A tensor-based formulation of hetero-functional graph theory

Amro M Farid; Dakota J Thompson; Wester Schoonenberg

doi:10.1038/s41598-022-19333-y

A tensor-based formulation of hetero-functional graph theory

Sci Rep. 2022 Nov 5;12(1):18805. doi: 10.1038/s41598-022-19333-y.

Authors

Amro M Farid^{1

2}, Dakota J Thompson³, Wester Schoonenberg¹

Affiliations

¹ Thayer School of Engineering at Dartmouth, Hanover, NH, USA.
² MIT Mechanical Engineering, Cambridge, MA, USA.
³ Thayer School of Engineering at Dartmouth, Hanover, NH, USA. Dakota.J.Thompson.TH@Dartmouth.edu.

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.