The dynamics of the resource R follow a differential equation of the form 1R˙=S(R)-∑nEnR,G1,n,…,GM,n, where S represents the reproduction and recharge and En the rate of loss from extraction and exploitation by resource user n, which is itself a function of R and interventions (e.g., regulations, subsidies, infrastructure) Gm,n by decision center m to either support or reduce each user's extraction. In the example system, S represents the natural net gain to the reservoir after natural inflows and outflows that are not delivered to any users, and E1 the total amount that agricultural users are able to extract. The effect of the intervention 2Gm,n=Gm,nYm,F1,m,nX1,…,FN,m,nXN, is then a function of the capacity of corresponding decision center and of efforts Fk,m,nXk by each actor k to influence policies or the enforcement of these policies (see the section on actors' strategies for more details). In Fig. , F1,2 is an example of such an effort that could represent urban users advocating for increasing the conveyance efficiency of the infrastructure delivering their water. These nested functions integrate the resource state, actors' ability to access the resource, decision centers' efforts to intervene in their access, and, in turn, actors' efforts to influence these interventions.

The resource users' organizational capacity Xn is modeled by 3X˙n=BnEnR,G1,n,…,GM,n+QnAnR,P1,n,…,PM,n+∑kCk,n+Wk,n+Xk-∑kCk,n-Wk,n-Xk-LnXn, where Bn represents user n's gain in capacity motivated by their ability to extract En. The function Qn is the analogous gain in capacity based on their non-extractive access to the resource, An. Depending on how these relationships to the resource are parameterized, these gain terms can represent different responses to resource access. This gain can represent actors becoming more agitated due to lack of access to the resource and thus more motivated to dedicate time and resources towards engaging with the institutions that determine their access. It can also represent actors becoming more invested in ensuring access to the resource as their use of the resource, and the value associated with it, increases. This parameter therefore encapsulates the importance of the resource to the users and the productivity of the system in determining their likelihood to self-organize, as described by Ostrom's social–ecological systems framework . The function Pm,n is analogous to Gm,n, representing interventions in their resource access and similarly affected by actors' efforts Hk,m,nXk.

Ck,n+ represents their gain in capacity from collaboration with or support from other actors that may, for example, provide information or resources about the institutions affecting their resource access or help connect them to these institutions . They experience loss in capacity based on other actors' efforts to undermine them (Ck,n-), through, for example, intimidation, misinformation, or demobilizing messaging and framing (W1,2 in Fig. ). Actors also experience a loss in capacity from attrition or turnover (Ln) due to a gradual loss in interest and participation among actors or their switching attention to issues external to the model domain.

Non-government organizations include non-profits, outreach, advocacy organizations, and other non-government organizations that typically are more public-facing and formal institutions than resource user groups and may receive funding or grants from external actors (e.g., donations or grants). These organizations play an important role in fostering and supporting collective action . Non-government organizations are modeled similarly to resource users by an equation of the form 4X˙n=UnXn+∑kCk,n+Wk,n+Xk-∑kCk,n-Wk,n-Xk-LnXn, where Un represents a gain in capacity from external sources.

Like resource users, these organizations have an objective related to the resource and are strategic but have a gain term based on their own capacity, which allows them to secure external support, and do not have a gain term dependent on the resource, reflecting their more established and less reactionary nature.

Decision centers, which can also be thought of as public infrastructure providers or venues for decision-making, intervene in resource users' ability to extract or access the resource, whether through provision of infrastructure, funding, or regulation. Their capacity Ym is modeled by 5Y˙m=Im+(Ym,K1,m+X1,…,KN,m+XN)-Im-(Ym,K1,m-X1,…,KN,m-XN), where Im+ represents their gain in capacity and Im- represents their loss in capacity. Both of these terms are functions of a decision center's existing capacity as well as actors' efforts at venue shopping, in which actors attempt to move decision-making authority to venues that are more favorable to them. These efforts are represented by Kk,m+Xk for supporting a venue and Kk,m-Xk for undermining a venue. K1,1 in Fig. , for example, may represent agricultural users' efforts to undermine the authority of the infrastructure provider to withdraw water to deliver to urban water users.

