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In this paper, a generalized high-dimensional Kuramoto model with directed graphs is investigated. A necessary and sufficient condition for equilibria is given and the synchronization is proved under weaker directed graph conditions and more general initial state constrains. Finally, an example is given to validate the theoretical results.doi:10.21203/rs.3.rs-394459/v1 fatcat:z4nq4direvb3jcdbyuugy2azba