Theoretical approach to a lower limit of KPIs for controlling complex organisational systems

Based on the very fundamental rules from Informatics, the dimensionality of the problem determines the degrees of freedom. The number of independently varying parameters q_j span a space open for the behaviour of the system, i.e. its static state vector. Hence, the number of orthogonal observables required to describe the situation is given by the dimension of this space. This equals the dimension of the describing adjacency matrix (Eber and Zimmermann 2018).

For each given static interaction between two or more elements, the degrees of freedom are reduced by one since each condition, i.e. dependency, inhibits one dimension from developing freely.

Hence, static systems constructed from more interactions than elements (ξN > N ⇒ > 1) are described by complexity α = ln(ξ + 1)/ln N > ln 2/ln N. Such systems represent overdetermined systems that have no solution if not degenerated.

Any non-static system involves interactions implying not states but modification of states. Such less-restrictively formulated interactions avoid overdetermination but lead to state vectors with dynamic components. This kind of interaction is introduced by arrows and edges given by differential equations (Haken 1983; Wiener 1992; Zimmermann and Eber 2017). 2 ∂qi∂t=f(qj). $${{\partial {q_i}} \over {\partial t}} = f({q_j}).$$

As long as interaction functions are sufficiently differentiable, they can be developed into a Taylor series. For short time intervals, first-order terms suffice. Then, interactions are no more linearly effectuating new values, but implement linear modifications to existing values over time. Linear equation systems become linear differential equation systems based on the linear weighted adjacency matrix c_i,j. The solutions of these systems are generally given by complex exponential functions allowing for escalating or dampening development, as well as respective oscillations (Eber 2021a, 2021b, 2022). 3 ∂qi(t)∂t~∑ ci,jqj⇒qi(t)~Aegt+Beg¯tg∈ℂ. \begin{matrix} \frac{\partial {{q}_{i}}(t)}{\partial t}\tilde{\ }\mathop{\sum }^{}{{c}_{i,j}}{{q}_{j}} & \Rightarrow & {{q}_{i}}(t)\tilde{\ }A{{e}^{gt}}+B{{e}^{\bar{g}t}} & g\in \mathbb{C}.\\ \end{matrix}

Against this background, development of these systems on the time axis is mostly escalating or oscillating, and, only in rare cases, stabilising (Re(g) < 0 and Im(g) = 0). Therefore, in general, dynamic systems are to be understood as mainly non-predictable. Due to the huge number of options to develop starkly, this is in fact one of the qualitative definitions of ‘complexity’ (Strogatz 2001; Newman 2003; Liening 2017).

Dynamic systems develop with time and inherently seek states of (stable) equilibrium, which takes time with durations out of a widely varying range up to infinity. Therefore, the only static state vectors occurring are given by states of equilibrium, in which the behaviour in close proximity of equilibrium as well as the dynamical paths towards equilibrium states play a role. Systems far off equilibrium show indeterminate (‘chaotic’) behaviour, in which investigating is limited to very general statements, in particular, mostly not offering detailed individual information (White et al. 2004; Liening 2017).

Clearly, only equilibrium states are predictable, and these too only if the systems rest at these state vectors or remain at least close to them. Sudden modifications pushing the system, in no time, away from stability initiate a period of unpredictable behaviour until stabilising again. Moreover, the newly achieved state of equilibrium is not necessarily a causal consequence of the previous situation but could be any stable state randomly approached and being caught in.

Referring only to a stable – or at least – metastable equilibrium situation, the state vectors are stable, again determined by N variables; the required number of KPIs also needs to cover this space. Systems with no existing equilibrium states or far off these are principally not predictable, hence not manageable and therefore of no interest.

The state vector is always in or very close to (stable) equilibrium states. Developing with time while remaining close to equilibrium requires relatively slow changes. Only then, this equals a static as well as a causal system and will principally allow deriving the number of observables.

This requires the stabilising mechanisms to be about one order of magnitude faster than any external modifications, be these perturbations or some deliberately induced steering input (Eber 2019a).

Stabilisation is only enforced by strong (hence, short) damping loops. All other existing loops are either clearly destabilising due to their parameters or of higher orders. So far, they certainly do not contribute to stabilising.

Short dampening loops inducing stability are necessarily local and can therefore be understood as locally limited units that are affected only very little by external changes (Figure 1). Furthermore, as they provide stabilised output, the transfer of modification is also starkly reduced. Treating these mechanisms as subsystems located outside the considered system therefore simplifies the situation and renders only the remaining system’s complexity to contribute to unpredictable behaviour (Bertalanffy 1969; Haken 1983).

