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A Unified Mathematical Framework for Distributed Data Fabrics: Categorical Hypergraph Models

T. Shaska, I. Kotsireas

TL;DR

The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.

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 \( Σ= (N, C) \), 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.

A Unified Mathematical Framework for Distributed Data Fabrics: Categorical Hypergraph Models

TL;DR

The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.

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 \( Σ= (N, C) \), 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.
Paper Structure (25 sections, 37 theorems, 83 equations, 9 figures, 2 tables)

This paper contains 25 sections, 37 theorems, 83 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. An Introduction to Data Fabrics
  3. Operations of Data Fabrics
  4. A Categorical Perspective
  5. Hypergraph Embeddings and Representations
  6. Vector Representations of Data Assets
  7. Construction of the Adjacency Structure
  8. Mapping to Modular Tensor Categories
  9. Braiding Action and Geometric Analogies
  10. Embedding and Applications
  11. Computational Challenges
  12. Heterogeneity
  13. Scalability
  14. Governance
  15. Real-Time Processing
  16. ...and 10 more sections

Key Result

Proposition 1

The schema distance $\mathop{\mathrm{dist}}\nolimits(S_i, S_j)$ defines a pseudo-metric on the set of schemas, with $\mathop{\mathrm{dist}}\nolimits(S_i, S_j) = 0$ if and only if there exists a bijective mapping $\pi: S_i \to S_j$ such that $\text{sim}(a, \pi(a)) = 1$ for all $a \in S_i$.

Figures (9)

  • Figure 1: Interactions between operations, illustrating data flow from integration to federated learning.
  • Figure 2: Structure of the hypergraph $G$, with vertices $v_i, v_m \in V$ (datasets or metadata) and hyperedge $e \in E$ encoding a multi-way relationship.
  • Figure 3: Braiding modeling symmetries.
  • Figure 4: Pareto front for real-time processing, balancing latency and loss trade-offs.
  • Figure 5: Relationships between consistency models, from strongest (linearizable) to weakest (eventual).
  • ...and 4 more figures

Theorems & Definitions (78)

  • Proposition 1
  • proof
  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 68 more