Time Evolution of the Degrees in a Network Formed by Traces of Random Walks

Traces of random walks form a mathematical model in which complex network structures emerge from random phenomena. By considering the local structure in which a random walker moves, we derive an equation that describes the time evolution of the degrees of the vertices in the network. In terms of the local structure of networks, this equation explains the "rich-get-richer" phenomenon of the time evolution of vertex degrees. The short-cuts created by the random movement of the walker create a highly clustered structure, and this local structure affects the global properties of the network, such as the distribution of vertex degree, by regulating the movements of the random walker.