# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 4 Numerical Experiments  
In this section, four numerical experiments are reported. Experiment A.1 is a time-dependent Markovian process of a second degree with randomness and no correction mechanism, to produce bifurcations and an avalanche. Experiment A.2 is of cellular automata with a grid morphology and a Markovian process with a spatial randomness additive; the goal is to produce multiplication and propagation. A.3 is a numerical experiment, where power function-scaled eigenvalue vectors propagate a second-degree Markovian system to produce a system collapse; the goal is to elucidate the effect of the DSP filters on information load and produce by signal processing generated overloads. Experiment A.4 portrays a modeled design information network that reorganizes in response to a redo order; the goal is to show morphology evolution, teleonomy, and centrality. The first experiment deals with an effect of randomness, the second adds morphology effects to randomness, the third deals with the inflation of the error signal overload, and the last depicts a social impact on networks; eventually all those impacts superimpose themselves within systems to create meta-complexity. **A.1** Discrete time (DT) numerical experiment with randomness and no correction mechanism. There are additive processes such as the DSP filters portrayed in A.3, processes involving multiplicity are presented in the Time section, and there are processes of multiplicity + additivity, such as the convolution process of sequential design and production. The mathematical operator used here is multiplicity; The multiplication rule is  
\[\mathbb{X}[n+1]=\mathbb{X}[n]\cdot\mathbb{X}[n-1]+q; \tag{40}\]  
the entering fault vector is in the magnitude of \(\delta\mathbb{[}\mathbb{N}\mathbb{]}=1\), \(N\leq 0\) (Figure 8). This experiment may portray a deterministic system with randomness caused by rounding rules. All iterations resulted in bifurcation and avalanche. The DT time unit in this work is a week. Therefore, after a few months, all systems with randomness and no correction collapse. All iterations exhibit the same phenomenon on different scales. The factors in this experiment are lenient considering the reality of design and construction: \"q\" is very small relative to site \"q\" (discussed in Figure 4); the operator chosen is multiplication rather than convolution; propagation is not by linear trajectories, but rather in multiplication loops (coupled, Case Study 2). The effect of a minor, almost negligible, random additive is the butterfly effect. This effect limits modeling and simulations' predictive value. **A.2** Cellular automaton--a bounded random walk. The morphology chosen is a grid; activities form one edge of the grid and time the other. There are m = 50 activities. There is a propagation for every discrete time \"n\". The experiments were stopped at 15-timeframe iterations; continuation was uninformative. The propagation rule used here is as follows:  
Propagation Rule:  
for i = 1:m  
z(n + 1,i) = z(n + 1,i) + z(n,i);  
j = i + 1;  
if j > m  
j = randi([2,(m - 1)]);  
end  
z(n + 1,j) = z(n + 1,j) + z(n,i);  
j = i - 1;  
if j < 1  
j = randi([2,(m - 1)]);  
end  
z(n + 1,j) = z(n + 1,j) + z(n,i);  
end  
Propagation is commanded to three adjacent neighbors. The relocation decision taken at the boundaries is random; thus, the random additive is spatial. There are no corrections. The operator is addition, and the effect of proliferation is multiplicity. With no correction, the accumulated possible effect is the proliferation ratio (here 3) in the power of time [\(n\)]. After a few iterations, all activities are affected (Figure 9b); an assumption of dissipative  
Figure 8: Time effects via repetitions [”bifurcation”]. q = 0.0002 (**a**,**b**,**d**), q = 0.2 (**c**). Legend: X—number of states. ergodicity suggests itself here. CSI systems form dissipative structures [60] or, in our case, have deliberately designed steady states, where time-averaged measurements converge to system averages [61].