Properties And Applications Of Self-Similar Complex Networks

This dissertation studies the spectral properties of several special but important self-similar networks and their applications on spanning trees, random walk or trapping. Extensive studies indicate that eigenvalue spectra associated with a network are closely related to its topological characteristics, and spanning trees, random walk, as well as trapping have a great variety of applications in many domains.First we study the trapping problem in the Vicsek fractals, an important class of regular fractals. We first consider the case where a perfect trap is fixed at the central node, and derive the exact closed-form formula for the average trapping time (ATT). Then we assume that the trap is distributed uniformly over all nodes. For this case, based on the known connections between first-passage time, effective resistance, and eigenvalues of Laplacian matrices, we obtain the exact solution of the global mean first-passage time (GMFPT). Our result shows that GMFPT grows superlinearly with the network size.Second we study the Laplacian spectra of Koch networks with scale-free and small-world properties. We calculate the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues, according to the relations between the Laplacian characteristic polynomials of Koch networks and their subgraphs at different iterations. Using these results, we obtain the number of spanning trees, Kirchhoff index, global mean first-passage time and average path length of Koch networks.Third we study the trapping problem on a family of treelike regular fractals with a trap fixed on a central node. We first obtain all eigenvalues and their corresponding multiplicities for a matrix governing the trapping process, with eigenvalues being provided through an explicit recursive relation. Based on these results, we evaluate the approximate value of the smallest eigenvalue and the ATT. Then we generalize the previous undirected fractals by introducing the directed edge weights dominated by a parameter. Using a similar method, we obtain the expressions for the smallest eigenvalue and the ATT. Result indicates that ATT is controlled by the weight parameter by modifying which ATT can scale superlinearly, linearly, or sublinearly with the system size.Finally we determine the number of spanning trees and entropy of Apollonian networks using a recursive enumeration of subgraphs. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-world, scale-free, Euclidean and space filling, modular and highly clustered, the study of their spanning trees is of particular relevance. This study can also provide some clues which may help us find a way to explicitly determine the Laplacian eigenvalues of Apollonian networks in the future.