Reconciling Formal, Multi-Layer, and Hetero-functional Graphs with the Hetero-functional Incidence Tensor

The developing consensus across a number of STEM fields is that each of the NAE game-changing goals is is characterized by an 'engineering system' that is analyzed and resynthesized using a meta-problem-solving skill set. Two fields in particular have attempted to traverse this convergence challenge: systems engineering and network science. Systems engineering has developed as a practical and interdisciplinary engineering discipline that enables the successful realization of complex systems from concept, through design, to full implementation based upon graphical modeling languages. In contrast, network science has developed to quantitatively analyze networks that appear in a wide variety of engineering systems but suffers from disparate terminology and a lack of consensus. This paper provides a tensor-based formulation of several of the most important parts of hetero-functional graph theory. More specifically, it discusses the system concept, the hetero-functional adjacency matrix, and introduces the hetero-functional incidence tensor for the first time. 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 an understanding of the relationships between HFGT and multilayer networks 'despite its disparate terminology and lack of consensus In so doing, this tensor-based treatment is likely to advance Kivela et. al's goal to discern the similarities and differences between these mathematical models in as precise a manner as possible.