A Tensor-Based Formulation of Hetero-functional Graph Theory

TL;DR

This paper introduces a tensor-based mathematical framework for hetero-functional graph theory, enhancing its analytical capabilities and connection to ontological concepts in systems engineering.

Contribution

It presents the first tensor formulation of HFGT, including new data structures like the hetero-functional incidence tensor, linking graph theory with MBSE concepts.

Findings

Introduces hetero-functional incidence tensor as a new data structure.

Provides analytical results relating HFGT to multi-layer networks.

Strengthens the theoretical foundation of HFGT with tensor mathematics.

Abstract

Recently, hetero-functional graph theory (HFGT) has developed as a means to mathematically model the structure of large-scale complex flexible engineering systems. It does so by fusing concepts from network science and model-based systems engineering (MBSE). For the former, it utilizes multiple graph-based data structures to support a matrix-based quantitative analysis. For the latter, HFGT inherits the heterogeneity of conceptual and ontological constructs found in model-based systems engineering including system form, system function, and system concept. These diverse conceptual constructs indicate multi-dimensional rather than two-dimensional relationships. This paper provides the first tensor-based treatment of hetero-functional graph theory. In particular, it addresses the ``system concept" and the hetero-functional adjacency matrix from the perspective of tensors and introduces the hetero-functional incidence tensor as a new data structure. The tensor-based formulation described in this work makes a stronger tie between HFGT and its ontological foundations in MBSE. Finally, the tensor-based formulation facilitates several analytical results that provide an understanding of the relationships between HFGT and multi-layer networks.