# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 2 Methodology and Methods  
The first goal is to establish a theoretical basis that will enable a universal understanding of complex design, build, and O&M systems; this objective is developed mathematically and by simulations. Following [2], a system, a set of interconnected elements that work to achieve a common goal, is defined by its transformation of stimuli to response. Its constituents are treated here as black boxes [38]; only their effects and interactions are studied. This definition implies communication and cybernetics [5], and, according to [35], also learning and adaptation. Systems have digital and analogical parts; here, we assume that the analogical are within the physical sections and that the physical systems are well understood by the engineer, and that for the most part, digital processes, in which we include information system processes, are faster in such a way that enables control over the analogical parts. This assumption and its collapse are dealt with in detail in the Discussion part. The direct use of analog systems for control is portrayed in [36]. Following [24], the systems are dynamical systems, i.e., when input is applied to them, the systems change state, and the output is generated from the new state. States hold all historical implications; therefore, memory is retained within states, and time has only a forward direction. The gained theoretical understanding is the basis for the development of a practical set of procedures that enable quality management under complexity. By the above discussion, it is apparent that a system theory must rely on system response, communication, cybernetics, learning, and adaptation. The theory is developed mathematically in Section 3; in this work, we developed an entropy estimator for directed graphs [39]. A \"practical knowing\" [40] maps the results to construction and infrastructure design and build projects. Numerical simulations and experiments (in Section 4) sustain the presentation, and case study results (Section 5) present analysis tools and applications. In experiment A.1, we iterated a system with randomness and no correction to present a mechanism for bifurcation and avalanche; this mechanism usually applies to defects and, at times, to faults. A cellular automaton was used in numerical experiment A.2 to simulate and explain propagation and multiplication. A third numerical experiment (A.3) simulated DSP effects on complex systems through feedback loop amplification towards the IC threshold and avalanche; this contributes to the factors causing information overload and another failure mechanism. Numerical experiment A.4 simulated a synthetic design network generated from a construction project to highlight effects related to a forceful outside intervention, such as stopping or limiting the traffic of data, payments, approvals, etc. [41], denominated here \"power plays\" and manifested in A.4 in triadic links severance [42], bridge formation, and centrality. A simulation of convolution (in Section 3.4) contributed to the list of factors generating information overload. In Case Study 1, multiple lists of faults produced by QC for 27 projects were digitized. The statistics of the log-normal PDF and power function magnitude distribution were validated, and the IC, avalanche, emergent traits, and propagation were explored. Statistics, descriptive statistics, network analysis, information constraint, and DSP data analysis tools provided insight, meaning, responsibility allocation, trajectories, and predictions. In Case Study 2, a scale-free morphology is validated, and information entropy is coupled with descriptive statistics tools. Descriptive statistics is a significant tool for the exploration of avalanches, correlations, IC, and trending; it is used here for projections and predictions. Statistics provide a phase parameter. Network presentation helps to find attractors and trajectories; it is used here to retrace propagation and reassign responsibility.