Can we detect clusters of chaotic dynamical networks via causation entropy?

It is known that chaotic dynamical systems in the coupled networks can synchronize, and they can even form clusters. Our study addresses the issue of determining the membership information of continuous-time dynamical networks forming clusters. We observe the output vectors of individual systems in the networks and reconstruct the state space according to Takens' embedding theorem. Afterward, we estimate the information-theoretic measures in the reconstructed state space. We propose the average integrated causation entropy as a model-free distinctive measure to distinguish the clusters in the network using the k-means clustering algorithm. We have demonstrated the proposed procedure on three networks that contain Chua systems. The results indicate that we can determine the members of clusters and the membership information from the data, conclusively.