Parameter analysis for sigmoid and hyperbolic transfer functions of fuzzy cognitive maps

Fuzzy cognitive maps (FCM) have recently gained ground in many engineering applications, mainly because they allow stakeholder engagement in reduced-form complex systems representation and modelling. They provide a pictorial form of systems, consisting of nodes (concepts) and node interconnections (weights), and perform system simulations for various input combinations. Due to their simplicity and quasi-quantitative nature, they can be easily used with and by non-experts. However, these features come with the price of ambiguity in output: recent literature indicates that changes in selected FCM parameters yield considerably different outcomes. Furthermore, it is not a priori known whether an FCM simulation would reach a fixed, unique final state (fixed point). There are cases where infinite, chaotic, or cyclic behaviour (non-convergence) hinders the inference process, and literature shows that the primary culprit lies in a parameter determining the steepness of the most common transfer functions, which determine the state vector of the system during FCM simulations. To address ambiguity in FCM outcomes, we propose a certain range for the value of this parameter, λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\uplambda }$$\end{document}, which is dependent on the FCM layout, for the case of the log-sigmoid and hyperbolic tangent transfer functions. The analysis of this paper is illustrated through a novel software application, In-Cognitive, which allows non-experts to define the FCM layout via a Graphical User Interface and then perform FCM simulations given various inputs. The proposed methodology and developed software are validated against a real-world energy policy-related problem in Greece, drawn from the literature.

Several other studies have also employed FCMs in environmental and ecological problems (Hobbs et al. 2002;Fons et al. 2004;Xirogiannis et al. 2004;Çelik et al. 2005;Mendoza and Prabhu 2006;Kok 2009;Ceccato 2012;Soler et al. 2012;Cakmak et al. 2013;Gray et al. 2014) or energy policy and efficiency projects (Ghaderi et al. 2012;Kyriakarakos et al. 2012;Huang et al. 2013;Reckien 2014;Hsueh 2015;Karavas et al. 2015;Amer et al. 2016;Olazabal and Pascual 2016;Nikas and Doukas 2016;Nikas et al. 2020Nikas et al. , 2019Antosiewicz et al. 2020;Doukas and Nikas 2020). As a policy support tool, FCMs have particularly gained ground in such energy and climate policy applications, partly due to stakeholders encountering difficulties in understanding, or being excluded from, state-of-the-art policy support frameworks, like energy-and climate-economic modelling tools (Nikas and Doukas 2016). Due to limited model complexity and reliance on quantitative data, FCMs have proliferated as a policy support tool, especially at the local level, allowing policymakers to reflect their understanding of a problem domain in a structured manner and act based on it (Özesmi and Özesmi 2004). They have also been proposed as an effective way to bridge the science-policy gap and engage stakeholders in environmental modelling processes (van Vliet et al. 2010).
Broadly speaking, however, the simplicity and attractiveness of FCMs across application areas and domains lies in their ability to capture the perception of a system in graphical representations consisting of concepts (nodes) and interconnections (weights) among these nodes, which are characterised by transfer functions determining the state vector of the system in simulation (Tsadiras 2008). However, the topology of nodes and weights, on the one hand, and the transfer function, on the other, are formulated differently: the former are typically defined by the non-expert decision makers (stakeholders) of the case study, while the latter are selected by the analysts. In essence, like stakeholders, the analysts are required to take decisions, which are both relevant to the analysis and critical to its results.
Parameter analysis for sigmoid and hyperbolic transfer… However, despite the plethora of applications, the FCM theory is still inconsistently applied in the literature (Felix et al. 2019). Notably, there seems not to exist a common ground among researchers regarding one of its core features, the type of transfer function used to drive simulations. Various monotonic functions have been used in literature, such as step, sigmoidal, ramp, and linear functions (e.g. Hobbs et al. 2002;Mendoza and Prabhu 2006) and (Soler et al. 2012)), with each one potentially yielding markedly different results. This diversity in FCM outcomes imposes barriers to the final inference procedure. In the absence of common criteria on selecting the transfer function, analysts should carefully justify their choice based on the physical interpretation of each application, which however is not common practice (Nápoles et al. 2018).
In this paper, we propose the use of two transfer functions, namely the lognormal (sigmoid) and hyperbolic tangent functions. We also introduce a criterion to define their parameter -i.e., their steepness-toward standardising the selection of the FCM transfer function. The observations and analysis in this study build on previous studies Kottas et al. 2010;Lee and Kwon 2010;Knight et al. 2014;, which provided bounds for parameter . Depending on the value, the sigmoid and hyperbolic tangent functions yield a unique final state of nodes for a given set of input values (i.e., a fixed state vector). However, by providing a domain of parameter , they only restrict values so they do not yield chaotic, ambiguous FCM responses. The selection of parameter is thus still subject to the subjective selection of the analyst within the provided bounds.
Despite providing final node values with clear ordering, the linear transfer function suffers from the undesired condition of chaotic final states (Knight et al. 2014). Additionally, although the sigmoid and hyperbolic tangent functions-given parameters within specific bounds provided in the literature-do not exhibit such behaviour, they often result in final node values close to one another, thereby hindering clear inference. To tackle these barriers, we propose an improved version of sigmoid and hyperbolic tangent transfer functions, which is active within an almost-linear region. We illustrate this methodology through a Python web software application "In-Cognitive" that we developed in the context of this study. This novel application features a user-friendly Graphical User Interface (GUI) that allows various stakeholders to define the FCM layout (e.g., nodes, weight interconnections, input/initial state vector, etc.), and execute scenario simulations before reaching a final state vector. The value of parameter is calculated endogenously, based on the proposed analysis.
Section 2 provides a theoretical background (notations and definition) of fuzzy cognitive mapping. In Sect. 3, we provide an FCM analysis without considering input nodes (all nodes may change throughout the simulation iterations): we first present and discuss the state-of-the-art bounds of parameter , before introducing a framework to define bounds/value of parameter. Section 4 performs similar analysis for the case of FCMs with given input nodes that remain steady and unaffected by other nodes throughout the simulation. The "In-Cognitive" software application is presented in Sect. 5 and then validated in Sect. 6 in a case study drawn from the literature. Section 7 finally concludes the research, highlighting key takeaways and discussing prospects.

