O'Clery Neave, Yuan Ye, Stan Guy-Bart, Barahona Mauricio
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Oct;88(4):042805. doi: 10.1103/PhysRevE.88.042805. Epub 2013 Oct 11.
Using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterize convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions, and characterize the asymptotic convergence to the synchronization manifold under nonuniform input signals. We also show that the EEP with respect to a node can reveal nodes in the graph with an increased rate of asymptotic convergence to the consensus value, as characterized by the second smallest eigenvalue of the quotient Laplacian. Finally, we show that the quotient graph preserves the observability properties of the full graph and how the inheritance by the quotient graph of particular aspects of the eigenstructure of the full Laplacian underpins the observability and convergence properties of the system.
利用外部公平划分（EEP）与图拉普拉斯算子谱特性之间的内在关系，我们刻画了网络上一致性动力学的收敛性和可观测性特性。特别地，我们建立了在不同初始条件下原始一致性动力学与商图相关一致性之间的关系，并刻画了在非均匀输入信号下到同步流形的渐近收敛性。我们还表明，关于一个节点的EEP可以揭示图中以商拉普拉斯算子第二小特征值为特征的、到一致性值的渐近收敛速率增加的节点。最后，我们表明商图保留了全图的可观测性特性，以及商图如何继承全拉普拉斯算子特征结构的特定方面，从而支撑了系统的可观测性和收敛性特性。
