INTRODUCTION

Science seeks to understand how systems are organized and function. A key element of such understanding is identifying the relationships between causes and effects, also referred to as causation (Woodward 2016, Pearl and Mackenzie 2018). Claiming causal relationships is so commonplace in everyday life that scientists rarely ask themselves what causation actually means and on what basis we make causal claims. The word “because” is omnipresent: we have day and night “because” of the earth's rotation; the plant will not grow “because” it does not get enough water. Such causal claims are simple and direct, and such simple claims are hardwired into our language and thinking.

For explaining phenomena in complex systems, however, simple claims are often not sufficient (Meyfroidt 2016). This is the case for social-ecological systems (SES), which are composed of decision-making interacting agents, such as humans and organisms, and their environment (Ostrom 2009, Ferraro et al. 2018). The key building blocks of these systems, the agents, often behave in elusive ways. Causal relationships between variables in these systems are often nonlinear and can involve more than two variables. Moreover, in SES, social and ecological processes are intertwined. All this leads to complex systems in which single causal relationships are difficult, if not impossible, to disentangle (Schlüter et al. 2019).

The consequences for trying to understand and, thereupon, influence or manage such systems can be dire. Decisions based on erroneously understood causal relationships, or too simple or narrow mental models, can lead to unintended consequences and potentially disasters (Merton 1936, Sterman 2006, Levin et al. 2013). This happened, for example, during the 2008 global financial crisis and the collapse of Atlantic cod stocks (e.g., Frank et al. 2016, Sguotti et al. 2019). Factors complicating the construction of useful models of causation in SES include the diversity of their building blocks, adaptive behavior, positive and negative feedback loops, indirect, delayed or path-dependent effects, stochasticity, and the interaction of processes on different spatial, temporal, and organizational scales (e.g., Meyfroidt 2019, Elsawah et al. 2020).

Being aware that simple causal claims and single, linear causal chains are often insufficient to understand what happens in SES, researchers have sought to identify and communicate more elaborate causal relationships. As the primary sense of humans is the visual one, visualizations play a key role in this endeavor. Their main purpose is to represent complex systems and phenomena with different foci, from different angles, and at different levels of detail. By decomposing and depicting parts of the system and their relationships, visualizations are used to elucidate how these systems function or how the phenomena emerge. This may happen at different stages throughout the research process, from visualizing initial hypotheses to eventually consolidated causal findings (Sheredos et al. 2013). Thus, visualizations are widely used to comprehend and to communicate causal relationships in SES. However, visualizations may also constrain our ability to capture causation, in particular in complex and intertwined systems such as SES. As graphical representations of structures and dynamics, they translate into mental models that may often suggest a level of simplicity in causal relationships that does not mirror reality. Likewise, even if we have better understood complex causation, using, for example, statistical analyses or mechanistic simulation models, our visualizations might not be able to adequately represent this understanding.

Although visualizations play an important role in exploring and communicating causation in SES, their inherent assumptions, strengths and weaknesses are rarely reflected upon (Spiegelhalter et al. 2011). The way we visualize, and hence conceptualize, complex systems affects our understanding of these systems and the conclusions based on this understanding. For example, the conceptual model of what is known, unknown, or uncertain about an agricultural SES, and how different variables in that SES influence and respond to each other, is virtually inconceivable without visualization, and in turn, shapes the development of climate change mitigation and adaptation strategies (Bentley and Anandhi 2020).

We therefore seek to raise awareness of how widespread types of visualization can shape our ability to properly identify and express causal relationships and provide some ways forward for using visualizations in SES research. We first identify challenges for visualizing complex causation, followed by a presentation of common types of visualization and a discussion of their scope, potential, limitations, and their suitability to address the challenges. We then highlight examples of visualizations in the literature on SES, specifically attempts to visually capture complex causation. Finally, we formulate caveats and recommendations for future use of visualizations. Although comprehensively visualizing causation in a complex SES with a single figure may be impossible, creative solutions for elucidating and communicating this complexity exist and should further be developed.

CHALLENGES FOR VISUALIZATIONS OF COMPLEX CAUSAL RELATIONSHIPS

To systematically analyze the visualization types’ potential and limitations for presenting and explaining causal relationships, we identify key challenges for visualizing causal relationships in SES. Although there is overlap with the general obstacles for studying causation in complex systems (e.g., Preiser et al. 2018, Schlüter et al. 2019), we focus on the following challenges that are specific to the task of visualization:

