Polymorphic Hypergraph Inference System (PHIS): A New Approach for Complex Relational Data Analysis

Godson Johnson

doi:10.20944/preprints202412.2452.v1

Submitted:

28 December 2024

Posted:

31 December 2024

You are already at the latest version

Abstract

This paper presents the Polymorphic Hypergraph Inference System (PHIS), a new framework for learning and in- ference on complex relational data structures. Hereby introducing a comprehensive system that employs dynamic hypergraph con- struction and refinement techniques to analyze and understand intricate relationships within datasets with hierarchical nature. The paper details the complete system architecture, mathematical foundations, algorithmic specifications, and Theoretical analysis of performance characteristics. The framework shows promising applications in various domains requiring sophisticated relation- ship analysis and pattern discovery, with theoretical guarantees and practical implementation guidelines

Keywords:

hypergraphs

;

inference systems

;

pattern match-ing

;

relational data analysis

;

graph algorithms

;

machine learning

Subject:

Computer Science and Mathematics - Artificial Intelligence and Machine Learning

I. INTRODUCTION

The analysis of complex relational data structures remains a significant challenge in modern computational systems. Traditional approaches often struggle with multi-dimensional relationships and dynamic pattern evolution. This paper introduces the Polymorphic Hypergraph Inference System (PHIS), which addresses these challenges through a novel combination of hypergraph theory and adaptive pattern recognition. The main contributions of this paper are:

A novel hypergraph-based inference system for complex relational data
Efficient algorithms for dynamic pattern discovery and rule generation
Mathematical framework with convergence guarantees

II. RELATED WORK

Recent advances in graph-based inference systems have shown promising results in various domains [1]. However, existing approaches often lack the flexibility to handle dynamic relationship evolution [2] or suffer from computational inefficiency when scaling to large datasets [3].

III. SYSTEM ARCHITECTURE

A.: Overview

The PHIS architecture comprises three main components: the pattern matching engine, rule generation system, and convergence controller. Figure 1 presents the system architecture and component interactions.

B.: Core Components

The system’s primary components include:

1) Pattern Matching Engine: The pattern matching engine implements efficient subgraph isomorphism detection with the following key features:

Incremental pattern matching with delta updates
Cached pattern validation using signature hashing
Parallel isomorphism checking across multiple cores
Early termination based on structural invariants

2) Rule Generation System: The rule generation process involves:

Pattern extraction using frequent subgraph mining
Transformation rule construction with preservation guarantees
Rule validation through formal verification
Pattern generalization using abstraction hierarchies

IV. MATHEMATICAL FRAMEWORK

A.: Fundamental Definitions

The core mathematical framework is defined by:

	V = ^[ {x_i,y_i} (x_i,y_i)∈D	(1)
	E_(t+1) = E_(t) ∪ τ_R(H_(t),f)	(2)
	( true if f : P_R,→ H	(3)
IsValidIsomorphism(f,P_R,H) ⇒
	false otherwise

B.: Convergence Metrics

System convergence is measured through:

V. ALGORITHM SPECIFICATION

A.: Core Functions

Algorithm 1 presents the core functions of PHIS.

B.: Main Procedure

Algorithm 2 presents the main PHIS algorithm.

VI. PERFORMANCE ANALYSIS

A.: Complexity Analysis

The system’s computational complexity is characterized by:

Pattern Matching: O(|V |^k) per pattern, where k is pattern size
Rule Application: O(|R|) per iteration
Consistency Evaluation: O(|D|·|E|) for full verification
Rule Generation: O(|D| · |V |) for basic patterns

B.: Space Requirements

Memory requirements include:

• Hypergraph	Storage:	O(\|V \|	+
\|HypergraphStorage	:O(—V—		+
—E—)forsparsegraphs
Rule Set Storage: O(\|R\|) with compression
Pattern Cache: O(k · \|V \|) for frequent patterns
Working Memory: O(\|V \|²) worst case scenario

VII. CONVERGENCE ANALYSIS

A.: Convergence Behavior

Figure 2 illustrates the system’s convergence characteristics across different phases.

VIII. CONCLUSION

The Polymorphic Hypergraph Inference System represents a significant advancement in graph-based learning and inference systems. Through its sophisticated pattern matching, rule generation, and convergence mechanisms, PHIS provides a robust framework for analyzing complex relational data structures. Theoretical aproximations were employed in terms of scalability, accuracy, and resource efficiency compared to existing systems. Future work will focus on extending the system’s capabilities and experimentation.

DECLARATION

Generative AI tools were utilized exclusively for editorial purposes and literature search in the preparation of this manuscript. These tools were not employed in the generation of ideas or the authorship of the content. The authors have reviewed and edited all material as necessary and assume full responsibility for the integrity and accuracy of the manuscript.

REFERENCES

J. Neville and D. Jensen, Relational Dependency Networks, Journal of Machine Learning Research, vol. 8, pp. 653–692, 2007.
L. Pan, C. Shi, and I. Dokmanic, A Graph Dynamics Prior for Relational Inference, Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, no. 8, pp. 10085–10093, 2023. [CrossRef]
X. Gao, W. Zhang, Y. Shao, Q. V. H. Nguyen, B. Cui, and H. Yin, Efficient Graph Neural Network Inference at Large Scale, arXiv preprint arXiv:2211.00495, 2022. [CrossRef]

Figure 1. PHIS System Architecture and Component Interactions.

Figure 2. Convergence Behavior Analysis.

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