# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 4 Numerical Experiments
A network was generated for time = 0, the time of commencement of works on site (Figure 11). The same procedure was carried out after encountering a design fault (Figure 12). The Matlab curve fitter app fitted degree probability distributions to the graphs; the relevant function is presented in Figure 12c. The function is a power function, therefore the networks are scale-free; its \(b\),\(b\) < 1, which is characteristic of networks that form one big cluster. Figure 11: Information net morphology at t = 0 (**a**). Nodes degree distribution (**b**), their degree distribution power function, R\({}^{2}\) = 0.9365. Legend: Nodes represent autonomous agents (such as designers and subcontractors). GC = general contractor, SE = structure engineer. In summation of the findings:  
Figure 11 has the morphology of one big cluster; finishing works form a subcluster [on the left], and all other design and construction works form another on the right. The color scale of outdegree highlights preferential attachment. The site engineer (SE) and the general contractor office (GC) had the same degree. This is in concert with the observation that all participants must communicate with the SE and all participants must communicate with the GC (payments, contracts, etc.). The \"SE\" and \"GE\" share the same location in the graph at t = 0. This is an outcome of their position as being a vertex for the two subclusters. The possible number of directed edges is \((\binom{n}{2}\cdot 2\ )\), where n = number of nodes; here, 66-65 = 4290 possible edges. The number of edges in the Figure 12 adjacency matrix is 629, that is, 15% of the number of possibilities. Yet, a small-world network with one cluster formation and a minimal radius was formed. The number of edges needed to form a giant cluster of random graph nodes is \(n/2\)[64], here 33; the max degree of separation within that cluster is \(\log(n)\). The minimal connected network of \(n\) nodes has \(n-1\) edges; adding \(n/2\) edges to assure small-world status can be achieved within 100 edges, well below our 629-edge graph. The added edges work to magnify centrality and synchronization; in Figure 12, for each added node an edge is formed to Sc and to Gc, this is [PERSON]'s [49] preferential attachment model with a probability of 1. Synchronized  
Figure 12: Information morphology evolution after a redo order originating in a design error (**a**). Degree histogram (**b**) and degree probability distribution (\(\mathbb{R}^{2}\) = 0.94) (**c**) and hub centrality (**d**). networks were found in [9] to be more stable at the cost of larger diameters (longer critical paths). Even within the confines of viability kernels, there is large room for permutations, making expertise a parameter of stability [9]. Self-adjustment to enlarge \"b\" reduces synchronization pressures; the formation of sub-graphs reduces the sync basin size [65]; synchronization is required within local clusters rather than globally. Information network morphology evolution after a redo order originating in a design error is portrayed in Figure 12. Figure 12 presents a considerable change. The number of participants grows to 67, and the morphology changes. Triadic connections are severed, bridges are formed. Information that passed directly in Figure 11 is communicated through relay activities denominated as **bridge activities**[36]. This is a form of learning, though far remote from the self-healing networks suggested by [64]. In Figure 12, subclusters are defined by bridge activities that relate to architect and constructor. The degree of distribution is now tilted towards the site engineer, because the site response is faster, and the site is central to a solution. Although the diameter is bigger and connections are severed, the network is still one big cluster, and \(b<1\). Hub centrality is another estimator of rank (Figure 12d). Hub centrality suggests that although of equal degree at time = 0, SE centrality is higher than GC.