A new framework of synchronous parallel processing systems called spatial networks is examined, in which the family of all cellular automata is included perfectly. This framework is free from the two restrictions of cellular automata of which one is the finiteness of the set of states of a cell and the other is the countability of an array space. Throughout this article, the relationships between function and structure of spatial networks are considered. First, the necessary and sufficient condition for spatial networks to be uniformly interconnected is given. That for spatial networks to be finitely interconnected is also given with a topological approach. The characterization theorem of cellular automata comes from these results. Second, it is shown that finitely and uniformly interconnected linear spatial networks can be characterized by the convolution form. Last, the conditions for their global mappings to be injective or surjective are discussed. 10 references.