What graph neural networks cannot learn: depth vs width (1907.03199v2)

Abstract: This paper studies the expressive power of graph neural networks falling within the message-passing framework (GNNmp). Two results are presented. First, GNNmp are shown to be Turing universal under sufficient conditions on their depth, width, node attributes, and layer expressiveness. Second, it is discovered that GNNmp can lose a significant portion of their power when their depth and width is restricted. The proposed impossibility statements stem from a new technique that enables the repurposing of seminal results from distributed computing and leads to lower bounds for an array of decision, optimization, and estimation problems involving graphs. Strikingly, several of these problems are deemed impossible unless the product of a GNNmp's depth and width exceeds a polynomial of the graph size; this dependence remains significant even for tasks that appear simple or when considering approximation.

• The paper demonstrates that GNNs can achieve Turing completeness when they have sufficient depth, width, and unique node attributes.
• The paper reveals that limiting the depth-width capacity significantly restricts a GNN’s ability to solve complex graph-theoretic problems.
• The study provides actionable insights for optimizing GNN design by highlighting critical trade-offs in model parameters for practical graph tasks.

Understanding the Expressive Power and Limitations of Graph Neural Networks

The paper, authored by Andreas Loukas, presents an in-depth examination of the expressive capacity and inherent limitations of Graph Neural Networks (GNNs) within the message-passing framework. The paper furnishes two pivotal insights into the capabilities of GNNs, exploring their theoretical boundaries and shedding light on crucial implications for both practical applications and future theoretical advancements in machine learning involving graphs.

Core Contributions

1. Turing Completeness of GNNs: The paper asserts that GNNs can achieve Turing universality under certain stringent conditions involving depth, width, node attributes, and layer expressiveness. Specifically, it posits that if these networks are sufficiently deep and wide, and nodes possess unique identifiers allowing them to distinguish themselves from each other, then a GNN can compute any function that a Turing machine could, given these criteria fulfill the requisites of Turing completeness.
2. Limitations Under Restricted Configurations: The paper advances a novel perspective by establishing that GNNs lose a substantial part of their expressive power when their depth and width are constrained. A key takeaway is the correlation between the product of a GNN’s depth and width, which the paper terms 'capacity', and the solvability of a variety of graph-theoretic problems. The research exemplifies that solving certain tasks is infeasible unless this capacity exceeds a polynomial threshold relative to the graph size.

Implications and Theoretical Insights

The implications of these findings are multifaceted:

• Guidance for GNN Design: Recognizing the conditions under which GNNs can exhibit Turing completeness provides important insights for the design and tuning of these models. This includes understanding the necessary depth and width to ensure GNNs can facilitate sophisticated computations without losing expressive power.
• Ganularity in Task Resolution: Highlighting problems that GNNs cannot resolve with restricted capacity is invaluable. It suggests focusing computational resources and model design efforts on instances where GNNs can operate efficiently, thereby optimizing their application in areas like graph-based optimization and classification.
• Trade-offs in Model Parameters: The research emphasizes the critical trade-off between depth and width for achieving certain computational tasks, hinting at potential optimization scenarios where increasing depth can compensate for reduced width, or vice versa.
• Benchmark for Future Models: These findings can serve as a benchmark for developing more nuanced GNN architectures and inspire further research into optimizing these networks within their theoretical limits.

Numerical Results and Empirical Context

The paper provides rigorous numerical thresholds and examples of graphs for various decision, optimization, and estimation problems, underscoring the boundaries where GNNs' performance begins to degrade. The correlation between the depth-width capacity and problem solvability is illustrated through specific problems like cycle detection, subgraph verification, and shortest path estimation, with lower bounds succinctly summarized in an accompanying table of results.

Future Directions

This exploration sets a foundational understanding of GNN's limitations and potential, paving the way for further exploration into alternative architectures or computational paradigms that may surpass these limits. Additionally, empirical testing and novel hypothesis generation will aid in harnessing the full potential of graph neural networks, particularly for complex and large-scale graph data manipulations.

In conclusion, Andreas Loukas's paper provides a thorough understanding of the expressive abilities and limitations of graph neural networks within the message-passing paradigm, offering insightful guidance for both current applications and future exploration within this domain.