Estimating topological entropy using ordinal partition networks

We propose a computationally simple and efficient network-based method for approximating topological entropy of low-dimensional chaotic systems. This approach relies on the notion of an ordinal partition. The proposed methodology is compared to the three existing techniques based on counting ordinal patterns - all of which derive from collecting statistics about the symbolic itinerary - namely (i) the gradient of the logarithm of the number of observed patterns as a function of the pattern length, (ii) direct application of the definition of topological permutation entropy, and (iii) the outgrowth ratio of patterns of increasing length. In contrast to these alternatives, our method involves the construction of a sequence of complex networks that constitute stochastic approximations of the underlying dynamics on an increasingly finer partition. An ordinal partition network can be computed using any scalar observable generated by multidimensional ergodic systems, provided the measurement function comprises a monotonic transformation if nonlinear. Numerical experiments on an ensemble of systems demonstrate that the logarithm of the spectral radius of the connectivity matrix produces significantly more accurate approximations than existing alternatives - despite practical constraints dictating the selection of low finite values for the pattern length.