  The pentagram map was introduced by R. Schwartz more than 20 years ago. In
2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville
complete integrability of this discrete dynamical system. In 2011, M. Glick
interpreted the pentagram map as a sequence of cluster transformations
associated with a special quiver. Using compatible Poisson structures in
cluster algebras and Poisson geometry of directed networks on surfaces, we
generalize Glick's construction to include the pentagram map into a family of
discrete integrable maps and we give these maps geometric interpretations. This
paper expands on our research announcement arXiv:1110.0472
