The eccentricity of a node v in a network is the maximum distance from v to any other node. In social networks, the reciprocal of eccentricity is used as a measure of importance of nodes in the network. Our work focuses on networks with prescribed structural properties, such as the number of nodes, the number of edges or the maximum degree. Specifically, we determine the maximum value of eccentricity centralization and (some) optimal networks for the families of bipartite networks with given partition sizes, networks with fixed maximum degree and fixed number of edges, tree networks with fixed maximum degree and fixed number of vertices, and networks with fixed number of nodes or edges. As a by-product, we introduce and study a new way of enumerating the nodes of a tree, which might be of independent interest.