1. Visualizing whether a relationship is causal. This challenge includes separating causation from mere covariance and visualizing confounding. Two variables covary if a change in one of them goes along with a change in the other, but the covariance does not reveal which variable is the cause and which is the effect. Even more important, they may also covary without a direct causal relationship if changes in both are caused by a third variable. These options are not mutually exclusive. Often, there is a causal relationship between two variables, but this relationship is confounded by a third variable, which affects both. This poses a problem for detecting whether two variables are causally related, and for visualizing it (Pearl and Mackenzie 2018).
2. Visualizing the characteristics of causal relationships. This challenge includes discriminating between positive and negative relationships. If two variables are cause and effect, increases in the first variable can increase (positive relationship) or decrease (negative relationship) the second variable. Moreover, the shape of their relationship is often nonlinear or may even change direction. It may also include discontinuities. Visualizing these characteristics of a causal relationship, especially in a quantitative manner, cannot be achieved by all visualization types.
3. Visualizing reciprocal causal relationships. This challenge includes illustrating that certain variables are causes and effects of each other. This is common in SES, for example, harvesting affects and is affected by the abundance of a target population or the decisions of suppliers affect and are affected by the market price. Such feedback may generate hysteresis or cyclic dynamics. Making these reciprocal relationships perceivable is a frequent demand when visualizing causation in complex systems.
4. Visualizing multiple causes. This challenge includes depicting that two or more factors, actors, processes or events can each lead to the same effect (alternative causes, or equifinality). Moreover, the relationship between a cause and its effect can be moderated by an additional factor or context, or it is the combination of causes that lets a certain effect emerge. The challenge of multiple causes also includes showing causal relationships at different resolutions of system elements. When two variables are causally related, a detailed analysis of their relationship at a higher resolution will often reveal that the relationship is indirect, meaning that the first variable has an effect, which itself is a cause of another effect, and so forth, until the second variable is affected. Visualizing these intermediate variables (mediators) and their relationship (causal chain) is challenging.
5. Visualizing temporal dynamics of causal relationships. This challenge includes showing that and how causal relationships change over time. It may also be that one or several causes lead to an effect with delay. If the temporal sequence of certain actions, processes, events, or the temporal development of a causal variable matter for the resulting effects, then these are legacy effects or path-dependent effects, which can be difficult to visualize. Finally, this challenge also includes displaying the occurrence of temporally discrete events or interventions and the consequences that follow them (Healy and Moody 2014).
6. Visualizing uncertainty about causal relationships. This challenge includes expressing uncertainty about causation and displaying stochastic relationships. Causal relationships between variables are often not deterministic, but changes in one or several variables change the probability of a certain effect to occur. Additionally, it is a challenge to visualize the uncertainty of a causal relationship that is depicted, be it uncertainty whether the relationship is causal at all or about the possible strength of an effect (Spiegelhalter et al. 2011, Hullman 2020).

TYPES OF VISUALIZATION OF CAUSATION

We present an overview of three main types of visualization of causation that are commonly used in the literature on SES and related fields, and we discuss each visualization type’s particular characteristics in the light of the challenges introduced above. On this basis, we assess whether a visualization type (1) cannot meet the challenge, (2) can partly meet the challenge, but requires specific adjustments, (3) broadly meets the challenge, or (4) is especially suited to meet the challenge (Table 1). The assignment to these categories is explained below and supported by a detailed consideration of how the different aspects of each challenge can be addressed by the different visualization types (Table A1.1, Append. 1). For illustrative purposes, we use a hypothetical SES as a main example (Box 1) and add examples from the literature where appropriate. It should also be noted that many visualizations mix these main types of visualization of causation and also may contain additional elements that do not fit into any of these types. We refer to examples of such combinations in below in the section “Visualizing Complex Causation in SES Research.”

Objects and Arrows

One of the most common ways of visualizing and explaining causation in complex systems is a diagram consisting of objects and arrows connecting them (e.g., Fig. 1). It visually represents a phenomenon in a system by its organized parts and relationships, proposing how these interact to bring about the phenomenon (Sheredos et al. 2013). When used for representing causal relationships, an arrow from object A to object B depicts that a change of A causes a change of B. If there is another opposite arrow from B to A (or the arrow is double-headed), the causation is reciprocal: there is a feedback between A and B, as a change of B also causes a change of A. These meanings of objects and arrows are very general and flexible. Yet they require a conceptualization of the system for which the causal relationships are visualized. In particular, the objects must be defined. They can be entities in the system (e.g., agents or other components), but also state variables or processes. It is also possible to connect different kinds of objects in one diagram.

The selection of objects and arrows to be included has important implicit consequences. The objects already determine which claims about causal relationships in the system are possible based on the diagram. The arrows depict which of these relationships are actually considered. Any further causal relationships the objects might be involved in, and any further objects not shown, are ignored. All details within an object are usually ignored too, i.e., the higher-resolution objects it is composed of and the causal relationships between them. This simplified representation of the real complex system (e.g., Fig. 1A) helps to focus only on elements and relationships that are considered essential for a given question (Starfield 1990, Grimm and Railsback 2005). But it is crucial to be aware of these decisions, and they should be made clear. Often they are not explicitly stated, but taken for granted with the diagram of objects and arrows. However, they guide and constrain all subsequent efforts based on this visualization. The diagram represents, so to speak, a certain worldview and leaves out elements that might in fact be important but are ignored in this worldview. This is not only a decision of where to draw the boundaries and how to choose the resolution of the SES representation. Also, what kind of objects are depicted in a diagram matters. In ecology, a good example is whether ecosystems are perceived either as being composed of organisms or of compartments containing energy and nutrients. This decision strongly influences for which real-world phenomena a causal understanding can be achieved (Grimm et al. 2017).

