Research On Degree-based And Eigenvalue-based Entropy Of Directed Complex Networks

Entropy is a method used to study the complexity of complex networks.Entropy was first proposed by German physicist Clausius in 1865,and then introduced into information theory by Shannon in 1948,which generally refers to the measurement of the internal disorder degree of a system.Since then,entropy has been widely studied and applied in various scientific fields.Rashevsky and Trucco introduced entropy into graph theory in 1955,Mowshowitz made a systematic and in-depth study of the graph entropy of undirected graphs.Hence,graph entropy has entered the vision of researchers,And the graph entropy of undirected graph based on different invariants such as such as degree power,distance,independent set and eigenvalue has been widely studied and fruitful results have been obtained.Graph entropy is an important research tool in complex network applications.Some of the real complex networks have obvious directions,such as food chain network and Web link network.Entropy is the discriminant method of network node importance and network similarity.Therefore,it is of great significance and application value to study the entropy and its application of directed networks and reveal the law of entropy between typical networks and real directed networks.Therefore,entropy of directed complex network is studied in this paper.The main research contents and achievements are as follows:1.This thesis studies the degree entropy and its extremum problem based on the in-degree/outdegree of the vertices of a digraph.For directed trees,unicyclic digraphs and tournament graphs,the upper and lower bounds of these digraphs based on degree entropy are obtained,and the construction algorithm of extreme digraphs reaching the upper and lower bounds is given,and the corresponding extreme digraphs are characterized.2.This thesis studies the entropy based on degree,and the real part entropy,imaginary part entropy and module entropy of eigenvalues based on adjacency matrix,Laplace matrix and signless Laplace matrix of directed k-nearest neighbor coupling networks,directed small-world networks,directed random networks and directed scale-free networks in directed networks.Through the calculation of the eigenvalue entropy of four kinds of directed networks and real directed networks based on three different matrices in the simulation experiment,the data show that the eigenvalue entropy based on matrix of the real European mail network is between the eigenvalue entropy based on matrix of the directed random network and the directed scale-free network,or close to the entropy of the directed scale-free network.The out-degree entropy of the real European mail network is between the entropy based on the degree of directed small-world networks and directed scale-free networks,and the in-degree entropy is close to the entropy based on the in-degree of directed scale-free networks.3.This thesis studies the entropy based on the degree,eigenvalues of adjacency matrix,Laplacian matrix and signless Laplacian matrix of directed bipartite networks.The construction algorithms and spectral distributions of directed bipartite nearest-neighbor coupled networks,directed bipartite scalefree networks and directed bipartite random networks are given.The entropy based on the degree and the real part entropy,imaginary part entropy and modulus entropy based on three different matrix eigenvalues for directed bipartite small-world network,directed bipartite scale-free network,directed bipartite random network and directed bipartite k nearest-neighbor coupled network and bipartite network of a real directed film scoring are simulated and calculated.The data show that the entropy based on matrix eigenvalues of real film scoring network is between the entropy based on matrix eigenvalues of the directed bipartite small-world network and the directed scale-free network,or close to the entropy of directed scale-free network.The entropy based on the out-degree of the bipartite network for real directed film scoring is between the entropy based on out-degree of the directed bipartite small-world network and the directed bipartite scale-free network,and the entropy based on in-degree is close to the entropy based on in-degree of the scale-free network.4.This thesis studies the entropy application of directed complex networks based on degree and eigenvalue,gives the application of directed complex network entropy in natural language processing and network representation learning,and proves that the equivalence between sentence and random walk of nodes in natural language processing and representation learning.