  In this paper, we use the network solution of the $A_r$ $T$-system to derive
that of the unrestricted $A_\infty$ $T$-system, equivalent to the octahedron
relation. We then present a method for implementing various boundary conditions
on this system, which consists of picking initial data with suitable
symmetries. The corresponding restricted $T$-systems are solved exactly in
terms of networks. This gives a simple explanation for phenomena such as the
Zamolodchikov periodicity property for $T$-systems (corresponding to the case
$A_\ell\times A_r$) and a combinatorial interpretation for the positive Laurent
property of the variables of the associated cluster algebra. We also explain
the relation between the $T$-system wrapped on a torus and the higher pentagram
maps of Gekhtman et al.
