  We investigate the origin of the ubiquitous existence of flat bands in the
network superstructures of atomic chains, where one-dimensional(1D) atomic
chains array periodically. While there can be many ways to connect those
chains, we consider two representative ways of linking them, the dot-type and
triangle-type links. Then, we construct a variety of superstructures, such as
the square, rectangular, and honeycomb network superstructures with dot-type
links and the honeycomb superstructure with triangle-type links. These links
provide the wavefunctions with an opportunity to have destructive interference,
which stabilizes the compact localized state(CLS). The CLS is a localized
eigenstate whose amplitudes are finite only inside a finite region and
guarantees the existence of a flat band. In the network superstructures, there
exist multiple flat bands proportional to the number of atoms of each chain,
and the corresponding eigenenergies can be found from the stability condition
of the compact localized state. Finally, we demonstrate that the finite
bandwidth of the nearly flat bands of the network superstructures arising from
the next-nearest-neighbor hopping processes can be suppressed by increasing the
length of the chains consisting of the superstructures.
