Unraveling Incommensurate Spatial Partitions: A Bipartite Graph Approach to School-Neighborhood Interactions and Their Impacts

Unraveling Incommensurate Spatial Partitions: A Bipartite Graph Approach to School-Neighborhood Interactions and Their Impacts Sergio J. Rey∗1, Elijah Knaap2, Zachary P. Neal3, and Clio Andris4 1San Diego State University 2San Diego State University 3Michigan State University 4Georgia Institute of Technology ∗Corresponding author email: [email protected] Abstract This paper investigates the challenges and opportunities arising from incommensurate spatial partitions (ISPs) in regional science and spatial econometrics, focusing on how processes with overlapping yet distinct boundaries, interact and influence each other. ISPs are prevalent in various domains, including housing markets, employment centers, voting districts, and educational institutions, often complicating spatial econometric modeling and analysis. Using the intersection of school catchment areas and neighborhoods as a primary case study, the paper introduces a novel methodological framework utilizing bipartite graphs. This approach reframes the relationship between different spatial units, allowing for the analysis of multi-process spatial contexts without needing harmonization of spatial supports. The paper also develops new spatial weights derived from the bipartite graph, facilitating both exploratory spatial data analysis and confirmatory spatial econometric modeling. These methods are illustrated through a case study of San Diego, California analyzing 198 neighborhoods and 370 public elementary school catchments. Introduction Consideration of the interdependencies between two, or more, processes with overlapping but not identical spatial partitions is an often encountered issue in regional science. Figure 1 contains a motivating example involving a set of geodemographic neighborhoods together with elementary school catchment boundaries. The incommensurate nature of the two partitions (schools and neighborhoods) complicates modeling efforts that seek to investigate how neighboorhoods influence schools (Garner & Raudenbush, 1991) as well as the question of how schools influence neighborhoods (Butler et al., 2022). Figure 1: Lemon Avenue Elementary School Catchment (red) intersecting three neighborhoods (filled polygons). Other catchments shown with green boundaries. The case of schools and neighborhoods is but one example of the problem of incommensurate spatial partitions (ISP). Other examples include the spatial mismatch between housing markets and employment centers 2 (Blumenberg & Siddiq, 2023), voting districts and neighborhoods (Kenny et al., 2023), ecological zones and regional economies (Anselin et al., 1990), as well as others. A number of solutions to the problem have been proposed in the literature (Goodchild et al., 1993). The dominant approach is to employ areal interpolation to transfer data from the polygons of one partition to those of the target partition (Logan et al., 2014). Closely related are interpolation approaches that aggregate the data to a higher level of geography (Martin et al., 2002). A third alternative is the use of a surface to model the data from both series (Bracken & Martin, 1995). Finally, lookup tables that assign values from one set of polygons to the other are sometimes adopted (Richards, 2014). This paper addresses two questions related to the problem of incommensurate geographical partitions in spatial analytical work. First, what are the challenges that multiple processes with incommensurate spatial partitions hold for exploratory spatial data analysis and spatial econometric modeling? Second, are there new approaches possible that exploit the opportunities that may be latent within these challenges? In the education literature, there are open questions as to how neighborhood, school, and district segregation relate to one another (Owens et al., 2016). The ISP problem reflected in Figure 1 has particular valence in this regard, as the incommensurate nature of most school and neighborhood boundaries has been under examined. The dominant approach to the ISP problem in the education literature has been to enforce harmonization of the two series (schools and neighborhoods) to the same partition through various areal interpolation methods. For example, Richards (2014) defines Voronoi polygons using the schools as generator points and then assigned census attributes to the resulting polygons based on containment of census tract centroids. Saporito (2017) also employs school-based Voronoi polygons, but uses areal interpolation to estimate family income and population in the resulting polygons. These approaches simplify the subsequent analysis of segregation over the imputed zones. Nevertheless, they present two constraints. First, interpolation may obscure complex spatial spillovers that work over the original partitions. Second, it may induce a form of spatial dependence in the interpolated series. Motivated by the need for more flexible and comprehensive approaches to model spatial processes with incommensurate spatial support, this paper makes the following contributions. First, we eschew the need for harmonization of the two series to the same spatial support by framing the relationship between spatial units from the two processes as a bipartite graph. Doing so affords the ability to rely on graph analytics to develop new measures summarizing the multi-process spatial context that have utility for comparative analyses. Second, we derive new spatial weights from the bipartite graph framing. We can employ these in both exploratory spatial data analysis and confirmatory spatial econometric modeling. In the latter case, we develop a family of model specifications that nest the traditional single equation with a square W matrix as a special case. This family also includes