  The pentagram map that associates to a projective polygon a new one formed by
intersections of short diagonals was introduced by R. Schwartz and was shown to
be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. Recently, M.
Glick demonstrated that the pentagram map can be put into the framework of the
theory of cluster algebras.
  In this paper, we extend and generalize Glick's work by including the
pentagram map into a family of discrete completely integrable systems. Our main
tool is Poisson geometry of weighted directed networks on surfaces developed by
M. Gekhtman, M. Shapiro, and A. Vainshtein. The ingredients necessary for
complete integrability -- invariant Poisson brackets, integrals of motion in
involution, Lax representation -- are recovered from combinatorics of the
networks. Our integrable systems depend on one discrete parameter $k>1$. The
case $k=3$ corresponds to the pentagram map. For $k>3$, we give our integrable
systems a geometric interpretation as pentagram-like maps involving deeper
diagonals. If $k=2$ and the ground field is $\C$, we give a geometric
interpretation in terms of circle patterns.
