Directed Network as a Chaotic Piece-Wise Linear One-Dimensional Map and Its Large-Deviation Properties

Abstract

A directed network such as the WWW can be represented by a transition matrix. Comparing this matrix to a Frobenius-Perron matrix of a chaotic piecewise-linear one-dimensional map whose domain can be divided into Markov subintervals, we are able to relate the network structure itself to chaotic dynamics. Just like various large-deviation properties of local expansion rates (finite-time Lyapunov exponents) related to chaotic dynamics, we can also discuss those properties of network structure. Recurrence time statistics and their relationships to double time correlation functions and to power spectra are also considered.