Potential. This visualization type can represent multiple causes by several arrows leading to an object. Thus, it can also indicate that a phenomenon emerges from the influence and interaction of these multiple causes. The visualization can also represent multiple effects by several arrows starting from an object. In addition to direct feedback between two objects (cf. above), indirect loops of causal relationships involving more than two objects can be visualized. An arrow starting from and looping back to an object can represent internal feedback. The causal relationship indicated by an arrow can be used to represent temporal order (i.e., termination of object 1 causes the start of object 2; e.g., in flowcharts). It can also represent flows of energy, matter, or information. The direction and strength of causal relationships can be visualized by using different widths or styles of arrows, or by labeling them. In any case, the absence of an arrow allows for a clear and even quantitative statement: there is no direct causal relationship between objects 1 and 2.

Limitations. Due to their versatility, visualizations with objects and arrows are used in many ways. This freedom can also be a drawback leading to confusion. Any intended meaning of specific shapes and styles of objects and arrows (e.g., Fig. 1A) should therefore be well explained in the figure captions. The meaning of arrows can remain unclear as they do not necessarily represent causal relationships. If they do, the possibilities of expressing different characteristics of these relationships are clearly limited. Even if the specific appearance of arrows can be varied to show differences, all these specifications are not inherent to the visualization. They might root implicitly in a presupposed common understanding of their meaning in the visualized context, for instance because this is established in a specific scientific discipline or methodological community. Preferably, they should be explicitly communicated in addition to the diagram, in particular with regard to the fact that SES research is interdisciplinary. Moreover, a single arrow does not specify the underlying mechanism(s) of causation. Additional objects involved in such mechanisms and any details within objects are not visualized according to the selected resolution or worldview. They are therefore also ignored. However, for showing these detailed objects and their relationships, such as causal chains, the same form of objects and arrows can be used too (e.g., Fig. 1B). This offers an important option for understanding and visualizing causal mechanisms.

Three subtypes of diagrams with objects and arrows are particularly relevant for causation in SES: conceptual diagrams, causal diagrams, and network diagrams.

Conceptual diagrams

Visualization by objects and arrows is very popular for conceptual diagrams, which are highly generic and flexible. They are ubiquitous in use for SES representations, visualizing conceptual models of the systems, but also frameworks and approaches to study them. Conceptual diagrams do not require strict formal rules. They may contain just a few objects and arrows (e.g., Fig. 1A) or many. To take into account the complexity of SES, different kinds of arrows, different kinds of objects (e.g., representing factors, actors, other system components, processes), multiple colors, layers, labels or pictures can be included, and diagrams can be nested (e.g., Lindkvist et al. 2020; Fig. A3.1, Append. 3). However, to be used meaningfully and unambiguously, these differences require careful explanation. Moreover, although very helpful for visualizing multiple causes and discriminating causal from non-causal relationships, the high flexibility comes at the cost of lower potential for conveying more specific or quantitative information, for instance on the shape, strength, or temporal changes of causal relationships (Table 1). As the numbers of objects and arrows increase, it can become difficult to grasp which causal relationships are indicated, and it becomes likely that only selected relationships will be grasped, ignoring how they are embedded and interact with additional system elements. Attempts to reflect the actual complexity of an SES can lead to a conceptual diagram containing plenty of objects and arrows in an almost arbitrary manner (i.e., everything is linked to everything). Although each object and link in such a diagram may be justified and have a meaning, the visualization is then of limited use for specifying and disentangling complex causal relationships. Nonetheless, it can be useful for demonstrating this complexity, showing that phenomena are intertwined and that simple, mono-causal thinking is inadequate (e.g., Walker et al. 2009; Fig. A3.2, Append. 3).

Formal causal diagrams

Certain diagrams of objects and arrows follow a more specific formalism to represent causal relationships, for instance the dot and arrow format (e.g., Fig. 2A). Using such causal diagrams to visualize what we assume (or know) about the causal relationships in an SES makes these assumptions about the system more explicit and transparent, and allows inferences about their consequences. Causal diagrams in the dot and arrow format, with dots representing variables and the additional restriction that only unidirectional arrows and no cycles are allowed, can be used as tools for causal inference (Pearl and Mackenzie 2018, Lederer et al. 2019). To this end, the paths between dots can be traced, and common challenges for causal inference can be identified because they go along with characteristic structures in causal diagrams. For example, for a given arrow between two variables, a backdoor path is a path that leads to both of them from a third variable, which is therefore a confounder of the relationship. An indirect path connects the two variables in the same direction as the direct arrow, but via a third variable, which is therefore a mediator (Textor et al. 2016, Lederer et al. 2019). Such paths can be embedded in relationships involving many more than just three variables. Detecting them, as well as more complicated generic structures with specific meaning for causal analysis, is greatly facilitated by causal diagrams. Thus, the visualization serves as a robust and powerful tool that provides the basis for well-justified decisions whether certain variables must or must not be controlled for in causal inference. Without visualization, this would soon become much more difficult with an increasing number of variables and causal relationships between them (Pearl and Mackenzie 2018).

