Analysis of coupling complexity in echo state networks via ordinal persistent homology

We study coupling complexity in multivariate time series generated by echo state networks subject to i.i.d. input signals using the ordinal persistent index as a coupling complexity measure. Coupling complexity is a notion of complexity focusing on the relations among components of a given system. Given a time segment of a multivariate time series, its ordinal persistent index is defined by taking the persistent homology of a filtered simplicial complex reflecting similarity among the ordinal patterns of individual time series. As the strength of input signals increases, the dynamics of echo state networks shift from asynchronous ones to more synchronized ones. We show that the original ordinal persistent index cannot capture such change in the synchronization behavior, but a generalized version of the ordinal persistent index is sensitive to the change: the latter takes relatively high values between the two extremes, namely when the strength of input signals to the echo state networks is within a certain range of intermediate values.