This paper studies delegation in a model of discrete choice. In the delegation problem, an uninformed principal must consult an informed agent to make a decision. Both the agent and principal have preferences over the decided-upon action which vary based on the state of the world, and which may not be aligned. The principal may commit to a mechanism, which maps reports of the agent to actions. When this mechanism is deterministic, it can take the form of a menu of actions, from which the agent simply chooses upon observing the state. In this case, the principal is said to have delegated the choice of action to the agent.

We propose a methodology to carry out vertex-frequency analyses of graph signals, with the goal of unveiling the signal's frequency occupancy over a localized region in the network. To this end, we first introduce localized graph signals in the vertex domain, by defining windows that are localized around each node by construction. Recent directed graph Fourier transform (DGFT) advances facilitate the frequency analysis of said localized signals, to reveal the signal's energy distribution in a way akin to a spectrogram in the vertex-frequency plane. We then learn a set of windows by applying gradient descent method to an optimization problem governed by penalty parameters in the spectral domain. We also argue about the tradeoff between the resolution in the vertex and frequency domains based on the said parameters. We evaluate the performance of the proposed windowed GFT approach through numerical experiments on synthetic and real-world graphs.

We address the problem of constructing a graph Fourier transform (GFT) for both undirected and directed graphs (digraphs), which decomposes graph signals into different modes of variation with respect to the underlying network. Accordingly, we seek orthonormal bases that yield maximally-spread frequency components in the graph spectral domain to better capture low, medium and high frequencies. To that end, we advocate a two-step design whereby we: (i) find the maximum directed variation (i.e., frequency on a digraph) a candidate basis vector can attain; and (ii) minimize a smooth spectral dispersion function over the achievable frequency range to obtain the desired spread GFT basis. Both steps involve non-convex, orthonormality-constrained optimization problems, which are efficiently tackled via a provably convergent, feasible optimization method on the Stiefel manifold. We illustrate the effectiveness of the novel GFT construction algorithm through numerical tests on synthetic and real-world graphs.