Higher pentagram maps, weighted directed networks, and cluster dynamics

The pentagram map was extensively studied in a series of papers by V. Ovsienko, R. Schwartz and S. Tabachnikov. It was recently interpreted by M. Glick as a sequenceof cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networkson surfaces, we generalize Glick's constructionto include the pentagram map into a family of geometrically meaningful discrete integrable maps.