multi-equation system specifications allowing for spillovers from and to each of the processes. Finally, we provide an illustration of these new analytics in a case study involving the school-neighborhood nexus for communities in San Diego California. Our paper proceeds as follows. We first introduce our approach to the formal representation of the schoolneighborhood nexus relying on bipartite graphs in section 2. From this framing, we introduce a set of 3 new global and local spatial analytics defined on the bipartite graph. We also introduce a family of model specifications for spatial spillovers using new spatial weights defined on the bipartite graph. We illustrate the application of these new methods to a case study involving 198 neighborhoods and 370 public elementary school catchments in San Diego California, in section 3. Section 4 concludes the paper with a summary of key findings and directions for future research. A Bipartite Graph Framing for Incommensurate Spatial Partitions We propose a reconceptualization of the school-neighborhood nexus from a spatial networks perspective. Viewing these connections from a graph theoretic perspective provides a more comprehensive representation of this critical nexus. Figure 2 contains a stylized example of our framework involving a community consisting of 14 school catchment zones and 10 neighborhoods. Panel A overlays the catchment boundaries (Blue) over the neighborhoods (Orange). Panel B displays the spatial congruence between each catchment and the neighborhoods it serves, while panel C shows the spatial congruence between each neighborhood and the catchments that serve that neighborhood. We define the measure of spatial congruence as |i∩K| |i∪K| where the numerator (denominator) is the area of the intersection (union) involving the focal/ego unit i (catchment or neighborhood) and K is the union of the alter units (neighborhoods or catchments) that intersect i . Darker shadings in Panels B and C indicate a closer congruence between the polygon for the catchment (neighborhood) and the alter-unit neighborhood (catchments) it serves (feeds). We suggest using the measures of school and neighborhood congruence to interrogate several questions. For example, one could examine the school congruence measure to determine if the catchments align with or split socioeconomic neighborhoods, potentially providing new insights into whether districts gerrymander school catchments (Richards, 2014; Saporito & Van Riper, 2016). Additionally, researchers could utilize the measures of congruence in spatial optimization work to examine whether modificatiion of catchment boundaries can reduce segregation (Wei et al., 2022). The elements of this framework enhance traditional spatial analysis methods and complement them with a key innovation that extends these spatial concepts using network methods. We represent these spatial relationships as a GeoGraph (Panel D), defined as the bipartite graph B = ( H, S, E ), consisting of two mode sets of nodes: neighborhoods ( H ) and schools ( S ). The bipartite condition permits edges ( E ) only between nodes from these two sets, not within them. Depending on the orientation, one can view one of the modes as an affiliation club attended by children from nodes in the other set. From the perspective of children living in neighborhoods, one could view schools as affiliation clubs that bring together students from different neighborhoods in cases where a catchment intersects multiple neighborhoods. Conversely, from the perspective of students living in the same neighborhood but attending different schools, neighborhoods can be seen as affiliation clubs. Thus, different spheres of activity can function as affiliation clubs, depending on the type of interactions being studied. This bipartite representation preserves a significant amount of information contained in Panel A, while 4 Figure 2: Incommensurate Spatial Partitions: [A] Catchments (blue), Neighborhoods (orange); [B] School congruence; [C] Neighborhood Congruence; [D] Bipartite Graph. 5 allowing its manipulation and analysis as a graph object. For example, neighborhood 1 (h01) in panel C has the highest level of neighborhood congruence in the example. This is due to the area of the intersection of this neighborhood with the three catchments constituting a large share of the union of the set of polygons including the neighborhood and these three catchments. Conversely, neighborhood 2 (h02) intersects five catchments, however, the area of these catchments that are outside of the neighborhood is large. From the schools’ perspective, one can consider a different form of congruence. In Panel B, school 2 (s02) has the highest level of congruence with the two neighborhoods it draws from, while school 0 (s00) which also draws from two neighborhoods intersects a much smaller share of the area of those two neighborhoods. Thus, the school congruence measures offer insights as to how dominant the school is for the neighborhoods it draw students from. Unipartite Projections Intersections between catchments and neighborhoods in Panel A from Figure 2 induce the edges in the bipartite graph B , with an adjacency matrix where the rows correspond to neighborhoods and the columns to schools. While the bipartite graph allows direct connections only between nodes belonging to different mode sets, a unipartite projection of the graph produces information about the relation structure among a particular set of nodes (Neal et al., 2021). In the resulting unipartite network, nodes belong to only one of the two mode sets. The network will connect two such nodes only if they share at least one common neighbor from the other mode set in the original bipartite network. One projection of a bipartite network B is P = BB′ where B′ is the transpose of B . P will be a square, symmetric matrix, where in our example from Figure 