  In this paper, we extend the notions of $\lambda$-cent-dian and
generalized-center from Facility Location Theory to the more intricate domain
of Network Design. Our focus is on the task of designing a sub-network within a
given underlying network while adhering to a budget constraint. This
sub-network is intended to efficiently serve a collection of origin/destination
pairs of demand. % rather than individual points.
  The $\lambda$-cent-dian problem studies the balance between efficiency and
equity. We investigate the properties of the $\lambda$-cent-dian and
generalized-center solution networks under the lens of equity, efficiency, and
Pareto-optimality. We provide a mathematical formulation for $\lambda\geq 0$
and discuss the bilevel structure of this problem for $\lambda>1$. Furthermore,
we describe a procedure to obtain a complete parametrization of the
Pareto-optimality set based on solving two mixed integer linear formulations by
introducing the concept of maximum $\lambda$-cent-dian. We evaluate the quality
of the different solution concepts using some inequality measures. Finally, for
$\lambda\in[0,1]$, we study the implementation of a Benders decomposition
method to solve it at scale.
