Message passing is the core operation in graph neural networks, where each node updates its embeddings by aggregating information from its neighbors. However, in deep architectures, this process often leads to diminished expressiveness. A popular solution is the use of residual connections, where the input from the current (or initial) layer is added to the aggregated neighbor information to preserve embeddings across layers. Following a recent line of research, we investigate an adaptive residual scheme in which different nodes have varying residual strengths. We prove that this approach prevents oversmoothing; particularly, we show that the Dirichlet energy of the embeddings remains bounded away from zero. This is the first theoretical guarantee not only for the adaptive setting, but also for static residual connections (where residual strengths are shared across nodes) with activation functions. Furthermore, based on an extensive set of experiments, this adaptive approach is shown to outperform the standard and state-of-the-art message passing mechanisms, especially on heterophilic graphs. To improve the time complexity of our approach, we introduce a variant in which residual strengths are not learned but instead set heuristically, a choice that performs as well as the learnable version.

We study how the stubbornness of social network users influences opinion polarization and disagreement. Our work is in the context of the popular Friedkin-Johnson opinion formation model, where users update their opinion as a function of the opinion of their connections and their own innate opinion. Stubbornness then is formulated in terms of the stress a user puts on its innate opinion.