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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.