**4. Discussion**

Our results show that (1) legal pluralism is present due to the evolution and convergence of multiple formal (e.g., water, agrarian, and municipal laws, indigenous peoples rights, and environmental law) and informal (e.g., indigenous institutions such as those related to community use and customs, such as a municipal agen<sup>t</sup> or water user assembly) institutions that co-exist at the local level. Even in the same legal system that regulates property rights over those of water, overlaps between national and state institutions at the local level are present along with concerns regarding the recognition of municipalities as responsible for the water supply system. (2) Legal pluralism has generated a grea<sup>t</sup> diversity of formal and informal stakeholders that are structured across multiple levels and are involved in various ways in the first (operative) and second orders of governance (institutional arrangements) of the water supply system for domestic use. (3) Diversity is associated with the four different modes of governance that exist and operate at the local level. The governance modes are determined by interactions (cross-level or within the jurisdictional level) between stakeholders in the first and second governance orders. They include the non-nested community-based mode, nested community-based mode, nested hybrid mode, and hierarchical–bureaucratic mode (municipality). (4) The municipality is nested by institutional design, unlike all indigenous communities and their respective modes (e.g., non-nested community-based mode) that manage the water system for domestic use through a municipal agen<sup>t</sup> or water committee, and water user assembly. (5) The diversity and creation of governance modes in the water supply system for domestic use in Oaxaca and possibly in the rest of Mexico result from the seven different trajectories of institutional change. (6) The seven trajectories of institutional change result from nestedness and the interplay between local water institutions (e.g., municipalities and indigenous communities) and national and state governmen<sup>t</sup> institutions during the centralization, decentralization, and current periods. Most of the diversity of stakeholders and institutions at the local level, modes of water governance, and trajectories of institutional change do not correspond to a single centralized plan but to the interplay of different stakeholders and institutions over time to secure water for households in the Mixtecan Alta region in Oaxaca, Mexico, and probably in the rest of the country.

These findings are consistent with empirical research, and in the case of nestedness and institutional change, they contribute new elements to our understanding of governance and institutional evolution [11]. The legal pluralism in water managemen<sup>t</sup> in this study is consistent with what has been reported in previous studies regarding the growing legal pluralism in many countries due to the decentralization of water managemen<sup>t</sup> [17–19], the existence of a multi-level process regarding the human right to water and sanitation that operates from the global to the local levels [20], the link between land and water rights in rural communities [18], and the recent recognition of indigenous rights and their traditional institutions [66]. Although it has not been viewed from a legal plural perspective, the diversity of stakeholders and governance modes agrees with what has been reported regarding the increasing number of stakeholders and novel institutional arrangements in the arena of water governance [67–69]. A novel institutional arrangemen<sup>t</sup> in the form of a hybrid mode has been suggested as being more likely to be present than other modes, such as the hierarchical–bureaucratic, market, or network modes [70]. The other modes of governance identified as non-nested community-based and nested community-based modes can be encased in the network mode [71]. However, we highlighted the critical role of the community and nestedness that reflects how the mode is affected by the system in which it is immersed [33]. Identifying nested and non-nested modes of governance allows us to identify new elements that shed light on how multi-level governance works. For example, the multi-level governance structure does not always imply a nesting of smaller organizations within larger organizations [72]. In the same structure, two subsystems can exist that function differently: one nested and another non-nested. From a functionalistic perspective, nestedness helps identify fragmented or coordinated governance in the water supply system [73]. We can observe that the existence of a non-nested mode of governance in the water supply system for domestic use in Oaxaca, Mexico, reflects the inability of national and state governmental institutions to reach specific communities, which implies a lack of multilevel coordination and a fragmented structure. It also gives us insight into the ability of a community to self-organize to meet the water demands of its inhabitants and to maintain both institutions and the household water supply over time. For its part, the trajectories of institutional change and the lack of change identified in this study contribute to filling the information gap regarding the longitudinal processes by which institutions are created and evolve [11], which were essential for explaining the structures of the modes of governance, particularly the emergence of hybrid modes.

From an overall perspective of the properties, legal pluralism, the diverse stakeholders and institutions, and nested or non-nested subsystems working in multilevel and dynamic properties due to institutional change establish the complexity of the governance of the water supply system for domestic use in Oaxaca. This complexity of the governance of the water supply system is confirmed if we consider the non-linear interactions (which are indirectly observed in the seven trajectories of institutional change) and emergence properties (new institutional arrangements such as those of hybrid modes) that were revealed due to the interrelatedness and complementarity between the diversity, nestedness, and dynamics properties. We empirically demonstrate that water governance becomes complex in structure and operation due to institutional evolution, with certain institutions aggregating, changing, adapting, and persisting over time while acting and interacting with stakeholders to supply water to people in different jurisdictional levels.

This study provides a replicable method to use the Water Governance Complexity Framework to understand the complexity of water systems. The framework considers diversity, nestedness, and dynamics at different scales (jurisdictional and temporal) and levels (national, state, and local) as well as periods of institutional change to address the complexity of water governance and the governance of the water supply system for domestic use in particular. This framework differs from the Kooiman IGF [14,39] by addressing complexity not only through a proxy (either legal pluralism [13] or nestedness [15], but as a property that encompasses diversity, nestedness, and dynamics. By themselves, each property provides an incomplete picture of water governance; however, taken together, they provide a more holistic understanding of the current structure and function of water governance, which can be complex. In the case of the governance of the domestic water supply system in Oaxaca, Mexico, we showed how this framework addresses the limitations of using a single variable or a set of separately viewed properties to understand water governance. Additionally, in contrast to the Kooiman framework that uses a more descriptive approach, we propose a variable-oriented approach that provides systematicity and replicability to describe the complexity of water governance in other regions. The Water Governance Complexity Framework joins recent efforts to advance our understanding of the past, present, and future of the institutions, and their interactions, and those of different frameworks, such as Power Polycentric Governance (PPG) [74] or a combination of PPG with other frameworks such as Institutional Analysis and Development (IAD) and the Socio-Ecological System (SES) Framework [11].
