  Graph hypernetworks (GHNs), constructed by combining graph neural networks
(GNNs) with hypernetworks (HNs), leverage relational data across various
domains such as neural architecture search, molecular property prediction and
federated learning. Despite GNNs and HNs being individually successful, we show
that GHNs present problems compromising their performance, such as
over-smoothing and heterophily. Moreover, we cannot apply GHNs directly to
personalized federated learning (PFL) scenarios, where a priori client relation
graph may be absent, private, or inaccessible. To mitigate these limitations in
the context of PFL, we propose a novel class of HNs, sheaf hypernetworks
(SHNs), which combine cellular sheaf theory with HNs to improve parameter
sharing for PFL. We thoroughly evaluate SHNs across diverse PFL tasks,
including multi-class classification, traffic and weather forecasting.
Additionally, we provide a methodology for constructing client relation graphs
in scenarios where such graphs are unavailable. We show that SHNs consistently
outperform existing PFL solutions in complex non-IID scenarios. While the
baselines' performance fluctuates depending on the task, SHNs show improvements
of up to 2.7% in accuracy and 5.3% in lower mean squared error over the
best-performing baseline.