In typical dynamical systems analysis, the functions in the equations above would then be assigned specific functional forms. However, generalized modeling is based on the recognition that computing steady states is computationally expensive, whereas determining stability around a given steady state is far less costly. Determining stability around a given steady state requires only the ability to parameterize the Jacobian, which is a linearization of the system at the steady state, and thus requires less information than specifying particular functional forms that would describe the system's evolution throughout its entire trajectory. Bypassing the need for functional forms is particularly useful in modeling social systems, where the functional forms of processes are difficult to quantify and highly uncertain. This approach allows for analyzing systems with a great degree of generality and without the computational constraints involved in modeling many different specific dynamical systems and has been used to analyze a wide variety of systems . Therefore, rather than assign specific functional forms and compute the steady state, the functions are normalized by the unknown steady state. For example, the normalized resource dynamics would be represented by 6r˙=S*R*s(r)-∑nEn*R*en(r,g1,n,…,gM,n), where S*, R*, etc., are the values of the corresponding functions or state variables at equilibrium, and s, xk, etc., represent the normalized functions or state variables. The normalization leads to the introduction of unknown factors S*/R* and En*/R*. However, these factors are constants and are treated as parameters, namely scale parameters. Scale parameters denote the magnitude of fluxes, such as turnover rates or the relative importance of different processes (Fig. ). We define ϕ:=S*/R*=∑nEn*/R*, which represents the overall turnover rate of the resource, and ψn:=(1/ϕ)(En*/R*), which represents the fraction of losses by each particular extractor at the steady state. The normalized resource dynamics can then be written as 7r˙=ϕs(r)-∑nψnen(r,g1,n,…,gM,n). An example of an entry of the Jacobian based on this equation can then be computed as 8∂r˙∂r=ϕ∂s∂r-∑nψn∂en∂r. The derivatives ∂s/∂r and ∂en/∂r are unknowns that are also treated as parameters, namely exponent parameters. These parameters are an indication of the sensitivity of the growth rate and the extraction rate to the resource state, respectively. In general, exponent parameters indicate the nonlinearity of a process at equilibrium. Once the Jacobian is parameterized, the stability can be determined by checking whether the real part of all eigenvalues is negative. Conceptually, this means that perturbations in the state variables close to the steady state will return to that steady state. Local stability therefore indicates that the system will return to a steady state under short-term shocks (e.g., a sudden change to an actor's political influence) but does not necessarily indicate how the system will respond to large perturbations from the steady state or long-term drivers that fundamentally change the system's functioning (e.g., altering how resource users benefit from or impact the resource).

A full derivation and description of all model parameters can be found in the Supplement.

Recognizing that actors in a governance system are strategic and self-organized, a quality that is unique among the complex systems for which stability has been studied in a systematic manner, the generalized model is coupled with an agent-based modeling component. Each actor allocates their limited organizational capacity among different actions in order to maximize their equilibrium extraction or access to the resource (or for non-government organizations, the access or extraction of another actor). Their strategies are thus subject to the constraint: 9∑k∑m|Fn,m,k|+|Hn,m,k|+∑k|Wn,k|+∑m|Kn,m|=1, where F, H, W, and K represent the proportion of their effort dedicated to lobbying or otherwise directly supporting or resisting policies, collaborating with or undermining other actors, or directly influencing the capacity of a decision center, respectively. In Fig. , for example, farmers divide their effort between undermining urban users' capacity (W1,2) and undermining the capacity of the infrastructure provider that conveys water to urban users and away from farmers (K1,1). The way actors allocate their effort among these actions is their strategy, which is calculated by finding a Nash equilibrium, in which no actor will want to unilaterally change their strategy.

While the generalized modeling approach means that the equilibrium extraction or access cannot be computed, the gradient of their extraction or access at the steady state can be calculated. As a simplified example of the method, take X˙=F(X,p), where p is a strategy parameter. To find how the steady-state X* depends on the strategy parameter, we can write this equation at the steady state with X* as a function of p: 100=F(X*(p),p). Taking the total derivative of both sides with respect to the strategy parameter gives 110=∂F∂XdX*dp+∂F∂p. We can then solve for 12dX*dp=-∂F∂X-1∂F∂p. In the full system, ∂F∂X-1 is the inverse of the Jacobian. Once we know how the steady state depends on the strategy parameters, it is straightforward to compute how the extraction or access depends on each strategy parameter. See the Supplement for the full calculation of the gradient.

A Nash equilibrium is calculated by computing the gradient of the equilibrium extraction or resource access and performing iterative steps of gradient descent for each actor in turn until the strategies converge. While there is no guarantee of a Nash equilibrium since the strategy space is not necessarily convex, the strategy optimization process ensures that even if optimality is not reached, actors are behaving in ways that are self-consistent and compatible with their goal of increasing their resource access. Modeling actors as behaving reasonably, if not necessarily rationally, ensures that the systems that are analyzed are feasible governance systems.

The strategy parameters computed by the agent-based modeling component and the sampled generalized parameters collectively provide all of the information needed to compute the stability of the system. Varying the generalized modeling parameters, as well as meta-parameters defining the number of each type of state variable and how densely connected they are, allows for exploring the stability of a wide variety of topological configurations and feedbacks among actors and decision centers.

