Graph homophily has been considered an essential property for message-passing neural networks (MPNN) in node classification. Recent findings suggest that performance is more closely tied to the consistency of neighborhood class distributions. We demonstrate that the MPNN performance depends on the number of components of the overall neighborhood distribution within a class. By breaking down the classes into their neighborhood distribution components, we increase measures of neighborhood distribution informativeness but do not observe an improvement in MPNN performance. We propose a Gumbel-Softmax-based rewiring method that reduces deviations in neighborhood distributions. Our results show that our new method enhances neighborhood informativeness, handles long-range dependencies, mitigates oversquashing, and increases the classification performance of the MPNN. The code is available at https://github.com/Bobowner/Gumbel-Softmax-MPNN.

Graphs with heterophily have been regarded as challenging scenarios for Graph Neural Networks (GNNs), where nodes are connected with dissimilar neighbors through various patterns. In this paper, we present theoretical understandings of the impacts of different heterophily patterns for GNNs by incorporating the graph convolution (GC) operations into fully connected networks via the proposed Heterophilous Stochastic Block Models (HSBM), a general random graph model that can accommodate diverse heterophily patterns. Firstly, we show that by applying a GC operation, the separability gains are determined by two factors, i.e., the Euclidean distance of the neighborhood distributions and $\sqrt{\mathbb{E}\left[\operatorname{deg}\right]}$, where $\mathbb{E}\left[\operatorname{deg}\right]$ is the averaged node degree. It reveals that the impact of heterophily on classification needs to be evaluated alongside the averaged node degree. Secondly, we show that the topological noise has a detrimental impact on separability, which is equivalent to degrading $\mathbb{E}\left[\operatorname{deg}\right]$. Finally, when applying multiple GC operations, we show that the separability gains are determined by the normalized distance of the $l$-powered neighborhood distributions. It indicates that the nodes still possess separability as $l$ goes to infinity in a wide range of regimes. Extensive experiments on both synthetic and real-world data verify the effectiveness of our theory.

Graph neural networks (GNNs) have shown great prowess in learning representations suitable for numerous graph-based machine learning tasks. When applied to semi-supervised node classification, GNNs are widely believed to work well due to the homophily assumption (``like attracts like''), and fail to generalize to heterophilous graphs where dissimilar nodes connect. Recent works design new architectures to overcome such heterophily-related limitations, citing poor baseline performance and new architecture improvements on a few heterophilous graph benchmark datasets as evidence for this notion. In our experiments, we empirically find that standard graph convolutional networks (GCNs) can actually achieve better performance than such carefully designed methods on some commonly used heterophilous graphs. This motivates us to reconsider whether homophily is truly necessary for good GNN performance. We find that this claim is not quite true, and in fact, GCNs can achieve strong performance on heterophilous graphs under certain conditions. Our work carefully characterizes these conditions, and provides supporting theoretical understanding and empirical observations. Finally, we examine existing heterophilous graphs benchmarks and reconcile how the GCN (under)performs on them based on this understanding.

A fundamental question in Computer Science is understanding when a specific class of problems go from being computationally easy to hard. Because of its generality and applications, the problem of Boolean Satisfiability (aka SAT) is often used as a vehicle for investigating this question. A signal result from these studies is that the hardness of SAT problems exhibits a dramatic easy-to-hard phase transition with respect to the problem constrainedness. Past studies have however focused mostly on SAT instances generated using uniform random distributions, where all constraints are independently generated, and the problem variables are all considered of equal importance. These assumptions are unfortunately not satisfied by most real problems. Our project aims for a deeper understanding of hardness of SAT problems that arise in practice. We study two key questions: (i) How does easy-to-hard transition change with more realistic distributions that capture neighborhood sensitivity and rich-get-richer aspects of real problems and (ii) Can these changes be explained in terms of the network properties (such as node centrality and small-worldness) of the clausal networks of the SAT problems. Our results, based on extensive empirical studies and network analyses, provide important structural and computational insights into realistic SAT problems. Our extensive empirical studies show that SAT instances from realistic distributions do exhibit phase transition, but the transition occurs sooner (at lower values of constrainedness) than the instances from uniform random distribution. We show that this behavior can be explained in terms of their clausal network properties such as eigenvector centrality and small-worldness (measured indirectly in terms of the clustering coefficients and average node distance).