2.3.2. Nestedness

A semi-structured interview was implemented with key stakeholders within the 13 communities to obtain data on cross-level interactions among stakeholders of different jurisdictional levels. Key stakeholders included two main groups: (1) current or past hydraulic network operators and (2) officials from the municipal council or municipal agency. These stakeholder groups were identified from the rapid survey. We identified the key stakeholders following the snowball method, which identified potential interviewees and then asked them for recommendations on whom to interview later [40]. The semi-structured interview contained a matrix in which the rows represented the ten activities belonging to the first and second governance orders (Table S2, Supplementary Materials). The columns represented the stakeholders directly or indirectly responsible for the water supply system for domestic use in the studied communities. We obtained a total of 79 semi-structured interviews (La Mexicana (4), Santa Cruz Capulalpam (4), San Francisco Teopan (5), El Enebro (7), San Antonio Abad (3), Santa Cruz Corunda (3), San Miguel Aztatla (7), Santiago Quiotepec (9), Santa Magdalena Jicotlán (10), Concepción

Buenavista (12), Santiago Ihuitlan Plumas (9), San Juan de los Cues (2), and Santiago Tepetlapa (4)).

Each stakeholder was characterized according to their respective jurisdictional level (Table S3, Supplementary Materials). Subsequently, the obtained matrix was analyzed in two ways. First, we conducted a descriptive analysis of the cross-level interactions reported by the interviewees from each community. Second, we implemented a metric analysis using the Nestedness based on the Overlap and Decreasing Fill (NODF) methodology proposed by Almeida-Neto et al. [41]. Recently, NODF has been used to analyze social and commercial networks [34]. According to Almeida-Neto et al. [41], NODF is based on two simple properties, decreasing fill (DF) and paired superposition, to calculate the entire nestedness of a binary matrix.

For this reason, the first matrix obtained in the first step with binomial presence (1) and absence (0) data (Table S3, Supplementary Materials) was split into two groups: one matrix for municipal seats (Table S4, Supplementary Materials) and another for municipal agencies (Table S5, Supplementary Materials). As mentioned earlier, municipal agencies are hierarchically subordinate to the municipal seat and should hypothetically be nested. The NODF analysis was carried out with the open-source online program NeD (Nestedness for Dummies) of the Joint Research Center (http://ecosoft.alwaysdata.net/ accessed on 15 February 2021) created by Strona et al. [42]. The NeD program provides information such as the nestedness index and the probability levels after comparing the matrix under evaluation with a certain number of null matrices. According to Ulrich and Gotelli [43], the null matrices can be obtained through five different null models: EE (equiprobable row totals and equiprobable column totals), CE (proportional row totals and proportional column totals), FE (fixed row totals and equiprobable column totals), EF (equiprobable row totals and fixed column totals), and FF (fixed row and fixed column totals). The null model chosen to test nesting significance is decisive with regard to the results obtained [42].

Finally, we compared the results obtained from both approximations to generate a complete analysis of nestedness and the advantages of each approximation. A descriptive approach allowed us to obtain an overview of the results without losing detail. For its part, an approximation based on a metric NODF can help shed light on whether it is nested and to what degree it is nested, decreasing the ambiguity of the descriptive approach.
