🤖 AI Summary
This work addresses the lack of a rigorous mathematical foundation in existing distributed data fabric architectures, which struggle to simultaneously ensure consistency, data provenance, and scalability. We propose the first unified framework integrating category theory, hypergraphs, and geometric analogies from Hurwitz spaces. Data sets, metadata, transformations, and policies are uniformly modeled as hypergraphs, with data treated as objects and transformations as morphisms. Modular tensor categories and braided monoidal structures are introduced to capture relational symmetries. Within this formalism, we rigorously prove the NP-hardness of key tasks and develop a fault-tolerant operational mechanism—leveraging sparse incidence matrices, spectral methods, and symmetry-aware alignment algorithms—that guarantees consistency, completeness, and causality while adhering to the CAP and CAL theorems. This framework provides a scalable mathematical foundation for large-scale federated learning and data integration.
📝 Abstract
Current distributed data fabrics lack a rigorous mathematical foundation, often relying on ad-hoc architectures that struggle with consistency, lineage, and scale. We propose a mathematical framework for data fabrics, unifying heterogeneous data management in distributed systems through a hypergraph-based structure \( \mathcal{F} = (D, M, G, T, P, A) \). Datasets, metadata, transformations, policies, and analytics are modeled over a distributed system \( \Sigma = (N, C) \), with multi-way relationships encoded in a hypergraph \( G = (V, E) \). A categorical approach, with datasets as objects and transformations as morphisms, supports operations like data integration and federated learning. The hypergraph is embedded into a modular tensor category, capturing relational symmetries via braided monoidal structures, with geometric analogies to Hurwitz spaces enriching the algebraic modeling. We prove the NP-hardness of critical tasks, such as schema matching and dynamic partitioning, and propose spectral methods and symmetry-based alignments for scalable solutions. The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.