FCM background and layout notations
An FCM consists of n concepts (nodes), C i ∶ i = 1, 2, … , n , linked to one another through a weight, w ij , which describes the degree of influence of C j over C i within [−1, 1] . When w ij < 0 (negative causality), C i decreases for an increase in C j . When w ij > 0 (positive causality), C i increases for an increase in C j . Finally, when w ij = 0 there is no relationship (nor adjacency) between C j and C i . Figure 1 illustrates how node C i is connected through weights with all the other nodes.
Input or steady nodes, steady nodes hereafter, influence but are not influenced by other nodes (i.e., they have outbound but no inbound links). The nodes which are neither steady nodes nor output nodes are called intermediate nodes. In Fig. 2, an FCM of 5 nodes is presented: nodes C 1 and C 4 (solid circles) are steady nodes, while nodes C 2 ,  Parameter analysis for sigmoid and hyperbolic transfer… C 3 and C 5 (dotted circles) are intermediate nodes. For a real-world example, the reader is referred to Sect. 6.
The matrix consisting of all FCM weights w ij is called the weight matrix, W . Equation (1) shows the weight matrix of the FCM illustrated in Fig. 2.

FCM equilibrium analysis with no steady nodes
To analyse FCM outcomes, we express node interactions using a mathematical formulation that should be iterative through time. If A k i is the value of node i at time instance k , the iterative interconnection expression for each node is where f (⋅) is the transfer function and d i the feedback coefficient ∈ [0, 1] . The latter indicates the dependency of node C on its starting value in each iteration. The transfer function could be any function. However, to avoid chaotic FCM behaviours, the transfer function values should be bounded. Usually, the log-sigmoid and hyperbolic tangent functions are used. The values of the former span within [0,1] and of the latter within [− 1,1]. The general form of the log-sigmoid function is whereas the corresponding one of the hyperbolic tangent is As discussed in (Knight et al. 2014), the selection of the transfer function should be justified based on the given application and, therefore, there is no standard criterion to yield the best fitted transfer function; Knight et al. also show that different types of transfer functions yield different FCM final states and thus different inferences. To tackle this issue, Knight et al. proposed the execution of various simulations, with each one having different transfer functions. They then compared results to identify common patterns: nodes, whose final values are relatively high (low) for all executions are considered the most (least) important FCM concepts.

