Guanyu Cui

Guanyu Cui, Yuhe Guo, Zhewei Wei et al.

Message-passing graph neural networks (MPGNNs) are commonly compared with the Weisfeiler-Lehman (WL) color-refinement procedure, but this comparison does not quantify the resource parameters a network needs to realize color refinement with bounded-size messages and finite numerical precision. We study the cost of simulating a single color-refinement step on unattributed graphs. We distinguish input-independent, or oblivious, simulation from instance-dependent simulation. In the former, the parameters, or their distributions in randomized models, are fixed before the input instance is known. Our results show that the local form of WL color refinement hides a global relabeling problem. In the oblivious setting, deterministic and zero-error randomized MPGNNs cannot solve this problem in the worst case using only shallow networks with small messages. We complement this lower bound with a nearly matching construction in a stronger rooted, port-aware model. By contrast, when the color set is large, bounded-error randomness can greatly reduce the cost, and a one-layer MPGNN with messages of logarithmic size and a logarithmic number of random bits suffices. We show that this logarithmic number of random bits is essentially necessary for shallow, small-message simulations. When the color set is small, we still obtain a rooted, port-aware simulation, but this construction requires more layers or larger messages. We also prove that this extra cost is partly unavoidable, as small color sets force a nontrivial trade-off between the number of layers and the message size. Finally, instance-dependent simulation can be much shallower, but the required instance-specific parameters are not necessarily easy to find. Together, these results reveal quantitative structure hidden behind the statement that MPGNNs match WL color refinement.

Guanyu Cui, Zhewei Wei, Kun He

Many works make the eye-catching claim that Transformers are Turing-complete. However, the literature often conflates two distinct settings: (i) a fixed Transformer system setting, in which a fixed autoregressive Transformer is coupled with a fixed context-management method to process inputs of different lengths step by step, and (ii) a scaling-family setting, in which a family of different models (with increasing context-window length or numerical precision) is used to handle different input lengths. Existing proofs of Transformer Turing-completeness are frequently established in setting (ii), whereas real-world LLM deployment and the standard notion of Turing-completeness correspond more naturally to setting (i). In this paper, we first formalize the fixed-system setting, thereby providing a concrete characterization of how real-world LLMs operate. We then argue that results proved in the scaling-family setting provide theoretically meaningful resource bounds but do not establish Turing-completeness, thereby clarifying a common misinterpretation of existing results. Finally, we show that different context-management methods can yield sharply different computational power, and we advocate the position that context management is a central component that critically determines the computational power of real-world autoregressive Transformers.

Guanyu Cui, Zhewei Wei

Graph Neural Networks (GNNs) have emerged as a prominent research topic in the field of machine learning. Existing GNN models are commonly categorized into two types: spectral GNNs, which are designed based on polynomial graph filters, and spatial GNNs, which utilize a message-passing scheme as the foundation of the model. For the expressive power and universality of spectral GNNs, a natural approach is to improve the design of basis functions for better approximation ability. As for spatial GNNs, models like Graph Isomorphism Networks (GIN) analyze their expressive power based on Graph Isomorphism Tests. Recently, there have been attempts to establish connections between spatial GNNs and geometric concepts like curvature and cellular sheaves, as well as physical phenomena like oscillators. However, despite the recent progress, there is still a lack of comprehensive analysis regarding the universality of spatial GNNs from the perspectives of geometry and physics. In this paper, we propose MetricGNN (MGNN), a spatial GNN model inspired by the congruent-insensitivity property of classifiers in the classification phase of GNNs. We demonstrate that a GNN model is universal in the spatial domain if it can generate embedding matrices that are congruent to any given embedding matrix. This property is closely related to the Distance Geometry Problem (DGP). Since DGP is an NP-Hard combinatorial optimization problem, we propose optimizing an energy function derived from spring networks and the Multi-Dimensional Scaling (MDS) problem. This approach also allows our model to handle both homophilic and heterophilic graphs. Finally, we propose employing the iteration method to optimize our energy function. We extensively evaluate the effectiveness of our model through experiments conducted on both synthetic and real-world datasets. Our code is available at: https://github.com/GuanyuCui/MGNN.