Study On Spectra And Related Indexes Of Three Kinds Of Weighted Networks

In recent years,with the further study of complex networks,more and more researchers focus on the study of the properties and network metrics.Inspired by previous work,we study the spectrum and related network metrics of three types of weighted networks in this paper.After consideration,we choose the distribution properties of the weighted evolving network including activity and inactivity,and Leader-follower coherence of the weighted recursive tree networks with an ordered assignment of leaders in the initial state,respectively,as the main network metrics to be studied in this paper.In the first chapter,we briefly introduce the background and current development of research related to complex networks.We also introduce the research on weighted networks,the relevant concepts and definitions that need to be used in the research process.Finally,we briefly explain the main work of this paper.In Chapter 2,combining subdivision-vertex and subdivision-edge neighbourhood corona,we construct the mixed weighted corona model by assigning weights to the edges of the network,and study the spectrum of this network with two different methods.We introduce super mixed corona matrix and calculate the eigenvalues by guessing its eigenvectors,analogously obtain the Laplace spectrum of the mixed weighted corona.Also,we derive the generalized adjacency spectra,Laplacian and signless Laplacian spectra based on the structural features of the network and the definition of eigenvalues and eigenvectors.In Chapter 3,we investigate the distribution properties of the weighted evolving network including activity and inactivity.Inspired by iterative methods and evolving network,we introduce a weighted evolving network using a modified corona product.Typically,the research of weighted evolving network focuses on weight distribution,statistical characteristics and iterative processes.To illustrate the model,we introduce inactive nodes that were created in the previous iteration and will stop iterating.Inactive nodes can be used to refer to people who stop spreading rumours after hearing them.We also distinguish between the active and inactive meanings of the edges,degrees of the network.By definition,the number of new nodes generated at each step depends on the active degree of active nodes.Then we derive analytically relevant properties of this weighted evolving network,including the degree distribution and the weight distribution.In Chapter 4,we study the leader-follower network coherence(i.e.The mean steady-state variance of the deviation from the static value of the leader nodes)in the weighted recursive tree networks with assigning the leaders orderly in the initial state.For the weighted recursive tree networks model,we study the characteristic polynomial of the Laplacian submatrix,and obtain the relationship for the eigenvalues in two successive generations.The analytical formula of the leader-follower network coherence related by the sum of the reciprocal of all eigenvalues of this submatrix is obtained.The result shows that more number of leaders and the greater weight factor lead to better consensus,respectively.