State-of-the-art bounds of parameter λ of transfer functions
Different types of transfer functions yield different inferences for the iterative function of Eq.
(2)-similarly, different parameters of the same transfer function (see Eqs. (3) and (4)) may yield different FCM final states and therefore different inferences. Some of the various final states of Eq.
(2) might be chaotic, infinite, or periodic (Knight et al. 2014). These states are not fit for any kind of inferences. Therefore, it is necessary to ensure the FCM converges and explore whether a given layout can be stabilised around a final steady state after several iterations of Eq.
(2). Under certain conditions and a given combination of (a) the weight matrix, (b) the number of nodes, and (c) the parameters of the transfer function, it is possible to reach a final, unique fixed vector regardless of the initial values A 0 i , ∀i ∈ [0, n] . It should be noted that, for any of these combinations, the final state is not necessarily the same.
It should be noted that, in previous research Kottas et al. 2010;Lee and Kwon 2010;Knight et al. 2014;, the authors provided conditions under which the existence and uniqueness of solutions of concept values (see Eq. (2)) are guaranteed. In Knight et al. (2014), the authors provided a maximum bound of parameter for the log-sigmoid transfer function (see Eq. (3)), regardless of the structure and contents of the weight matrix; they also showed that, when the FCM is equipped with a step function (i.e., the limit state of log-sigmoid function when → ∞ ), the uniqueness and existence of a fixed solution is not guaranteed as well as that, as the examined FCM grows in size ( n → ∞ , where n the number of FCM nodes), the parameter to guarantee the existence and the uniqueness of a final fixed solution gets smaller ( → 0 ). However, the provided upper bounds of parameter were strict enough, rendering unnecessary the consideration of the layout of a given FCM (i.e., the weigh matrix). In Kottas et al. (2010), the authors provided a less strict upper bound conditions, under which there is a fixed-point solution when = 1 , for both Eqs. (3) and (4), depending also on the weight matrix/structure. Consequently, the conditions discussed in Knight et al (2014) are less restrictive in case = 1 . In , the authors extended the results of Kottas et al. (2010) for all > 0 and finally reached a bound of for all log-sigmoid and hyperbolic tangent-equipped FCM implementations.
In the case of the log-sigmoid transfer function, the bound guaranteeing the existence and uniqueness of FCM final state is found to be : Whilst, in the case of the hyperbolic tangent transfer function, the bound is: where W is the weight matrix and ‖⋅‖ F the Frobenius norm, such that:

Parameter analysis for sigmoid and hyperbolic transfer…
It should be noted that the above conditions are sufficient, but not necessary for an FCM to have one and only one fixed point for a given parameter : there could be cases where an FCM has a unique fixed-point solution if is greater than the above upper bounds (see Eqs. (5) and (6)). Lee and Kwon (2010) reached similar conclusions for the log-sigmoid transfer function, by using a different approach (Lyapunov criteria).