Although causal diagrams can be used for depicting stochastic causal relationships, the stochasticity itself is not visualized (Table 1). This means that it is not clear whether an arrow indicates, for example, that changes in one variable will sometimes, often, or always lead to changes in another variable that is linked by an arrow. Or an arrow connecting two events need not mean that the first event always causes the second. It may just modify the chances for the second event to occur.

Causal diagrams are also used to visualize structural equation models (SEM), a popular method for analyzing statistical associations between variables that then serve for causal interpretation (Shipley 2000, Grace 2006, Asah 2008, Fan et al. 2016). In these diagrams, in addition to arrows showing the direction, path coefficients indicate the strength and sign of relationships between variables (e.g., Palomares et al. 1998; Fig. A3.3, Append. 3). When applying SEM, the relationships that are included (i.e., the structure of the diagram) can be varied to test different models and select the preferred one based on fitting to data. Thus, the causal relationships that form the underlying conceptual model of the system would not be determined a priori but as a result of the system analysis (Eisenhauer et al. 2015). However, it is important to notice that the causal relationships cannot be derived from the associations between variables alone. Rather, these relationships need to be provided to the SEM as causal assumptions, and subsequent findings rely on careful interpretation (Bollen and Pearl 2013).

Causal loop diagrams are another popular type of visualizing causal relationships (e.g., Fig. 2B). They are a key tool in systems dynamics approaches to SES research (Elsawah et al. 2017, Radosavljevic et al. 2020). In these diagrams, arrows are labeled with “+” or “-” signs to indicate positive or negative relationships between the connected variables, respectively. The particular focus of the diagrams is on closed loops involving two or more variables. If the number of negative signs along a loop is even, the loop is reinforcing, meaning that increases (decreases) in one variable feed back positively through the loop and further increase (decrease) that variable. If the number of negative signs along the loop is odd, the loop is balancing, meaning that increases (decreases) in one variable feed back negatively through the loop and decrease (increase) the variable. These features make causal loop diagrams highly suitable to visualize and characterize reciprocal causal relationships (Table 1).

However, when combining stocks and flows (as variables) in a causal loop diagram, the visualization is prone to misinterpretation because the “+” and “-” labels can have two different meanings: additive or proportional change. An additive arrow denotes that the first variable adds to the second variable (or, for the negative sign, subtracts from that variable). But two variables linked by a positive arrow do not necessarily change in the same direction (and vice versa for a negative arrow). For instance, fish reproduction adds to the fish population (Fig. 2B), but it is possible that fish reproduction decreases whereas the fish population increases (e.g., when the population approaches its maximum capacity). For proportional arrows, the two variables do always change in the same (or opposite) direction and thus are positively (or negatively) correlated (see Richardson 1997, Lane 2008 for details on this ambiguity of causal loop diagrams). Hence, thorough reflection and clarification of these meanings is strongly advocated when using causal loop diagrams.

Network diagrams

Network diagrams visualize objects and their pairwise relations as networks of nodes (objects) and edges (lines linking objects) between them (e.g., Fig. 3). The networks (also called graphs) can be used as abstract descriptions of structural aspects of studied systems, including SES (Borgatti et al. 2009, Dale 2017, Will et al. 2020). The nodes and edges can have flexible meaning. For instance, nodes may represent specific organizational units, such as individuals, populations, species, or spatial areas. Although nodes typically represent elements of the same type, this need not always be the case. For example, actor-network theory focuses on associations between different types of elements to investigate, among other things, how social processes influence a studied phenomenon (Latour 2005, Langley and Tsoukas 2016). Edges in network diagrams may represent proximity, flows, (potential) interactions or any other kind of associations between the elements. They can be undirected (edges only) or directed (edges with arrows), and they can have weights that, for example, express the distance between objects or costs, durations, or intensities of flows or interactions. This flexibility can lead to ambiguity. For example, when visualizing foodwebs with network diagrams, arrows can be used to represent predation (directed from predator to prey), flows of biomass (from prey to predator), or reciprocal causation between predator and prey abundance (bidirectional arrows). Edges without arrows are also common, which can represent any of these options.

Network representations of SES focus on structural aspects of the systems and thereby implicitly assume that interaction structure plays a key role in the systems’ causations (Borgatti et al. 2009, Scott 2011, Levine et al. 2017). To analyze this structure, graph theory provides a variety of tools and metrics to assess, among others, the spatial or functional connectance, density, nestedness, vulnerability, or node centrality of networks (Bollobas 1998, Butts 2009, Thébault and Fontaine 2010). The structure that is visualized in a network diagram often determines the framework for identifying causal relationships, for example, the potential interactions. But it is not equivalent to the actual processes operating in an SES, for example, the realized interactions.

