brain network.The main work of the dissertation is as follows:(1)In order to accurately calculate the fractal dimension,a maximum-minimum ant colony box counting algorithm for calculating the minimum number of boxes was proposed.Three types of heuristic information were redefined according to the characteristics of complex network structure,and the pheromone accumulation method and the maximum-minimum limit range were determined.The calculation results under different heuristic rules and pheromone evaporation rates were compared.And the comparison with the classic combustion algorithm illustrated the advantages of this algorithm in calculation results and running time.The proposed algorithm provided a data basis for the subsequent fractal feature analysis.(2)Aiming at the problem of the coexistence mechanism of fractal and small world in large networks,based on the Box-based Preferential Attachment(BPA)and Inverse Renormalization Procedure(IRP)principles,a randomly growing fractal complex network model was proposed.In the large-world fractal model,by adding short-distance connecting edges inside the box,the overall small-world feature emerged in the model,providing a theoretical framework for the transition and coexistence of fractal and small-world features.Based on these analyses,the fractal characteristics of the complex brain network during the renormalization process and under the coverage with different scales were studied.It was found that the overall box state conformed to the characteristics of the box state when the fractal and the small world coexist in the BPA model.This provided an explanation based on the local perspective for the coexistence of the relative independence of various functions and the global small-world characteristics of the complex brain networks.(3)Aiming at the relationship between the connection mode and vulnerability of the fractal complex network,the connection ratio of the box center nodes in the fractal network was defined.It was found that when the neighbor nodes of the central node were evenly distributed inside and outside the box,the network had better stability.And taking the BPA fractal model as an example,the relationship between the average connection ratio and the fragility of the model was illustrated.It was found that using smaller diameter boxes during the renormalization process of the dual-task complex brain network,the number of boxes in the interval of different connection ratios presented the structural characteristics of normal distribution.At the same time,it was found that the connections pattern of the central nodes of the dual-task complex brain network also showed a uniform distribution.(4)Aiming at the quantitative relationship between fractal dimension and network vulnerability,the prediction of the power function relationship between fractal dimension and network vulnerability was proposed,and the rationality of the prediction was verified through fractal structure analysis and experiments.First,the expression of the power function relationship was given,and the experiment results showed that using two different vulnerability measurement ways to test two different fractal models,the fractal dimension and vulnerability presented power function relationships.The research on the fractal dimension and vulnerability of the dual-task complex brain networks showed that the vulnerability exponent during the renormalization process was much larger than that of the fractal network generated by the BPA model,indicating that the vulnerability of the dual-task complex brain network was more sensitive to the fractal dimension.In summary,based on the study of BPA fractal modeling and the relationship between connection pattern,fractal dimension and the vulnerability,the dual-task complex brain network was analyzed from the perspective of local connections,which has the structural features of fractal and small world coexistence and extremely high stability.It provides new ideas for in-depth understanding of complex brain network structure.