We investigate how three types of features of complex governance correspond with stability: (1) the scale and exponent parameters, which are used to understand the importance of different processes and functional forms, as well as the importance of variance in these processes, representing, for example, heterogeneity in actors' response to resource access conditions, diversity in interventions, or inequity in actors' abilities to influence different governance processes; (2) the diversity of the system, indicated by the total number of actors and decision centers and how densely connected they are; and (3) the relative number of decision centers and actors, which is used to understand, for example, whether diverse stakeholder interests may have a different effect on stability than diversity in decision centers.

The parameter correlation results reveal that for a smaller system (Fig. a), stabilizing factors include greater capacity gains from external support and gains motivated by resource access conditions, as opposed to gains from collaboration, and greater losses based on attrition rather than being undermined. Conversely, a greater capacity gain from collaboration and capacity loss from undermining by other actors and a greater proportion of actors' efforts spent on collaboration or undermining are both destabilizing factors. This suggests that stronger interactions and greater interdependence among actors is destabilizing, while greater autonomy is stabilizing.

In both the smaller and larger systems (Fig. b), the strategy parameters emerge as important in determining stability. Since the strategies are computed rather than sampled as the other parameters are, the causal effect of the strategies on stability is not clear. However, the results suggest that a greater effort put toward influencing the capacity of decision centers, or venue shopping, corresponds with stability, while greater effort put into the other strategies corresponds with reduced stability. In the example system, agricultural users are engaging in venue shopping by reducing the infrastructure provider's influence over the infrastructure (K1,1); if there were other decision centers in the system, they may try to move that authority to a venue that favors agricultural interests. Venue shopping has gained interest as a possible mechanism through which less powerful actors can enact fundamental policy changes . This distinction suggests that venue shopping may arise as a desirable strategy in a different context than other political strategies or that it changes the system in a fundamentally different manner from other strategies and does so in a lasting manner.

To understand the effect of heterogeneity among actors' relationships to the resource or to institutions on stability, we look at the variation in the parameters that define, for example, their sensitivity to changes in resource accessibility, or their share of total resource extraction. We find that higher variation in the sensitivity of actors' extraction to governance is destabilizing. This variation corresponds with heterogeneity among resource users in terms of the ease with which their extraction or access can be monitored or regulated. It can also represent institutional diversity, in which decision centers pursue a variety of policies or approaches, which has been hypothesized to increase adaptive capacity . A higher variation in the sensitivity of actors' gain in organizing capacity to their resource access is also destabilizing. Differences in this parameter correspond to different relationships with resource use: actors with low resource requirements, particularly if they are not involved in a profit-driven activity, may experience the largest capacity gains when their ability to extract is low. In contrast, some actors may become more invested and gain greater resources with which to mobilize as their extraction increases. In the example system, for example, urban interests will likely become less engaged once they have sufficient access to water (an inverse relationship between capacity and resource access), whereas agricultural users, particularly industrial agriculture operations, might become less engaged once the available water, and thus profitability of farming, drops below a certain threshold. The presence of both of these relationships to the resource similarly signify heterogeneity and potentially inequity among actors, leading to a greater tendency for contestation and change.

The number of different groups in the system, whether actors or decision centers, has a strong effect on stability, while connectance has no noticeable effect on stability (Fig. ). This suggests that diversity in actors, a feature of complex and polycentric governance systems, is a destabilizing force. This is consistent with the idea that the inclusion of a greater diversity of actors in governance processes leads to greater flexibility and adaptability, as well as with findings for other complex systems such as ecosystems . The absence of an effect of connectance on stability, however, is in contrast to other complex systems where connectance is destabilizing . This may be because each of the interactions in these systems can influence processes by either increasing or decreasing their effect, unlike in natural systems, where interactions may all push the system in the same direction, leading to greater potential for destabilizing feedbacks. Thus, while greater connectivity in the form of stronger interactions is a destabilizing force, as found in the parameter correlation experiments (Fig. ), the presence or absence of interactions is not as important for determining stability in governance systems.

Finally, looking at the effect of the relative proportions of different entities reveals that stability is determined not just by total diversity but also the diversity in decision centers in particular (Fig. ). A greater number of decision centers is destabilizing, while a greater proportion of non-government organizations is stabilizing. The proportion of resource users does not have a strong effect on stability. Whether the resource users are extractive users of the resource also does not have an effect on stability (Fig. S3). This result thus supports that polycentrism causes governance systems to be more prone to change, likely because they offer more opportunities for actors to influence the system . Non-government organizations may have a stabilizing effect because of their role in supporting, and thus having aligning goals, with other actors in the system, reducing contestation and helping other actors develop longer-lasting and more durable institutions .