  We consider chains of generalized submanifolds, as defined by Gualtieri in
the context of generalized complex geometry, and define a boundary operator
that acts on them. This allows us to define generalized cycles and the
corresponding homology theory. Gauge invariance demands that D-brane networks
on flux vacua must wrap these generalized cycles, while deformations of
generalized cycles inside of a certain homology class describe physical
processes such as the dissolution of D-branes in higher-dimensional D-branes
and MMS-like instantonic transitions. We introduce calibrations that identify
the supersymmetric D-brane networks, which minimize their energy inside of the
corresponding homology class of generalized cycles. Such a calibration is
explicitly presented for type II N=1 flux compactifications to four dimensions.
In particular networks of walls and strings in compactifications on warped
Calabi-Yau's are treated, with explicit examples on a toroidal orientifold
vacuum and on the Klebanov-Strassler geometry.