Networks can be used in two ways for deciphering causation in an SES. On the one hand, the structure has certain causes, that is, properties and dynamics of the system elements that led to the structure. On the other hand, the structure also has consequences as it enables, facilitates, or constrains processes in the system. In both cases, the network structure and its metrics can be regarded as patterns, and these patterns hint at the system’s causal relationships (Bodin et al. 2019). However, such causal implications of the network structure are not conveyed by the visualization itself. They require interpretation and additional analyses and explanations. Nonetheless, the causal understanding is considerably supported by network diagrams, and its derivation can be straightforward. A network diagram of a foodweb implies that an abundant predator population causes a pressure on its prey population. A weighted network diagram of trading routes makes visible that closing a certain route could cause higher costs for trading partners using that route and higher usage of alternative routes. Here, the weights also serve to characterize the causal relationships.

X-Y Plots

This very common visualization type comprises two-dimensional line, point, bar, or similar plots relating two variables X and Y (e.g., Fig. 4). In this way, an X-Y plot reduces system complexity to just two chosen factors. The plot displays how different values of X are associated with different values of Y. Given that Y is not constant, this means that changes in X are related to changes in Y. Very often, this visualization type is interpreted as indicating causation, meaning that changes in X cause changes in Y. This suggested causal relationship is mirrored by a common terminology, calling X the independent and Y the dependent variable. X-Y plots are appealing as they convey a simple message, which is easy to grasp and memorize. However, they also strongly evoke mono-causal thinking, ignoring complexity and context. X-Y plots often suggest a simplicity and generality that do not exist in reality.

There are exceptions of X-Y plots for which the visualized relationship is clearly not meant to indicate causation. The independent variable may be a mere criterion for discriminating different cases, such as the names of different agents or places, or the ranks of different items according to their associated Y value (e.g., species rank-abundance plots, May et al. 2016; Fig. A3.4, Append. 3). Also time series, plots in which time is the independent variable on the X axis (e.g., Fig. 4A), do not imply that time is the direct cause for change of the dependent variable (although time is usually involved in such change).

Potential. This visualization type is standard and extremely useful for depicting the relationship between one potential cause and one effect. In X-Y plots, the detailed characteristics of such a relationship can be made visible right away, such as its direction, strength, nonlinearity, or non-monotony, and also critical thresholds (Table 1). The relationship can be shown at high resolution of the variables’ values. Variations, uncertainty, or confidence can be incorporated, for instance by individual values underlying a mean (Fig. 4C), error bars, intervals around the plotted curve, or symbols and labels for the results of statistical tests (e.g., tests whether data from different treatments have a common mean). In time series, the time points/periods of occurrence of events/treatments/interventions can be highlighted. This facilitates inspection whether those changes, which are not plotted themselves, are related to changes in the plotted Y variable. For the same purpose, one can plot two time series, one for the (potential) causal and one for the effect variable, or even more to consider multiple effects. Generally, using multiple lines/points/bars differing in style or color can overcome the restriction to only one Y variable (sometimes also using two different Y axes). This strategy can be used to visualize different instances of an additional variable (cf. “Multiple Plots of the Same Visualization Type” below). But within one plot, these potentially dependent Y variables all need to be related to the same independent X variable.

Limitations. Apart from depicting the relationship to only one (potential) cause, the most important limitation of X-Y plots is that it is not possible to visually discriminate causation from mere covariance (Table 1). For instance, two variables might show a strong relationship not because X is the cause of Y, but because both are affected by a common cause C that confounds their relationship but is not shown. This is crucial and makes the plot amenable to mis- or over-interpretation. It adds to the problem that depicted relationships, even if they are causal, can be modified in complex systems by additional factors. Information about such factors and context, as well as mechanistic interpretations of causation behind depicted relationships, is not conveyed by the plot itself. These aspects require additional explanation. The same is true for underlying statistical models and assumptions, if uncertainty is included in the visualization.

It is also possible to plot the values of an independent variable X together with the change of that variable (dX/dt as Y variable). This allows visualizing the feedback of a variable X on itself (e.g., Fig. 4D). However, it also separates this self-feedback from its context and suggests ignoring the additional causes for changes of the X variable.

To visualize the reciprocal relationships between two variables, phase space plots can be used. These X-Y plots show the trajectories of combinations of two state variables (value pairs of X and Y), denoting changes over time in the phase space of possible states of a system (e.g., Fig. 5A). They usually imply that X and Y are causally related in both directions. In addition to visualizing possible combinations of X and Y, a trajectory also shows that for a given value of one variable, various values of the other variable are possible.

