[Paper Review] On the Equivalence between Node Embeddings and Structural Graph Representations

Balasubramaniam Srinivasan, Bruno Ribeiro|arXiv (Cornell University)|Oct 1, 2019

Advanced Graph Neural NetworksComputer Science65 references16 citations

TL;DR

This paper establishes a theoretical equivalence between node embeddings and structural graph representations using invariant theory, showing they are interconvertible and functionally equivalent for all downstream tasks. It resolves long-standing confusion by proving transductive/inductive learning is independent of representation type and offers improved guidelines for generating and using node embeddings.

ABSTRACT

This work provides the first unifying theoretical framework for node embeddings and structural graph representations, bridging methods like matrix factorization and graph neural networks. Using invariant theory, we show that the relationship between structural representations and node embeddings is analogous to that of a distribution and its samples. We prove that all tasks that can be performed by node embeddings can also be performed by structural representations and vice-versa. We also show that the concept of transductive and inductive learning is unrelated to node embeddings and graph representations, clearing another source of confusion in the literature. Finally, we introduce new practical guidelines to generating and using node embeddings, which fixes significant shortcomings of standard operating procedures used today.

Motivation & Objective

• To unify the theoretical understanding of node embeddings and structural graph representations, which are often treated as separate paradigms.
• To resolve confusion in the literature regarding the relationship between transductive and inductive learning and the choice of representation type.
• To provide a formal framework that demonstrates the functional equivalence of node embeddings and structural representations across all tasks.
• To identify and correct significant shortcomings in current standard operating procedures for generating and using node embeddings.

Proposed method

• Applying invariant theory to formalize the relationship between structural representations and node embeddings as analogous to a distribution and its samples.
• Proving that any task solvable by node embeddings can also be solved by structural representations, and vice versa, under the same invariance constraints.
• Formalizing the concept of structural representations as invariant functions over graph neighborhoods, enabling a unified view of representation learning.
• Deriving theoretical conditions under which node embeddings can be derived from structural representations and vice versa, ensuring equivalence in expressiveness.
• Introducing a new framework that decouples representation type from learning paradigm, clarifying that transductive and inductive learning are independent of the choice of representation.
• Proposing practical guidelines for generating and using node embeddings based on the theoretical equivalence, improving robustness and generalization.

Experimental results

Research questions

• RQ1 What is the theoretical relationship between node embeddings and structural graph representations, and can they be considered equivalent in expressive power?
• RQ2 How does the concept of transductive versus inductive learning relate to the choice of node embeddings versus structural representations?
• RQ3 Can all tasks performed by node embeddings also be performed by structural representations, and vice versa, under the same invariance constraints?
• RQ4 What are the fundamental invariants that govern the equivalence between different types of graph representations?
• RQ5 How can current standard procedures for generating node embeddings be improved using the theoretical insights from this work?

Key findings

• Node embeddings and structural graph representations are theoretically equivalent in their expressive power, meaning any task solvable by one can be solved by the other.
• The relationship between structural representations and node embeddings mirrors that of a probability distribution and its samples, providing a new theoretical lens for understanding representation learning.
• Transductive and inductive learning are independent of the representation type, resolving a persistent source of confusion in the literature.
• The proposed framework enables the derivation of node embeddings from structural representations while preserving task performance, validating the equivalence.
• The new practical guidelines for generating node embeddings significantly improve robustness and generalization by aligning with the theoretical invariance principles.