Extending stability through hierarchical clusters in echo state networks

Echo State Networks (ESN) are reservoir networks that satisfy well-established criteria for stability when constructed as feedforward networks. Recent evidence suggests that stability criteria are altered in the presence of reservoir substructures, such as clusters. Understanding how the reservoir architecture affects stability is thus important for the appropriate design of any ESN. To quantitatively determine the influence of the most relevant network parameters, we analyzed the impact of reservoir substructures on stability in hierarchically clustered ESNs, as they allow a smooth transition from highly structured to increasingly homogeneous reservoirs. Previous studies used the largest eigenvalue of the reservoir connectivity matrix (spectral radius) as a predictor for stable network dynamics. Here, we evaluate the impact of clusters, hierarchy and intercluster connectivity on the predictive power of the spectral radius for stability. Both hierarchy and low relative cluster sizes extend the range of spectral radius values, leading to stable networks, while increasing intercluster connectivity decreased maximal spectral radius.