When X depicts an environmental driver and Y depicts a state variable, the effect that the driver has on the system state can depend on the current state of that system (e.g., Fig. 5B). Thus, both X and Y have an effect on changes in Y, leading to alternative stable states for the same value of the driver variable X. There may also be tipping points (Milkoreit et al. 2018), which means that the state variable changes abruptly when the driver exceeds a certain threshold. Even if the driver variable changes back, potentially very far from the threshold, the state variable does not reach the same values as before. It is possible that two tipping points exist, meaning that the state variable changes back (abruptly) only if the driver exceeds another threshold. This phenomenon of path-dependence causing alternative stable states (hysteresis) is common for SES, and X-Y plots are popular for visualizing it (May 1977, Scheffer et al. 2001, Hughes et al. 2013, Sguotti et al. 2019). Typically, the trajectories of the different stable states are connected with dashed or dotted lines to visualize that the corresponding combinations of X and Y values belong to unstable states (cf. Fig. 5B).

X-Y-Z Plots

This visualization type includes three variables. Here, X and Y are two independent variables, and Z is the dependent variable shown for different combinations of values of X and Y (e.g., Fig. 6). This is realized, for example, by two-dimensional contour or raster plots, or by three-dimensional surface or point plots. Thus, two potential causes and their effects on Z can be visualized together. This is an important step toward visualizing complex relationships in the system represented. The downside of this added complexity is that the plots can be more difficult to grasp compared with X-Y plots. And similar to X-Y plots, there remains the pitfall of perceiving depicted relationships as causal even though this may not be the case (Table 1).

Potential. The visualization of two independent variables enables inspecting and presenting effects that emerge from their interaction. The characteristics of their relationship to the dependent variable get visible, including nonlinearity or critical thresholds. Even highly multifaceted relationships can be quantitatively shown. Similar values of Z for different combinations of X and Y values show that different regimes are similar in terms of their associated effect on Z. This can point to trade-offs between two causes and to options for buffering or compensating for negative effects of one factor by changing a second factor. In particular, the second factor can be a moderator or a variable representing the context and, thus, the dependence of the relationship between X and Z on this second factor Y is visualized. If time is the Y variable, changes of a relationship over time are depicted, although time is not considered a direct cause for changes of the dependent variable (cf. above). X-Y-Z plots can also be used to visualize the possible combinations of three variables that describe a system state and may all affect each other (e.g., Radosavljevic et al. 2020; Fig. A3.5, Append. 3). Or, comparable to X-Y plots (Fig. 5B), they can visualize that and how the relationship between two variables changes from bistability to a single stable state depending on the value of a third variable (the “cusp” model in catastrophe theory, e.g., Petraitis and Dudgeon 2016, Sguotti et al. 2019; Fig. A3.6, Append. 3).

Limitations. Also in X-Y-Z plots, causation cannot be visually discriminated from mere covariance. Moreover, it can be difficult to accurately discern the characteristics of relationships. Transferring visual differences of colors into quantitative differences of Z values is susceptible to errors (Fig. 6B). Exactly determining the positions of points or a surface in a three-dimensional space, which must nevertheless be plotted as a two-dimensional figure, is not always possible (Fig. 6A). Visualizing uncertainty about relationships is difficult and usually requires an additional plot, for example of the variation of Z values for different combinations of X and Y values. If time is chosen as the second independent variable Y, the benefit of showing the temporal development of the relationship between X and Z comes at the cost of not showing the interaction with a second potential cause.

VISUALIZING COMPLEX CAUSATION IN SES RESEARCH

The presented characteristics, potentials, and limitations of the different visualization types provide a sound basis for choosing the appropriate visualization of causation in an SES depending on the intended purpose. Nonetheless, this means that a decision needs to be taken as to which—often substantial—aspects of the complexity of the SES will be disregarded and which aspects will be brought into focus. For example, recent advances developed ways to visualize multiple and complex causal relationships that bring about an emergent phenomenon of interest in an SES, such as a fisheries collapse, through elaborate conceptual diagrams with objects and arrows (Fig. 7). A distinctive feature here is that several levels of organization are visualized: from single actors and ecological entities over the action situations they participate in as well as networks of action situations, to the overall emergent phenomenon. Implicitly, such a figure also communicates that (and why) the phenomenon is complex as it is produced by multiple interacting processes. The approach to visualize networks of action situations can be applied to a specific case, for example the hypothetical SES used for the DemoViz model (Fig. 8, cf. Box 1). Based on such visualizations, the analytical framework can be used to develop hypotheses about causal mechanisms that generate SES phenomena (Schlüter et al. 2019, Orach and Schlüter 2021). These possible explanations can then be further examined through empirical research or modeling.

If the focus of a visualization, however, lies in supporting causal inference of relationships between variables in an SES, more formal causal diagrams are often the appropriate choice (Fig. 2). On the other hand, the focus may also lie in presenting specific causal relationships, ignoring the wider SES context they are embedded in for the sake of visualizing the particular shape, tipping points, or reciprocity of a relationship in an X-Y plot (Figs. 4, 5) or, for instance, interactions of two causal variables in an X-Y-Z plot (Fig. 6).

