An Impossibility Theorem for Node Embedding

TL;DR

This paper proves that no node embedding method can satisfy three desirable properties simultaneously, highlighting fundamental limitations in graph representation learning, and proposes relaxed axioms to enable feasible embeddings.

Contribution

It establishes an impossibility theorem for node embedding axioms and introduces relaxed conditions to overcome these fundamental limitations.

Findings

No embedding method can satisfy all three axioms simultaneously.

Relaxing axioms allows for feasible node embeddings.

Highlights inherent challenges in graph representation learning.

Abstract

With the increasing popularity of graph-based methods for dimensionality reduction and representation learning, node embedding functions have become important objects of study in the literature. In this paper, we take an axiomatic approach to understanding node embedding methods, first stating three properties for embedding dissimilarity networks, then proving that all three cannot be satisfied simultaneously by any node embedding method. Similar to existing results on the impossibility of clustering under certain axiomatic assumptions, this points to fundamental difficulties inherent to node embedding tasks. Once these difficulties are identified, we then relax these axioms to allow for certain node embedding methods to be admissible in our framework.