Vectors vs Graphs: How Topology and Geometry Are Transforming Machine Learning

The debate between vector embeddings and graph representations has become increasingly prominent in AI and machine learning circles. We see constant arguments about which is superior, or proposals for hybrid approaches combining both. But I’ve come to an intriguing realization: perhaps we don’t need to choose between them at all. What we actually need are different kinds of embeddings for different purposes, and the key to understanding this lies in topology and geometry.

Beyond Euclidean: The Rise of Geometric Embeddings

The current excitement around topological and geometric embeddings represents a fundamental shift in how we think about representing knowledge. Instead of forcing everything into flat vector spaces, we’re learning to embed information in geometric structures that naturally capture the relationships we care about:

Multidimensional boxes for bounded, orthogonal features

Multidimensional spheres for normalized, directional relationships

Hyperbolic spaces for hierarchical and nested structures

That last one is particularly fascinating. Hyperbolic embeddings excel at capturing complex hierarchical relationships that are awkward or inefficient to represent in standard Euclidean vector spaces. The exponential growth of hyperbolic space perfectly matches the branching structure of taxonomies, organizational charts, and knowledge hierarchies.

This isn’t just theoretical mathematics — geometric machine learning has emerged as a substantial field addressing real limitations in classical approaches. We still need embeddings, and vectors remain valuable, but the critical question has shifted from “vectors or graphs?” to “what kind of embedding for what kind of structure?”

Graphs as Topological Structures

Here’s where things get truly interesting: when you work with graphs, you’re already doing topology and geometry, whether you realize it or not. A graph is fundamentally a non-Euclidean space, and this perspective opens up powerful analytical approaches:

Spectral Analysis: You can apply spectral methods to graphs, treating them as operators on vector spaces. The eigenvalues and eigenvectors of graph Laplacians reveal clustering, connectivity, and community structure in ways that local graph traversal cannot.

Simplicial Complexes: Every graph can be viewed as a simplicial complex — a topological structure built from points (0-simplices), lines (1-simplices), triangles (2-simplices), and higher-dimensional analogs. Some graphs have rich higher-order structure with many triangles and tetrahedra; others are sparse, consisting mainly of points and edges.

This simplicial perspective enables topological data analysis (TDA), where we can:

Search for holes and voids in the data structure

Measure connectivity at different scales

Identify persistent topological features that remain stable across perturbations

These topological characteristics reveal properties of your data that purely graph-based or vector-based approaches might miss entirely. The topology matters — it’s not just an abstract mathematical curiosity but a practical lens for understanding structure.

Geometry, Embeddings, and Graph Structure

The relationship between geometry and graph structure is bidirectional and rich with implications:

From Geometry to Graphs: Geometric embeddings can induce graph structures. Points close in hyperbolic space naturally form hierarchical clusters. Embeddings on spheres create graphs with angular relationships. The geometry constrains and suggests the connectivity.

From Graphs to Geometry: Graph structure can be embedded into geometric spaces that preserve important properties. This is the domain of graph embedding techniques like node2vec, but it extends far beyond random walks:

Riemannian geometry for graphs with curvature

Hyperbolic embeddings for trees and hierarchies

Product spaces combining multiple geometric primitives for graphs with mixed structure

The key insight is that different graph topologies have natural geometric homes. A social network might live comfortably in Euclidean space, while an organizational hierarchy demands hyperbolic space, and a knowledge graph might require a product of multiple geometric spaces to capture its diverse relationship types.

Metagraphs and Topological Structure

Now we arrive at an even more sophisticated level: metagraphs and their topological properties. A metagraph extends beyond simple graphs by allowing edges to connect not just nodes, but sets of nodes, or even other edges. This creates a nested, compositional structure that’s extraordinarily expressive.

Metagraphs as Higher-Order Structures: In topological terms, metagraphs naturally encode higher-order relationships. Where a graph gives you 1-simplices (edges) connecting 0-simplices (nodes), a metagraph can represent:

Hyperedges connecting arbitrary sets of nodes

Meta-relationships between relationships

Hierarchical compositions of subgraphs

Topology of Metagraphs: The topological analysis of metagraphs opens fascinating possibilities:

Persistent Homology at Multiple Scales: Because metagraphs have inherent hierarchical structure, we can analyze topology at different levels of abstraction. What looks like a hole at one level might be filled at another.

Category-Theoretic Structures: Metagraphs naturally form categorical structures where morphisms between graphs themselves form higher-level graphs. This creates a tower of increasingly abstract topological spaces.

Sheaf Theory Applications: The nested structure of metagraphs makes them ideal candidates for sheaf-theoretic approaches, where local data on different parts of the graph must be consistently glued together globally.

Practical Implications: For AI and knowledge representation, metagraph topology enables:

Capturing not just “A relates to B” but “this entire pattern relates to that entire pattern”

Representing meta-knowledge about knowledge structures themselves

Modeling how different levels of abstraction interact and constrain each other

Detecting structural patterns that exist only at compositional levels

The topological lens transforms metagraphs from merely “graphs with extra features” into genuine multiscale geometric objects with rich internal structure.

Topology: The Bridge Between Discrete and Continuous

Perhaps topology’s most profound contribution is that it provides a bridge between discrete and continuous representations. In topology, you work with continuous spaces that can undergo continuous transformations, yet these spaces can be built from discrete elements (simplicial complexes) or can discretely sample continuous phenomena.

This duality is exactly what we need in machine learning:

Discrete data points (observations, entities, events)

Continuous structure (relationships, similarity, transformation)

Topology doesn’t force you to choose. It provides the mathematical framework for moving fluidly between discrete graphs and continuous manifolds, between symbolic reasoning and geometric intuition.

The Future: 2026 as the Year of Geometric and Topological Learning

I have a strong intuition that 2025–2026 will mark the emergence of geometric and topological learning as mainstream tools in AI and knowledge representation. The pieces are falling into place:

Mature geometric deep learning frameworks

Accessible topological data analysis tools

Growing understanding of hyperbolic neural networks

Recognition that different structures demand different geometries

The war between vectors and graphs can end — not because one side won, but because we’ve transcended the dichotomy. We can talk about the topology and geometry of information itself:

What is the shape of this knowledge domain?

Does it have holes, boundaries, curvature?

How does it transform under different operations?

What geometric space naturally hosts it?

The combination of graphs, metagraphs, and topology creates an extraordinarily rich framework. Hypergraphs — where edges can connect arbitrary numbers of nodes — are already a major part of topological data learning. When you add metagraph structure on top, you get compositional, hierarchical, topologically analyzable knowledge structures that can finally capture the true complexity of real-world information.

Conclusion

The question was never really “vectors or graphs?” It was always “what geometric and topological structure best captures this information?”

Topology provides the unifying framework we’ve been searching for — a way to think about discrete and continuous, local and global, structure and transformation all within a single coherent mathematical language. As we move forward, the most powerful AI systems won’t be purely vector-based or purely graph-based. They’ll be topology-aware systems that fluidly move between geometric representations as the data and task demand.

The future of machine learning is geometric, topological, and beautifully multidimensional.