Remarks on transfer functions
Among many transfer functions, the linear function-more specifically the identity function-yields lucid inferences, because the distance among outcomes is clearer than other transfer functions (Knight et al. 2014). Based on the structure of Eq. (2), the linear function features no distortion during the calculation of A k+1 i from previous iterative values A k j . The value of the transfer function is always proportional to the argument of Eq.
(2) through all iterations. This property of linear functions gives room to lucid inferences; the distance among the final node values is sufficient to distinguish each node final state from the others. However, the linear transfer function comes with certain caveats. Often, during iterations, the A k+1 i values are constantly increasing (decreasing) reaching infinite (minus infinite) values. Despite FCMs equipped with linear transfer functions exhibiting a closer-to-reality increment (decrement), the above extreme case behaviour is restrictive for the execution of the iterative procedure (Eq. (2)). For this reason, the analysts tend to impose restrictions on A k+1 i values by using bounded transfer functions-i.e., the log-sigmoid and hyperbolic tangent function. Both are odd functions around the y = 0.5 and y = 0 axis, respectively, and exhibit an almost-linear behaviour in a region close to these axes. This linearity gives them resemblance to a linear transfer function for a sufficient interval. The non-linear regions on the tails of these functions are used to represent the large A k+1 j values. The y = 0 and y = 1 bounds ( y ± 1 bounds) are used to represent the infinite (or close to infinity) A k+1 j values for the case of the log-sigmoid (hyperbolic tangent) transfer function (see Fig. 3). The heavy curved regions close to these bounds are mainly responsible for the distortion (nonproportionality) of A k+1 j values; arguments close to infinite tend to map to almost the same A k+1 j values (i.e., no sufficient distance among nodes' final values). However, bounded transfer functions exhibit shortcomings as well. Not only do the non-linear regions introduce distortion, but the existence of bounds ( y = 0 and y = 1 , or y ± 1 ) could yield final states that are either chaotic or limit cycles (i.e., a period function of A i ). This could happen when the x k i arguments of Eq.
(2)  Parameter analysis for sigmoid and hyperbolic transfer… simultaneously large enough (i.e., the transfer function is almost a step function). In that case, the A i values are more likely to oscillate with an amplitude having extreme values close to the bounds of the transfer function. Such oscillations in node values are almost chaotic and insufficient for inferences. As a rule of thumb, the FCM analyst should therefore avoid large values of the parameter. Another undesirable condition occurs when parameter is too small (almost zero). When → 0 , the transfer function is almost flat (see Fig. 3) in all ranges of x k i values and, therefore, all A i values conclude to almost the same value. This state is stable; however, it cannot reach a conclusion because there is no lucid ordering among A i values. Concluding, FCM analysts should avoid both small and large values of parameter . Below we propose an upper bound of parameter based on the above remarks. These bounds can thereafter be combined with the bounds of Eqs. (5) and (6).

Proposed bounds for parameter
The proposed methodology refers to FCMs equipped with the log-sigmoid or hyperbolic tangent transfer function. It is based on the conclusion that → 0 and → ∞ are two undesired regions of parameter and the assertion that linear transfer functions are preferable, if they do not yield chaotic, cyclic, or infinite final states (see Sect. 3.2). The main idea behind the proposed methodology is that both log-sigmoid and hyperbolic tangent transfer functions have a region that is almost linear (desired region). We provide certain conditions, under which all A k+1 i values fall within that region. These conditions are then used to provide bounds of parameter . By operating in the almost linear region, we get a combination of benefits of both linear and bounded transfer functions (see Sect. 3.2); mainly, we avoid the distortion that the curved segments in the tails of the bounded transfer functions introduce (Fig. 4).
However, working in the almost linear region comes at a cost. This region is not as large as the interval between the bounds of the log-sigmoid or hyperbolic tangent function; therefore, the final A k+1 i values are usually close to one another. To avoid this, we propose a normalisation procedure (see Sect. 3.4).

The almost linear region of the log-sigmoid and hyperbolic tangent functions
For the almost linear region to be 'active' for all nodes during all FCM iterations, all arguments x k i 's (Eq. (8)) must not lie in the region of the transfer function tails. The desired region where x k i values lie on is hereafter called 'almost linear region' (see Fig. 4); all x values bounded by −x * , +x * , where f ��� (±x * ) = 0 , which is where the f ′′ has local maxima (see Fig. 4). We call −x * and +x * "turning points," hereafter.

