A network-based Structural Complexity Metric for Engineered Complex Systems Kaushik Sinha Engineering Systems Division Massachusetts Institute of Technology Cambridge, MA, USA Olivier L. de Weck Engineering Systems Division Massachusetts Institute of Technology Cambridge, MA, USA Abstract — The complexity of today’s highly engineered products is rooted in the interwoven architecture defined by its components and their interactions. Such structures can be viewed as the adjacency matrix of the associated dependency network representing the product architecture. To evaluate a complex system or to compare it to other systems, numerical assessment of its structural complexity is mandatory. In this paper, we develop a quantitative measure for structural complexity and apply the same to real-world engineered systems like gas turbine engine. It is observed that low topological complexity implies centralized architecture and it increases as one marches towards highly distributed architectures. We posit that the development cost varies non-linearly with structural complexity. Some empirical evidences of such behavior are presented from the literature and preliminary results from simple experiments involving assembly of simple structures strengthens our hypothesis. Keywords — Structural Complexity, topological complexity, matrix energy, development cost, real-world engineered systems, gas turbine engine. I. INTRODUCTION Today’s large-scale engineered systems are becoming increasingly complex due to numerous reasons including increasing demands on performance levels, improved lifecycle properties. As a consequence, large product development projects are becoming increasingly challenging and are falling behind in terms of schedule and cost performance. For example, in 13 aerospace projects reviewed by the US Government Accountability Office (GAO) since 2008, large development cost growth of about 55% was observed. Such large development cost overruns/failures of large-scale system development projects can largely be attributed to our current inability to characterize, quantify and manage associated complexity [11]. With increasing complexity of engineered systems, typically the associated Life Cycle Cost also increases. The challenge of quantifying and managing complexity is also central to many research areas occupied with engineering. A particular concern with the work done in the area of complexity estimation is that less than one-fifth of the studies even attempted to provide some degree of objective quantification of complexity. An objective and quantifiable measure of structural complexity is imperative for systematic search and optimization of system architecture. In particular, the consideration of dependency network attracts attention in various scientific works because dependency-based system structures affect system characteristics and behavior. A system consisting of many components that are linked to each other and the interaction between these parts influences the system’s behavior [5, 7]. In the context of product design, the quantification and efficient management of complexity has increasingly gained importance. There is a difference between the “complexity of technical systems” and the “complexity of components” [5]. The complexity of components or objects is characterized by the parameters like quantity of variables, completeness of understanding about the object, etc. The complexity of technical systems depends on the heterogeneity and quantity of different elements and their connectivity, and is a measurable system characteristic. This internal product architecture can be represented by complex networks, which are graph-theoretic representation of complex systems. The nodes, representing components of the systems, are connected by links if there exists an interaction between any pair of components. The product functionality is bounded by the underlying architecture. A perpetually occurring theme is the complexity of product architecture. It is often perceived that as we stretch the limits of efficiency and attempt to design more robust system, we tend to make architectures more complex. There are several underlying common themes to complexity theories and it remains to be somewhat ambiguous concept in practice. In this paper, a rigorous and quantitative structural complexity metric for architecture evaluation and optimization, incorporating the fundamental underlying characteristics of product architecture, is proposed. This would lend objectivity to the process of product architecture selection and design, which at this date, is still an art. This work proposes a simple, quantifiable measure for structural complexity that captures the underlying system connectivity structure. Subsequently this measure is applied to a pair of jet engine architectures to measure and compare their structural complexities. Next, we posit that the development cost varies non-linearly with structural complexity. Some empirical evidences of such behavior are presented from the literature. This hypothesis is further buttressed by preliminary results from simple experiments involving assembly of simple structures. II. STRUCTURAL COMPLEXITY QUANTIFICATION The structural complexity of technical systems depends on the quantity of different elements and their connectivity structure and is a measurable system characteristic. This representation should include contributions coming from the internal complexities of the components of the system; the