# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 3 Development
### Entropy  
Information entropy is a measure of the amount of information needed to define the state of the system; it is a measure of (a normalized) change in the \"internal energy\" of the system (Equation (17)). Graph entropy is a measure of the morphological information content of a graph [50]. Shannon's entropy (Equation (14)) is an extensive variable \(S=S(k)\), where \"\(k\)\" is the number of states. When a system is presented as a network, the degree of nodes is a possible candidate for state variable (information functional [51]) and its proportional likelihood distribution an estimator of the degree of probability distribution. Entropy has maxima for a total number of states \"\(k\)\", with an equally distributed probability:  
\[S(x)=-\sum_{1}^{k}p(x)log(p(x))=-\sum_{k}\frac{1}{k}log\frac{1}{k}=-\frac{k}{k }log(k)=-log(k) \tag{14}\]  
This is the maximum possible entropy for \"\(k\)\". \"\(k\)\" itself is not bound; it inflates entropy. A move to the right on Figure 4 entails information gain. It entails an investment of energy, and it is possible only in subsystems. \(h(p(x)=-\log(p(x)\) is the information content of an event [29]; it is the surprise in its occurrence. The generation of entropy from degree distribution highlights the Figure 5 log power function as a presentation of the magnitude of the surprise in the event. The space G(M,E,T) is a graph where the nodes [M] are activities, agents, or locations, and the edges [E] are their information links. In Case Study 2, network nodes represent activities, and in the numerical experiment A.4, network nodes represent design agents. In directed networks, edges present precedence, therefore directed networks are a timeline snapshot. Networks are in constant flux; the morphological changes' trend is to inflate the graph:  
\[\frac{dG}{dt}=\frac{\partial G}{\partial M}\frac{\partial M}{\partial t}+ \frac{\partial G}{\partial E}\frac{\partial E}{\partial t}\geq 0 \tag{15}\]  
Figure 6: Log–log graph of scaled power values, \(b=1.946\). For example, in infrastructure such as HVAC systems, nodes may be machines and in drainage systems, conduits and manholes--nodes are added, and links (physical and info-links) are generated, never to be deleted. In construction projects, activities generate a multiplicity of procurement activities, QC, redesigns, and claims; each ripple causes ever more activities, ever more edges. Entropy is introduced to the system externally (\"\(\epsilon\)\") or produced internally (\"\(i\)\"):  
\[\frac{dS}{dt}=\frac{d^{\epsilon}S}{dt}+\frac{d^{i}S}{dt} \tag{16}\]  
In synchronous production, constant efforts are invested at keeping \(\frac{dS}{dt}=0\), design changes are not allowed, and barriers are high to minimize \(\frac{d^{\epsilon}S}{dt}\). Internal faults are corrected in each discrete unit of time \(\frac{d^{\epsilon}S}{dt}=0\), \(t\in\mathbb{N}\). In synchronous production lines, entropy production is at its minimum. From a thermodynamic point of view, this is possible because energy is invested and information gained in a persistent effort to correct faults (Equation (17)). Attempts to benefit site production from industrial synchronized low entropy, by transferring production to off-site prefabrication factories, enlarge total entropy. This is an outcome of the first law of thermodynamics (Equation (17)): there is a conservation of energy law that reduction of energy in one part of a system is but an energy transfer elsewhere; here entropy is transferred from a subsystem to the system, i.e., from factory to site. The commonsense explanation of this phenomenon is that the number of possible states on site (to be designed and coordinated) increases. The thermodynamic concept of a closed system is not directly applicable here; on the one hand, complex projects are partially open to an influx of incoming information, and on the other, the export of data entropy does not reduce its systemic content.