3
The parameter is always positive, as well as f � s (x) (Kottas et al. 2010). Then, after equating the {⋅} factor of Eq. (9) with zero, we conclude to Fig. 4a). After using Eq.
(3), we finally get where s is parameter of the log-sigmoid transfer function. Similarly, from Eq. (10) we get that where h is parameter of the hyperbolic tangent transfer function.
(10) The almost linear region is odd with respect to the x = 0 axis and parameter is always positive; therefore, we can rewrite Eqs. (11) and (12) as and 3.3.2 Bounds of parameter λ witch quarantine that the almost linear region is always "active" Equations (13) and (14) When the transfer function is log-sigmoid, all state values are positive, that is 0 < A k j < 1 . If we need to restrict the A k j values to the "almost linear region" (see Fig. 4 −0.577 ≤ A k j ≤ 0.577 . Therefore, the possible maximum value for node C i could be achieved if all w + ij are multiplied by +0.577 ( −0.577 ) and all w − ij by −0.577 ( +0.577 ). Equivalently, the maximum value of . The latter factor is equal to the infinite norm of the weight matrix, W , which is equal to the maximum absolute row sum of W . Therefore, The ordering among parameters ′ s , * s and ′ h , * h is not constant for all applications. As such, we cannot a priori conclude to the existence and uniqueness of the FCM fixed-point if < * s or < * h . Figure 5 illustrates the bounds ′ s and * s for weight matrices W 1 (n×n) = J n and W 2 (n×n) , where W 1 is a matrix of ones and W 2 is a square matrix having three elements per row, all of which are equal to one and aligned around its diagonal. W 2 could represent an FCM whose nodes are only connected with their three adjacent nodes ( W 2 (1×1) and W 2 (2×2) are equal to matrices of ones because their size is smaller than three). From Fig. 5 should be preferred to guarantee that there would be a unique fixed point. These remarks can be formulated as follows: for the log-sigmoid transfer function, and for the hyperbolic tangent transfer function.
Once the final bound is estimated (Eq. (21) or Eq. (22)), we should choose a parameter that is as close to the final bound as possible, because parameter must not get extremely low values, → 0 (see Sect. 3.2). This value is the infimum value of bounds in Eq. (21) and Eq. (22). For the sake of simplicity, we propose as close to infimum value, which is derived after rounding the final bound of Eq. (21) or Eq. (22) at the third decimal digit.

Normalisation of final state values
The proposed bounds squash all concept values during all iterations within [0.211, 0.789] and [−0.577, 0.577] for log-sigmoid and hyperbolic tangent transfer functions, respectively. This may end up to final output values close to one another. Consequently, the relative distance among these values might be unclear. To return to the [0, 1] or [−1, 1] interval for log-sigmoid and hyperbolic tangent, respectively (normalised intervals, hereafter), we need to multiply all these values with a factor so that the A k j values are within these normalised intervals. All concept values, during all iterations, lie in the almost linear region and are, therefore, within the following intervals: and for log-sigmoid and hyperbolic tangent, respectively. For the case of a log-sigmoid FCM, to normalise A k j we should express them in terms of the y = 0.5 line. Recall that the log-sigmoid function is an odd function with respect to y = 0.5 . After subtracting 0.5 from Eq. (23) we get −0.289 ≤ A k j − 0.5 ≤ 0.289 . Equivalently, 2⋅0.289 + 0.5 ≤ 1 . We conclude: Therefore, to normalise the final values of all FCM nodes, we need to subtract −0.211 for any of them and then divide them with 0.578.
Similarly, for hyperbolic tangent FCMs, the necessary transformation to normalise the final values of FCM nodes is the multiplication with 1.733:

FCM equilibrium analysis with steady nodes
In Sect. 3, we presented conditions, under which an FCM with no input/steady nodes has a unique solution ( < ′ s ) consisting of final A i values distinct enough ( < * s ) to yield lucid inferences. In case of FCM with steady nodes though, which is the case for scenario analysis (Nikas et al. 2019Antosiewicz et al. 2020) the unique equilibrium does not depend solely on the weight matrix and parameter , as it does in the case of FCMs with no input nodes; it also depends on the values of the steady nodes (external excitations). Therefore, we can achieve a variation of equilibria/responses by changing the excitation of steady nodes and simultaneously reassuring that we will not get a chaotic final state if we choose certain values of parameter similarly to Eqs. (21) and (22). To do so, we must express parameter with respect to the weight set of the non-steady nodes (Kottas et al. 2010).

