  The Swift-Hohenberg equation (SHE) is a partial differential equation that
explains how patterns emerge from a spatially homogeneous state. It has been
widely used in the theory of pattern formation. Following a recent study by
Bramburger and Holzer [2], we consider discrete SHE on deterministic and random
graphs. The two families of the discrete models share the same continuum limit
in the form of a nonlocal SHE on a circle. The analysis of the continuous
system, parallel to the analysis of the classical SHE, shows bifurcations of
spatially periodic solutions at critical values of the control parameters.
However, the proximity of the discrete models to the continuum limit does not
guarantee that the same bifurcations take place in the discrete setting in
general, because some of the symmetries of the continuous model do not survive
discretization.
  We use the center manifold reduction and normal forms to obtain precise
information about the number and stability of solutions bifurcating from the
homogeneous state in the discrete models on deterministic and sparse random
graphs. Moreover, we present detailed numerical results for the discrete SHE on
the nearest-neighbor and small-world graphs.
