Topological Data Analysis In Complex Networks

Complex network is an effective tool to abstract complex systems in reality.Abstracting individuals in a complex system into nodes,and abstracting a certain relationship between individuals into connections,a complex network model is established.Common statistical indicators in complex network include degree and degree distribution,average path length,agglomeration coefficient and betweenness,etc.Traditional network research is based on the idea of graph theory,but nowadays,looking at complex networks from the perspective of algebraic topology has become a new trend in network science.This thesis innovatively designs an algorithm for finding high-order topological structure in complex networks.A clique is a topological structure with fully connected nodes in the network.Higher-order cliques can be derived from lower-order cliques.A series of vector spaces are generated respectively based on the clique,and then the boundary operators are formed according to the boundary relations between the groups of adjacent orders to establish the association of the vector spaces.As a result,concepts in algebraic topology can be introduced into complex networks.According to the Betty number formula and the nature of the topological structure,a 0-1 integer programming model is established,and each solution corresponds to a high-order topological structure.This thesis uses the latest connection data(2020 version)of Caenorhabditis elegans(C.elegans)neurons for empirical analysis.C.elegans has a simple nervous system and is an important model organism for studying information transmission and behavior control in the biological nervous system.Searching for and studying the topological structure of the nervous system can help reveal the behavioral control of organisms and how errors in neural connections can lead to diseases.The neurons in the C.elegans nervous system are abstracted as network nodes(0-clique),and chemical connections and electrical connections are abstracted as undirected and unpowered edges(1-clique)to construct the C.elegans pharyngeal neural network and the nonpharyngeal neural network of hermaphrodite C.elegans and male C.elegans.This thesis conduct statistical analysis of topological data and compare the statistical features in the neural network and the special high-order topological structure.