Multiple Plots of the Same Visualization Type

An important option for widening the focus and visualizing complex causation is bringing together several plots of the same type in one figure. This allows increasing the number of causes to be shown, the number of effects, or both. For example, multiple X-Y plots in one figure depicting the same relationship between one causal and one effect variable for different instances of a second cause (i.e., different contexts) can show how two causes interact (Fig. 9, different columns). At the same time, the precise quantitative visualization of the depicted relationship provided by X-Y plots is maintained. Moreover, depicting the relationships between one causal and several effect variables in multiple X-Y plots further adds complementary information to these relationships, respectively (Fig. 9, different rows). This may, for instance, visualize trade-offs between different effects of changing one cause. The principle of multiple X-Y plots can be extended, for example, by using the X axes, different columns, different rows, and different line colors to show the effects and interactions of four different causal variables (e.g., Banitz 2019; Fig. A3.7, Append. 3). However, it needs to be mentioned that the data needed for such comprehensive visualizations of an effect variable for all combinations of relevant values of the causal variables are rarely available, except for computational models of SES, where conditions can easily be controlled and resulting system state variables obtained (Schulze et al. 2017, Schlüter et al. 2019).

The same principle of multiple plots can also be used with other visualization types. With X-Y-Z plots that already show the interaction of two causes, a third cause or context can be added (Fig. 10). Similarly, this can be achieved by nested rows and columns for multiple causes in a table and visualizing their combined effects by colors (e.g., Ferguson et al. 2020; Fig. A3.8, Append. 3). Combining multiple network diagrams in one figure can be used to show how networks, and thus their associated causes and effects (cf. “Network Diagrams” above), change in different contexts (Fig. 3; see also Bodin 2017; Fig. A3.9, Append. 3).

Combining Visualization Types

Combining different visualization types in one plot is another common and useful option for visualizing complex causation in SES. This strategy allows benefiting from the respective strengths of each visualization type (cf. “Types of Visualization of Causation” above). For example, X-Y plots showing the temporal development of an SES state and of underlying causal variables can be amended by objects and arrows diagrams showing the internal SES organization for different points in time (Fig. 11). Thus, the state of system entities and the causal relationships between them are integrated into the visualization and support causal understanding of the depicted temporal developments. Vice versa, X-Y plots can be inserted in a diagram of objects and arrows to illustrate the specific shape of the visualized relationships between different system variables (e.g., Banitz et al. 2020; Fig. A3.10, Append. 3). In another example from the SES literature, the combination of X-Y plots with objects and arrows visualizes causal relationships between trophic groups in a modeled marine ecosystem and quantitatively shows certain preconditions for these trophic interactions to occur as well as certain effects they have (van Leeuwen et al. 2013; Fig. A3.11, Append. 3).

It is also possible to combine different X-Y-Z plots and thus visualize the interaction of two causes both in three dimensions and—in a more precise quantitative manner—in two dimensions (Fig. 12). However, a disadvantage of such nonstandard multifaceted visualizations is that they may easily get very complicated. They require extensive explanation, and grasping the depicted causal relationships is not always intuitive. If used, readers should therefore always be guided through the visualizations in order to fully explain the intended causal claims.

The visualizations presented so far can also be combined with additional types. One example are set diagrams that visualize sets by closed areas (e.g., circles). Set diagrams are widely and flexibly used, for instance, to visualize an SES conceptualization for studying poverty traps, according to which economy is part of a society, and this society is part of the biosphere (e.g., Lade et al. 2017; Fig. A3.12, Append. 3; see also Folke et al. 2016). In Venn diagrams, overlap of two areas represents overlap of sets, that is, elements that belong to both sets. The same principle works for more than two sets. Integrating a Venn diagram in an objects and arrows visualization can be used to illustrate complex causation in an SES, for instance, the contribution of different causes to an effect. A typical situation is that different factors alone are insufficient to cause an effect. However, they necessarily belong to a particular combination of factors that causes the effect. As this combination is not the only way to bring about the effect, it is sufficient but unnecessary (e.g., Fig. 13A). Such “insufficient but necessary parts of unnecessary but sufficient” (INUS) conditions were defined by Mackie (1965) and are employed to disentangle complex causation in social and social-ecological research (Mahoney 2008, Morgan 2013, Meyfroidt 2016, Carlson et al. 2018). The combination of a Venn diagram with a network diagram can also be used to integrate visualizing subsets of potential factors that are included in an analysis and visualizing causal relationships between these factors (e.g., Fig. 13B).

CONCLUSION

We have highlighted that visualizing causation in SES poses significant challenges, but there are promising ways to overcome them. A prerequisite for using visualizations to present and explain complex SES phenomena is to be aware of the key visualization challenges (“Challenges for Visualizations of Complex Causal Relationships” above) and the strengths and weaknesses of the different types (“Types of Visualization of Causation” above). In summary, the high flexibility of diagrams of objects and arrows is excellent for visualizing complex causation, but can easily lead to ambiguities, so consistency and explanation are critical. Moreover, the biggest limitations of these diagrams are their poor capability to show the characteristics of singular causal relationships and to visualize temporal dynamics and uncertainty. X-Y plots and X-Y-Z plots are more suitable for these challenges, but strongly limited in discriminating causal from noncausal relationships and visualizing multiple causes (Tablea 1, A1.1). Our assessment enables the informed selection of the appropriate visualization type depending on the research question, the purpose (e.g., visualizing assumptions, visualizing results, or visualizing a conceptual framework of a causal analysis), the knowledge about causal relationships in the SES, and the properties and part(s) of the system that shall be the focus of a figure (cf. specific exemplary recommendations at the beginning of “Visualizing Complex Causation in SES Research” above).