Fig. 1:

The complexity is given by the inherent parameter α derived from the number of interactions, as well as the virtual contribution via recursiveness β and localisation ϑ. All of these are constructed from tight coupling, which leads to the requirement to decouple a significant number of ties (Separation/Separability) (White at al. 2004; Eber 2020). In contrast to static systems, where the degrees of freedom are reduced with interactions, here, stability is gained with defunctionalising ties. Then, at least, the dynamic components are driven to inactivity, and the dimension of the solution space is again determined by the parameters alone. Operationally, this can easily be achieved by carefully introducing control loops and respective overproduction/overtime (Eber 2019a).

Cross impact analysis traditionally uses well-defined transition probabilities between states as the adjacency matrix to be investigated. As these are not available, a well-determined value representing the unidirectional coupling of two nodes needs to be identified.

From the differential equation of control (Eber 2019b), the time constant τ_contr of repairing measures is known as a function of the resources available for control and the local strength of control. 5 R(C)i,j=∂qi∂t=−Δqi/τi,contr⇒τi,contr=| Δqi |/R(C)i,j. \begin{matrix} {{R}^{(C)}}_{i,j}=\frac{\partial {{q}_{i}}}{\partial t}=-\Delta {{q}_{i}}/{{\tau }_{i,contr}} & \Rightarrow & {{\tau }_{i,contr}}=|\Delta {{q}_{i}}|/{{R}^{(C)}}_{i,j}.\\ \end{matrix}

Relating this to the specific time reserve t_Res available between the participating nodes allows defining the coupling parameter χ_i,j = τ_contr/t_Res of this particular tie (see Figure 2). This parameter is normalised in a two-fold way: χ_i,j. equals one if the control mechanisms allow ruling out the expected deviations down to a factor of e⁻¹ within the time reserves. A value of zero indicates no coupling at all, while higher values respectively represent the most critical interactions whereby local deviations are widely carried into the network. Hence, according to Zimmermann and Eber (2014), χ_i,j is also referred to as the ‘criticality’ of the respective interaction; however, this term is not used in the context of this paper.

Fig. 2:

Clearly, this value χ_i,j represents no transition probability but can easily be understood respectively by transforming into a linear parameter χ^i,j${\hat \chi _{i,j}}$:

The integral over the decreasing function Δq_i(t) Δq_0,ie^−t/τ_i,C from the point of incidence zero to infinity, i.e. the total deviation over the complete time frame is as follows: 6 Δtotal=∫0∞Δq0,ie−t/τi,Cdt=−Δq0,iτi,Ce−t/τi,C|0∞=Δq0.iτi,C. $${{\rm{\Delta }}_{total}} = \mathop \smallint \limits_0^\infty {\rm{\Delta }}{q_{0,i}}{e^{ - t/{\tau _{i,C}}}}dt = - {\rm{\Delta }}{q_{0,i}}{\tau _{i,C}}{e^{ - t/{\tau _{i,C}}}}|_0^\infty = {\rm{\Delta }}{q_{0.i}}{\tau _{i,C}}.$$

In contrast, the remaining total deviation after the time reserve t_Res = τ_i,C /χ_i,j is then 7 Δtotal=∫τi,C/χi,j∞Δq0,ie−t/τi,Cdt=−Δq0,iτi,Ce−t/τi,C|τi,C/χi,j∞=Δq0,iτi,Ce−1/χi,j=Δtotale−1/χi,j. \begin{array}{*{35}{l}} {{\Delta }_{total}} & =\underset{{{\tau }_{i,C}}/{{\chi }_{_{i,j}}}}{\overset{\infty }{\mathop \int }}\,\Delta {{q}_{0,i}}{{e}^{-t/{{\tau }_{i,C}}}}dt=-\Delta {{q}_{0,i}}{{\tau }_{i,C}}{{e}^{-t/{{\tau }_{i,C}}}}|_{{{\tau }_{i,C}}/{{\chi }_{_{i,j}}}}^{\infty }\\ {} & =\Delta {{q}_{0,i}}{{\tau }_{i,C}}{{e}^{-1/{{\chi }_{i,j}}}}={{\Delta }_{total}}{{e}^{-1/{{\chi }_{i,j}}}}\\ \end{array}.

Hence, the share of total deviation, which, not covered with a certain value of χ_i,j, is e^−1/χ_i,j, identifying χ^i,j=e−1/χi,j${\hat \chi _{i,j}} = {e^{ - 1/{\chi _{i,j}}}}$ as a sensible parameter describing the deviation share transported over a certain tie.

