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We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.

Non-local network dynamics via fractional graph Laplacians