No abstract

No abstract

Recently, topology structures of many social, biological and technological networks have been discovered to display a scale-free property. For a network, a community is a natural division of network nodes into groups in which there are more links between nodes within the groups than to nodes outside of it. Many methods of community finding have been proposed to seek a fast, feasible and reasonable partition algorithm for the whole network nodes. In this chapter, we introduce the topology of the network to evaluate the feasibility and correctness of a community finding algorithm. A relationship between the rough number of communities and the magnitude of the number of hub nodes in the network is given in detail firstly. Then, an algorithm based on Laplace matrix spectral decomposition is proposed and its key technology, threshold selection of Euclidean distance between nodes, is discussed. Based on the scale-free topology of complex network, the evaluation criterion of community finding algorithm including three conditions is obtained. Numerical results show that the algorithm of community finding is an effective one and the evaluation criterion is feasible, fast and easy to operate.

Many of the topological and dynamical properties of a network are related to its Laplacian spectrum; these properties include network diameter, Kirchho index, and mean first-passage time. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. We choose a family of uniform recursive trees as our model, and propose a method to calculate the first-and second-order network coherence. Using the tree structures, we identify a relationship between the Laplacian matrix and Laplacian eigenvalues. We then derive the exact solutions for the reciprocals and square reciprocals of all nonzero Laplacian eigenvalues. We also obtain the scalings of network coherence with network size. The scalings of network coherence of the studied trees are smaller than those of Vicsek fractals and are not related to its fractal dimension.

No abstract