Cellular materials possess size-dependent mechanical behavior. This work examines the causes of these size effects in periodic thin-walled cellular materials, modeled as a network of beam elements. The literature has attributed stiffening size effects to the local beam bending behavior of these materials; there is an additional stiffness connected to the rotation of the beam nodes at lattice vertices that causes a stiffening size effect. This work decomposes the strain energy of the beams in the lattice into energy from axial stretching and beam bending and shows that size effects are connected to bending strain energy in certain situations. Other literature has shown a softening size effect in stochastic foams caused by damaged or incomplete cells on free surfaces. This work uses a strain map of the lattice model along with the micropolar constitutive law to show that these edge softening effects can appear in periodic cellular materials when surface cells are neither damaged nor incomplete. This edge softening effect is due to the truncation of repeated unit cells found in the interior of the body. This effect is only observed for certain lattice topologies and is quantified and connected to a global size effect. In conjunction with the beam bending size effect, this edge effect is able to explain the origin of size effects for a variety of lattice topologies and boundary conditions.