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Recent studies revealed an important interplay between the detailed structureof fibration symmetric circuits and the functionality of biological andnon-biological networks within which they have be identified. The presence ofthese circuits in complex networks are directed related to the phenomenon ofcluster synchronization, which produces patterns of synchronized group ofnodes. Here we present a fast, and memory efficient, algorithm to identifyfibration symmetries over information-processing networks. This algorithm isspecially suitable for large and sparse networks since it has runtime ofcomplexity $O(M\log N)$ and requires $O(M+N)$ of memory resources, where $N$and $M$ are the number of nodes and edges in the network, respectively. Wepropose a modification on the so-called refinement paradigm to identifycircuits symmetrical to information flow (i.e., fibers) by finding the coarsestrefinement partition over the network. Finally, we show that the presentedalgorithm provides an optimal procedure for identifying fibers, overcoming thecurrent approaches used in the literature.

Fast algorithm to identify cluster synchrony through fibration symmetries in large information-processing networks