Obviously, this parameter is approximately linear around χ^i,j=1${\hat \chi _{i,j}} = 1$ according to Figure 3. Development into a Taylor Series around χ^i,j=1${\hat \chi _{i,j}} = 1$ yields a useful approximation, which is certainly accurate to a sufficient degree: 8 χ^=e−1/χ≃[ e−1/χ ]χ=1+(χ−1)[ ∂∂χe−1/χ ]χ=1=e−1+(χ−1)[ e−1/χ/χ2 ]χ=1; \begin{array}{*{35}{l}} {\hat{\chi }} & ={{e}^{-1/\chi }}\simeq {{[{{e}^{-1/\chi }}]}_{\chi =1}}+(\chi -1){{[\frac{\partial }{\partial \chi }{{e}^{-1/\chi }}]}_{\chi =1}}\\ {} & ={{e}^{-1}}+(\chi -1){{[{{e}^{-1/\chi }}/{{\chi }^{2}}]}_{\chi =1}}\\ \end{array}; 9 χ^≃e−1+(χ−1)e−1=e−1[ 1+(χ−1) ]=χe−1. $$\hat \chi \simeq {e^{ - 1}} + (\chi - 1){e^{ - 1}} = {e^{ - 1}}[1 + (\chi - 1)] = \chi {e^{ - 1}}.$$

Fig. 3:

From this, the adjacency matrix is given as χ^i,j${\hat \chi _{i,j}}$. This clearly points out that any organisational system can only be assessed sensibly if all elements without exception are precisely known and all interactions are well investigated. Since the scaling factor will in no way change the principal results, the coupling parameters χ^i,j${\hat \chi _{i,j}}$ and χ_i,j can be used as the normalised measure for the given interaction.

Cross impact analysis refers to analysing the adjacency matrix χ^i,j${\hat \chi _{i,j}}$ that describes the weighted interaction of a number of elements. First-order investigation, hence, implies the properties of the χ^i,j${\hat \chi _{i,j}}$ directly, while higher-order analysis takes into account indirect interactions via many steps and loops by analysing higher powers of χ^i,j${\hat \chi _{i,j}}$ as well (Zimmermann and Eber 2014). Since parallel paths of different lengths are cumulative, the total matrix to be investigated is as follows: 10 χ^i,j(m)=∑k=1mχ^i,jk⇒χ^i,j(∞)=∑k=1∞χ^i,jk, \begin{matrix} {{{\hat{\chi }}}_{i,j}}^{(m)}=\underset{k=1}{\overset{m}{\mathop \sum }}\,{{{\hat{\chi }}}_{i,j}}^{k} & \Rightarrow & {{{\hat{\chi }}}_{i,j}}^{(\infty )}=\underset{k=1}{\overset{\infty }{\mathop \sum }}\,{{{\hat{\chi }}}_{i,j}}^{k},\\ \end{matrix} where each element represents the cumulated weight of all paths that the respective node participates in. This is added up to path lengths of m, forming the m-th-order analysis and theoretically valid for infinite powers, which is, however, numerically not sensible. In the end, the characteristics extracted are as follows:

The ‘active sum (AS)’, which is the cumulation of all interactions where i is the source, i.e. a measure of the degree to which the node i influences the remaining system. 11 ASi(m)=∑jχ^i,j(m). $$A{S_{_i}}^{(m)} = \mathop \sum \limits_j {\hat \chi _{i,j}}^{(m)}.$$

The interpretation of the roles for particular nodes follows from the characteristics plotted on a graph using a position given by AS vs. PS (see, e.g. Figure 4; Gordon and Hayward 1968; Vester 1995; Zimmermann and Eber 2014).

Fig. 4:

Nodes located in the active area (top left) are highly influencing while not being significantly influenced by other nodes. Thus, they are effective levers to manage the system.

Reactive nodes (bottom right) represent the opposite character, being mainly influenced by the system, yet themselves not strongly influencing. They are useful as indicators of the current state.

Buffering nodes (bottom left) serve as inert volume, mainly not participating in the dynamics of the system.

Finally, the so-called ‘critical’ section (top right) comprises risky positions, likely to cause instabilities. These influence the system as strongly as they are influenced by the system. Carefully note that the term ‘critical’ used here is different from the definition of the coupling χ^i,j${\hat \chi _{i,j}}$ that forms the adjacency matrix. The nodes located in this area strongly contribute to the unstable character of a system since any modification at these points tends to initiate further modifications, instead of reducing consequences and, hence, calming the system down.

The Q index Qi(m)=ASi(m)/PSi(m)$Q_i^{(m)} = AS_i^{(m)}/PS_i^{(m)}$ stands for the angle differentiating between an active and reactive character, while the P index Pi(m)=ASi(m)⋅PSi(m)$P_i^{(m)} = AS_i^{(m)} \cdot PS_i^{(m)}$ distinguishes between inert buffering characters and being critical.