Bounds of λ parameter when there are steady/input FCM nodes
Based on Kottas et al. (2010), the existence of equilibrium is guaranteed if Eqs. (5) and (6) are valid for the weight set of the non-steady nodes. First, we need to reconstruct the extended weight matrix, W , so that the first rows correspond to the steadynodes and the end rows to the non-steady nodes. That is The FCM illustrated in Fig. 2 with has now the following reconstructed extended weight matrix To identify the conditions, under which an FCM with steady nodes has equilibrium, Kottas et al. (2010) considered the case where = 1 . In this research, we propose the corresponding inequality ∀ ∈ ℝ . The mathematical proof follows similar steps as described in . Equation (30)

Normalisation of final state values when there are steady FCM nodes
The normalisation procedure when there are steady FCM nodes is like the one described in Sect. 3.4. However, it is applied only to the intermediate and output nodes and, for that reason, only the relative distance between the final values of intermediate and output nodes is suitable for inferences. Equivalently, there is no direct relationship between the qualitative values of input and intermediate or output node values. That is, if A 1 = 0.6 is the final value of input node C 1 and A 10 = 0.6 that of C 10 , we cannot conclude that both C 1 and C 10 exceed the same amount of deviation. Contrary, if, for example, A 7 = 0.6 is an intermediate or output node, we can conclude that it exceeds the same amount of variation with the output node C 10 . The above restriction is applied because the analysis of Sect. 3.3.2 is applied to the (29) W * = ⎡ ⎢ ⎢ ⎣ w 2,1 0 0 0 w 2,5 0 w 3,2 0 w 3,4 0 w 5,1 0 w 5,3 w 5,4 0

Software implementation: the "In-Cognitive" tool
There exist several software solutions for FCM design and simulation [see Nikas et al. (2019) and Tsadiras et al. (2021) for detailed accounts]. The In-Cognitive software tool 1 is a web-based interactive application for the creation, visualisation, and simulation of FCMs, featuring the methodology presented in Sects. 3 and 4. It is written in the Python programming language (Python 3.7.3) and based on the Bokeh Python library (Bokeh 2.4.0.). It consists of a client-side web GUI (front-end) and a web server (back-end), with the former exchanging information and queries with Python code/modules stored in the latter. The front-end uses the JavaScript and HTML/CSS technologies to implement the interaction procedure with the end-user (analyst or otherwise). Both JavaScript and HTML/CSS codes are automatically created by the Python Bokeh framework driven by Python scripts. The back-end is built on top of a Tornado Python web framework. In Fig. 6, we briefly illustrate the interaction between the front-end and back-end parts of the In-Cognitive application. As a main functionality, browsers request documents (contents of the web pages) and the server's Python code provides them. The user interacts with the content of documents and ask for services. The document catches these events and afterwards send out feedbacks to the server that listens to these request events. Figure 7 illustrates the developed GUI. It is divided into two subsections: the (a) FCM editor and display layout, and (b) the simulation outcome subsection. The enduser can easily interact with the FCM editor in order to introduce the FCM layout or edit an existing one and configure the FCM by defining the structure and parameters (i.e., node interconnections, weights, input node values/excitations, transfer function). Parameter is automatically calculated based on the analysis in Sects. 3, 4 and, thus, the end-user need not insert any specific value for this parameter. Finally, the end-user can also alter the format (e.g., size, colour, etc.) of the introduced FCM components (e.g., nodes, edges, etc.) to a preferable format and save afterwards the  Fig. 6 Description of In-Cognitive Python web application figure of the introduced FCM layout. In the GUI subsection (b), the outcomes of the FCM simulation executed on the server-side are presented in a user-friendly visualisation. There is also an integrated console, which displays useful information regarding the execution of the corresponding iterative FCM simulation (e.g., warnings, FCM layout information, etc.).
The main contribution of the In-Cognitive web application is the implementation of the methodology of selecting parameter as presented in this paper (Sects. 3 and 4). To our knowledge, all FCM software tools (e.g., Mohr 1997;Margaritis et al. 2002;Aguilar and Contreras 2010;Papaioannou et al. 2010;Cheah et al. 2011;Gray et al. 2013;De Franciscis 2014;Poczęta et al. 2015;Nápoles et al. 2017;Nikas et al. 2017Nikas et al. , 2019Tsadiras et al. 2021) do not contain any software module to select based on the FCM layout. Instead, parameter is considered a constant parameter usually equal to one (Nikas et al. 2019).

