Spectrum Calculations Of Multi-Layer Networks And Applications In Network Coherence

The spectrum of a network plays an important role in the network structures and dynamics,which describes the topology and the processes of the studied network.Presently network coherence quantified by the spectrum in single-layer networks has attracted increasing attention because it characterizes robustness of the network topology to the noise.Due to the complexity of multi-layer networks,calculating the spectrum of multi-layer networks faces a technical challenge.In this dissertation,the new methods for calculating the spectrum of multi-layer networks are proposed and are applied to the network coherence.Based on the results of the spectrum,we study the influence of the weights and number of layers on the coherence and investigate the relationship between the leader-follower coherence and distributions of the leaders.The detailed works are as follows:1.The relationship between coherence and weighted factors for a family of duplex weighted networks is studied.The analytical expressions of the coherence with respect to weight factor and network size are obtained by using the knowledge of algebraic matrix.The results show that under the fixed topology of the network,the smaller the network size and the larger the weight factor,the better the coherence of the network.In addition,the higher the connected edges between duplex networks,the better the coherence.2.Three multi-layer network models generated by star network,koch network and neighbor network are constructed,and the influence of multi-layer networks connection mode on coherence is studied.By using an algebraic graph theory method,the exact results of the spectrum of multi-layer networks are obtained,and the expression of coherence are given.In the case of fixed network size,the multi-layer network generated by the neighbor network has the worst coherence,and the number of layers of the multi-layer network also affects its coherence.3.The role of leader distribution on leader-follower coherence in multi-layer networks is explored.Multi-layer networks with different numbers of nodes are constructed,and leader distributions with various combinations are selected to obtain an expression for leader-follower coherence with respect to the number of leaders.Leader-follower coherence in multi-layer networks is closely related to the distribution of leaders,with optimal coherence depending on the number of leaders per node at each layer.