Mechanisms for Network Growth that Preserve Spectral and Local Structure

We introduce a method that can be used to evolve the topology of a network in a way that preserves both the network's spectral as well as local structure. This method is quite versatile in the sense that it can be used to evolve a network's topology over any collection of the network's elements. This evolution preserves both the eigenvector centrality of these elements as well as the eigenvalues of the original network. Although this method is introduced as a tool to model network growth, we show it can also be used to compare the topology of different networks where two networks are considered similar if their evolved topologies are the same. Because this method preserves the spectral structure of a network, which is related to the network's dynamics, it can also be used to study the interplay of network growth and function. We show that if a network's dynamics is intrinsically stable, which is a stronger version of the standard notion of stability, then the network remains intrinsically stable as the network's topology evolves. This is of interest since the growth of a network can have a destabilizing effect on the network's dynamics, in general. In this sense the methods developed here can be used as a tool for designing mechanisms of network growth that ensure a network remains stabile as it grows.