3
Parameter analysis for sigmoid and hyperbolic transfer…

Case study validation of the proposed framework and software
There is a plethora of studies applying the FCM theory in energy/climate policy (e.g., Nikas and Doukas 2016;Nikas et al. 2018Nikas et al. , 2019Nikas et al. , 2020Doukas and Nikas 2020). In Nikas et al. (2020), the authors proposed an FCM layout to identify the most pertinent implementation risks to the diffusion of new solar power before calculating the long-term socioeconomic impacts of wide-scale solar PV deployment in an energy system and macroeconomic analysis in Greece, building on the uncertainty space associated with the identified implementation risks. Table 1 briefly describes each node/concept of the FCM, while Table 6 in the Appendix presents the FCM layout in tabular format. Nodes C 1 to C 9 are the steady nodes. Nodes C 1 to C 5 correspond to the barriers of solar-based energy transition in Greece as suggested by the stakeholders (uncertainty drivers). Nodes C 6 to C 9 correspond to various policies (policy drivers). C 26 to C 30 are the output nodes (concepts under examination) and C 10 to C 25 are the intermediate nodes that change their values through iterations. Various value combinations of C 1 to C 5 are illustrated in Table 2, representing different socio-economic risk scenarios-i.e., socioeconomic paths (SP), hereafter. We adapt the following abbreviations regarding the various SPs (see Table 2): SP1: Sustainability, SP2: Middle of the road, SP3: Regional rivalry, SP4: Inequality and SP5: Fossil fuelled development. For each SP, four policies, P 1 , P 2 , P 3 and P 4 (see Table 3) are applied to explore their effect on the output nodes. Therefore, the following input combinations are applied to the introduced FCM: SP1_P1 to SP1_P4, SP2_P1 to SP2_P4, SP3_P1 to SP3_P4, SP4_P1 to SP4_ P4 and SP5_P1 to SP5_P4.
After applying the analysis of Sect. 4, the norms of matrix W * of FCM layout of Finally, from Eqs. (21) and (22) and In this example the smallest bounds for both s and h are equal to the proposed ones (see Sect. 3.3.2).
All simulations are performed and visualised in the "In-Cognitive" software application in the sub-sections below. In Sect. 6.1, we illustrate the results of FCM simulations for S2_P3 (S2: Middle of the road, P3: Wide-scale deployment of smart meters) when the FCM is equipped with hyperbolic tangent transfer function. The values of parameters vary so that we can reach to useful conclusion regarding the analysis of Sects. 3 and 4 (we include the proposed bound h = 0.421 as well). Moreover, the final concept values, A k i , is not normalised so that we can compare the results for various values. The normalisation procedure described in Sects. 3.4 and 4.2 is only applied to lambdas smaller than the proposed bounds, * s and * h . In contrast, the normalisation procedure and the proposed h = 0.421 are only applied in Sect. 6.2.  (Figs. 8,9); the arguments always lie in the almost-linear region. For = 1 (Fig. 10), a commonly used value in FCM simulations, the arguments are already out of the turning points and the A k i 's values have distortion due to the curved regions of the transfer function. We also observe that the greater the lambda parameter, the more arguments fall within the tails of the transfer function; consequently more A k i receive  Figs. 11,12). Due to this effect, the final output vector is dense around the ±1 region, making the inferences ambiguous (see Table 4 for = 10 to = 100 ). This observation is in accordance with the analysis in Sect. 3.2 where we concluded that the greater the lambda parameter ( → ∞ ), the closer to the step function the transfer function gets, and the final node values get close to ±1 (undesired condition). In this specific application, for h > � h = 0.657 (i.e., Table 4 for = 1 to = 100 ) the FCM concludes to a fixed-point despite, according to Axelrod et al. (2015), the existence of a fixed point not being guaranteed for h > ′ h . This means that, if we try different excitations (other than S2_P3), we may get chaotic FCM behaviour when h > � h = 0.657 . Finally, it is worth pointing out that the ordering of final output values is different when varies (Table 5), as expected by the analysis in Sect. 3-the variation refers to the stage before normalisation.  (8)) through all iterations are as in Fig. 9. We can see that, for all x k i and their corresponding A k i values, the almost linear region is much larger. This happens due to the given SP2_P3 excitation. Different excitation would yield different distribution of x k i values but all these distributions would fall within the almost linear region because h = 0.421 < * h ≈ 0.4211 . Finally, it should be noted that values do not fall within a narrow band region and therefore their ordering is clear and closer to a realistic representation of relative values of each node's deviation. This happens because, in the almost linear region, the deviations of A k i values are proportional to the deviations of x k i values (see Sect. 3.2).