Ideally, figures speak for themselves and their messages are straightforward and fully comprehensible. But a correct and complete understanding of a figure may also rely on certain implicit assumptions. These may be familiar to researchers with a common scientific background. However, tacitly assuming specific knowledge and interpretation along with a visualization risks confusing readers who are unfamiliar with them, even more so because SES research is highly interdisciplinary. For these reasons, information that is needed to correctly understand a figure, or that is needed to communicate an aspect of causal analysis not conveyed by the figure, should be explicitly added with explanatory text. The respective caption should aim at clearly guiding readers through a figure, almost as one would do in presentations. This guidance can be critical to get the meaning and causal implications of a visualization unambiguously understood. One practical option to support this is to use numbered labels and, in the figure caption, make readers grasp the figure sequentially by “walking” from element to element (e.g. Lindkvist et al. 2020; Fig. A3.1, Append. 3).

For diagrams of objects and arrows, such guidance can also help readers to mentally animate visualizations and, thereby, comprehend how an SES phenomenon is caused by the operation of several processes. This works particularly well for sequential processes. But also in the case of simultaneous processes that let a complex phenomenon emerge, the visual summary of these processes can support causal reasoning and provide a sound basis for the design of analytical tools such as computational models (Jones and Wolkenhauer 2012, Sheredos et al. 2013).

Besides relying on mental animation, a recent additional option is to create actual animations of visualizations (e.g., Anderson 2013, Grossman et al. 2016). Similar to their use during a presentation, animations can be used to visually tell a story about how an SES is organized and functions, in other words, to develop a causal narrative (cf. Shepherd and Suddaby 2017). This option can be appropriate for all three main types of visualization presented. Applied to X-Y and X-Y-Z plots, animations can also be used to visualize the changes of relationships over time, overcoming the challenge of visualizing temporal dynamics while retaining other strengths, such as the precise characterization of relationships (cf. Table 1). Most journals allow for animated visualizations in the supplement, whereas including them in the main manuscript is rarely possible. However, there are attempts to encourage the latter and keep the manuscript still comprehensible when printed on paper (e.g., with additional static figures as placeholders; Grossman et al. 2016).

Most visualizations used in science aim for simplicity so that the key message of a figure is easy to grasp and memorize. For complex causation, however, this design principle can be counterproductive as it suggests a level of simplicity that plainly does not exist. We therefore see a particularly promising way for overcoming the challenges of visualizing complex causation in SES (“Challenges for Visualizations of Complex Causal Relationships” above) in the combination of multiple visualizations in one figure.

Multiple plots of the same visualization type (“Multiple plots of the same visualization type” above) can be used to show specific causal relationships between variables in different contexts or for different values of additional variables. Thus, they visualize interactions and common emergent effects of multiple causes. The approach can also be used to visualize multiple effects of the same cause(s) within one figure (e.g., Fig. 9). As this principle of adding causes and effects by multiple instances of the same visualization works for all types, their specific potential for visualizing different aspects of complex causation in SES can nonetheless be exploited.

An alternative way to exploit this potential are well-designed combinations of different visualization types in one plot (“Combining visualization types” above). If one is careful not to overcomplicate these combinations, they can provide otherwise impossible comprehensive pictures that simultaneously tackle several of the presented visualization challenges. This considerably helps to disclose the complexity of SES phenomena and prevents unduly simple causal interpretations. However, especially when using creative and less common ways of combining different visualization types, careful and consistent design and thorough guidance in accompanying captions are crucial. Otherwise, they risk being of little value: although containing a large amount of heterogeneous information, they do not serve their main purpose of conveying causal insights to a broad interdisciplinary readership.

Thus, we assert that visualizing causation in complex SES remains an often difficult task and simultaneously addressing many, let alone all, of the identified challenges is virtually unfeasible. A single figure will not capture and characterize all causal relationships that are relevant for the fate and functioning of an SES. Simple figures remain important and useful, for example when putting the visualization focus on subsystems, on specific aspects of complex phenomena, on selected causes and effects. But they should be complemented by figures with combinations of visualizations to remind ourselves that causation in SES is more complex than our common way of thinking in terms of simple causal relationships and linear causal chains—and more complex than common visualization types may suggest. By being aware of this pitfall, and of the strengths and weaknesses of the different types of visualization, we will be well equipped to use visualizations to do justice to the complexity of SES and support a better and more comprehensive causal understanding.

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