Calculations Of Network Coherence In Deterministic Tree-like Networks Based On Laplacian Spectrum

Complex networks have been proved as a powerful tool to describe a lot of complex systems in reality,which involves with the topological structures and dynamics among the interactions of elements.In the recent decade,the research of complex networks has attracted increasing attention from the field of science,such as computer,physics,mathematics etc.The main issues of complex networks are modeling,dynamics and its applications.The network modeling is the initial topic,e.g.,the small-world and scale-free networks.During the network modeling,deterministic network is defined in an iterative form.The advantages are that analytically obtaining some properties and verifying the conclusions and providing research ideas for random graphs.Therefore,it has become a hot topic in the complex network science.This dissertation mainly studies the network coherence characterized as Laplacian spectrum and investigates the relationship between the scaling and fractal dimension.We choose a family of deterministic tree-like networks as our model,calculate its Laplacian spectra and obtain the scalings of network coherence with regard to network size.Finally we compare the scalings with those of existing tree networks.In details,the contents are as follows:Chapter Ⅰ introduces the research background and research status throughout the world,and gives the definitions of deterministic network,fractal network,Laplacian spectrum,and consensus dynamics.In Chapter Ⅱ,we investigate the relationship between the scalings and fractal dimension.We construct a family of recursive trees and calculate the sum and square sum of reciprocal of all nonzero eigenvalues of the considered Laplacian matrix based on the regular topology.We also obtain the scalings of first-and second-order coherence with regard to network size.The obtained results show that the coherence of our network is not relevant to its fractal dimension and the consensus dynamics is better.In Chapter Ⅲ,we study the effect of topology structure on the network coherence.Based on the second chapter,we change the initial state of that model and form another family of recursive trees by introducing a controlled parameter.We then propose to a new method to calculate the expression of the coherence.The results show that these two family of recursive trees have same scalings of network coherence though they have different structures.Finally,we investigate the effect of the network parameter on the coherence and see that the coherence is worse with increasing of parameters.The obtained results are verified by figures and numerical simulations.Chapter Ⅳ summarized the conclusions and gives further studies in the future.