Against the background of a respective cross impact analysis, any organisational system can be assessed correctly and be modified accordingly to become manageable.

Complex and therefore unpredictable behaviour is obviously dictated by critical nodes, which are determinedly initiated by causal loops. These are not necessarily direct causal loops but, according to the understanding of cross impact analysis, reflect the overall tendency to return to volatility if modified in a blurred way all over the system. Nevertheless, these nodes are sensitive to their input and strongly effectuate consequences. Therefore, it is necessary to subject them to respective local control systems that compensate for their sensitivity and stabilise the output. Thereby, likewise-treated nodes will move from the critical position to the buffering or reactive range.

Besides this effect, the recursive character of the system is reduced inevitably, and the system becomes manageable. The final indicator for a stable system is given by vanishing recursiveness of all nodes and respectively by the vanishing trace of the cumulated adjacency matrix: 14 Trχ^i,j(m)=∑​iχ^i,i(m)≃0.

Ideally, in the end, all nodes are either purely active or reactive if not completely decoupled and are therefore buffering.

As soon as all loops are eliminated successfully, a purely causal, i.e. a rankable structure, is left. Using the most basic approach, one source and one sink are available, expanding and converging causally with interoperability or impact ξ = ζ over the ranks r = 0…Γ, as depicted in Figure 5.

Fig. 5.

Then, the AS of each node is AS_i(r) = ξ^Γ−r, rising potentially with the ranks as all consecutive nodes throughout the expanding tree structure are equally affected.

The PS is likewise PS_i(r) = ξ^r, representing the influential structure of the converging tree.

This purely causal structure leads to a distribution of node characters, which clearly signals the causality, as shown in Figure 6:

Fig. 6.

The remaining degree of criticality δ derived from the causal structure is given by the origin cutoff share of the diagonal, which is elaborated from the AS and respective PS at Γ/2: 15 ASi(Γ/2)=PSi(Γ/2)=ξΓ/2. $$A{S_i}({\rm{\Gamma }}/2) = P{S_i}({\rm{\Gamma }}/2) = {\xi ^{{\rm{\Gamma }}/2}}.$$

Normalising, we obtain 16 AS(N)i(Γ/2)=PS(N)i(Γ/2)=ξΓ/ξΓ/2=ξΓ/2, $$A{S^{(N)}}_i({\rm{\Gamma }}/2) = P{S^{(N)}}_i({\rm{\Gamma }}/2) = {\xi ^{\rm{\Gamma }}}/{\xi ^{{\rm{\Gamma }}/2}} = {\xi ^{{\rm{\Gamma }}/2}},$$ and, hence, for the relative share δ of the diagonal, we get 17 δ=2/AS(N)2+PS(N)2=2/ξΓ+ξΓ=2/2ξΓ=ξ−Γ=ξ−Γ/2. \begin{array}{*{35}{l}} \delta =\sqrt{2}/\sqrt{A{{S}^{(N)2}}+P{{S}^{(N)2}}} & =\sqrt{2}/\sqrt{{{\xi }^{\text{ }\!\!\Gamma\!\!\text{ }}}+{{\xi }^{\text{ }\!\!\Gamma\!\!\text{ }}}}\\ {} & =\sqrt{2/2{{\xi }^{\text{ }\!\!\Gamma\!\!\text{ }}}}=\sqrt{{{\xi }^{-\text{ }\!\!\Gamma\!\!\text{ }}}}\\ {} & ={{\xi }^{-\text{ }\!\!\Gamma\!\!\text{ }/2}}.\\ \end{array}

Clearly, with substantial networks (ξ > 1.3), the remaining criticality given by the causal system is neglectable, with longer causal chains (Γ ≈ 30…90) being even more so (Figure 7).

Fig. 7.

Causal systems with more active nodes and respective reactive indicators can easily be modelled by operating a number of identical networks in parallel cumulating the number of nodes per rank. This approach models a similar number of source nodes and sink nodes. As long as the interoperability is still described by the average number of in-ties or out-ties, nothing else is to be observed. Then, only the number of nodes per rank is scaled, not their location on the AS/PS graph. Therefore, the remaining criticality is not affected.

Hence, we state that the remaining causal structure after eliminating all loops in order to create a manageable system does not contribute significantly to the complexity. Thus, aiming at the least possible criticality converts the original organisational system into a manageable system, operated by the remaining most active nodes while the operation is indicated by the remaining most reactive nodes.