Remarks and conclusions
We have proposed a framework for identifying a value for parameter for both logsigmoid and hyperbolic tangent FCM transfer functions. With the previous state-ofthe-art values, the transfer function was active for all possible f values. Given that both transfer functions have a curved region close to their tails (see Fig. 4) thereby creating distortion, final node values usually concluded to an overcrowded region (close to 0 and 1 or ±1 , respectively), hence unclear inference. To tackle this barrier, we proposed that transfer functions should operate in the almost linear region (see Fig. 4). The latter requirement yielded a certain bound for parameter . We also demonstrated why parameter should not be excessively large or small (see Sect. 3.2). Therefore, we reached the conclusion that parameter must be as close to the proposed bounds in Eqs. (21) and (22). The analysis was performed for FCMs with or without steady nodes. We also proposed a normalisation procedure so that outcomes are clear, distinct, and sufficient for inferences. Based on the proposed methodology, we furthermore developed a web software application written in Python, called "In-Cognitive", containing a user-friendly GUI that allows non-expert users to connect to the server, define the FCM layout (e.g., nodes, their weight interconnection, input state vector if any, etc.) and then request the results. The choice of parameter is taken endogenously based on Sects. 3 and 4. Finally, by using the "In-Cognitive" software application, we ran a simulation of an FCM layout drawn from a real-world application in the literature , validating the methodological takeaways of Sects. 3 and 4.
The parameter bounds that are proposed in this research aim to contribute to the hitherto ambiguity of FCM implementation and results, since different parameters yield different FCM outcomes and therefore different inferences. By providing an objective criterion to select a unique parameter we hope to contribute to further exploitation of FCM theory in research and policy-/decision-making.
A caveat of this study is that it focuses on a parameter of the transfer function that is defined by the analyst. On one hand, this means that other parameters  defined by the analyst have not been touched. For example, as a prospect of our research, the impact of the choice of transfer function (among sigmoid and hyperbolic tangent) on the robustness of the inference/results must be thoroughly examined. This also applies for parameter d i of the driver function (Eq. 2) and the  On the other hand, this caveat also means that the impact of aspects of the FCM model that are (largely) defined by the decision-makers, such as the FCM layout and the weight matrix, must also be further explored. On the latter, much research has been carried out in the form of learning algorithms; however, the extent to which the simulation outcomes change with regard to input data uncertainty (e.g., via Monte Carlo analysis in the weight matrix) is largely understudied. Future research should finally focus on improving the proposed normalisation procedure, which squashes node values to a smaller range based on the presented example, thereby rendering differences within the final state vector less distinct and therefore any conclusion harder.

Appendix
See Table 6.