2, the rows and columns represent the neighborhoods in Band cell Pi,j contains the number of school catchments that both neighborhoods iand jintersect. For our example we have: B′=                            0000001100 0010000100 0010100000 1100001010 0010110100 0100001010 0000011011 0010000100 0010011100 1000000011 0000011111 0000010001 0000001100 1101000010                            6 BB′=                   3201001031 2301002030 0050221400 1101000010 0020210100 0020153323 1210037442 0040134711 3301024163 1000032134                   Figure 3 displays the graphs resulting from the projections for the schools and neighborhoods, with the projection for the neighborhoods being BB′, while the projection for the schools graph is B′B. (a) Neighborhoods (b) Schools Figure 3: Unipartite graphs for neighborhoods and schools Focusing on the projection for the neighborhoods (Panel a), there is an edge between neighborhoods h0 and h1, as these two neighborhoods intersect with two of the same catchments (row 0 column 1 of BB′ ). Neighborhood h0 connects to a total of five other neighborhoods (h1, h3, h6, h8, h9) through shared intersections with catchments. The edges in the neighborhood graph represent the possible channels of diffusion of school-based exposure (e.g., curricula unique to a single school) across neighborhoods. For example, students attending school s13 from both h0 and h3 will be directly exposed to the curricula at that school, creating a form of dependence between the student communities in the two neighborhoods. While all the children in h1 attend s3, this is not the case for h0. h0 intersects three other catchments giving rise to the connections with four other neighborhoods (besides h1). Thus, collectively, students residing in h0 are potentially exposed to a wider set of school-based information through interactions in their neighborhood, then are students living in h3. An edge in the unipartite graph for the schools denotes that the two involved schools draw students from the same set of neighborhoods. Such a projection represents the possible channels of diffusion of school-based exposure (e.g., curricula unique to a single school) from one school to another via unstructured neighborhoodbased interactions. For example, it indirectly captures the potential for a child from neighborhood h2 to encounter a unique idea at school s2, then share it with another child from the same neighborhood attending 7 school s4, thus spreading the idea from school s2 to s4. The neighborhood interactions also allow innovations to flow in the reverse direction. The edges suggest similarities in student socioeconomic and ethnoracial diversity at the involved schools, indicating that students at those schools experience similar neighborhood contexts. Exploratory Analytics Framing the school-neighborhood nexus as a bipartite graph provides several methodological benefits. Like other spatial analyses, we can distinguish between global (macro) and local (micro) analytics. We can use the Macro properties of the graph to compare the spatial structure of the school-neighborhood nexus across different districts or cities. For instance, the density of the bipartite graph in Figure 2 is: Density = m/(nS+nH) with m the number of edges in the graph, and nS and nH the number of school and neighborhood nodes, respectively. In the example from Figure 2 there are |E| = 43 edges, with 24 nodes (10 neighborhoods and 14 schools) resulting in a density of 1.79. The mean degree of the nodes in the graph serves as a measure of incomensurability between the catchments and the neighborhoods. A mean degree of 1 would reflect perfect comensurability between the two partitions. As incomensurability between the two partitions increases, so would the mean degree. The 24 nodes in the example graph have a mean degree of 3.58. Other global measures reflect the clustering embodied in the graph. Measures of graph clustering provide insights into the tendency of nodes to form closely-knit, interconnected communities within a larger network. Robins & Alexander (2004) define one such measure as four times the number of four-cycles C4 divided by the number of three paths L3: CRA =4C4 L3 which gives a value of 0.39 for our example graph. Local measures of the bipartite graph can identify key nodes in the spatial system. Table 1 reports three such measures, together with their ascending ranks, for our example. For node v in the two bipartite sets S (schools) with nS nodes and H (neighborhoods) with nH nodes, the sum of the fraction of all-pairs shortest paths that pass through the node vis its betweenness centrality. Closeness centrality for vis: cv=nS+ 2(nH−1) d,∀v∈H or cv=nH+ 2(nS−1) d,∀v∈S where dis the sum of the distances from vto all other nodes. 8 betweenness rank closeness rank degree rank s0 0.02 7.50 0.58 15.50 0.20 5.50 s1 0.01 3.50 0.51 9.50 0.20 5.50 s2 0.01 2.00 0.42 3.00 0.20 5.50 s3 0.11 15.00 0.60 19.00 0.40 18.00 s4 0.13 18.00 0.58 15.50 0.40 18.00 s5 0.05 12.00 0.58 15.50 0.30 12.50 s6 0.09 14.00 0.67 22.50 0.40 18.00 s7 0.01 3.50 0.51 9.50 0.20 5.50 s8 0.14 19.00 0.67 22.50 0.40 18.00 s9 0.04 11.00 0.51 9.50 0.30 12.50 s10 0.24 22.00 0.75 24.00 0.50 23.00 s11 0.01 5.00 0.50 7.00 0.20 5.50 s12 0.02 7.50 0.58 15.50 0.20 5.50 s13 0.11 17.00 0.51 9.50 0.40 18.00 h0 0.02 10.00 0.44 5.00 0.21 9.50 h1 0.02 9.00 0.42 4.00 0.21 9.50 h2 0.11 16.00 0.48 6.00 0.36 14.50 h3 0.00 1.00 0.35 1.00 0.07 1.00 h4 0.01 6.00 0.39 2.00 0.14 2.00 h5 0.15 20.00 0.57 13.00 0.36 14.50 h6 0.27 24.00 0.64 21.00 0.50 23.00 h7 0.24 23.00 0.62 20.00 0.50 23.00 h8 0.23 21.00 0.59 18.00 0.43 21.00 h9 0.07 13.00 0.52 12.00 0.29 11.00 Table 1: Bipartite Local Centrality Measures Degree centrality : dv=deg(v) m,∀v∈H or dv=deg(v) n,∀v∈S where deg ( v )is the degree of node v . Note that in the bipartite case, the maximum possible degree of a node in a particular node set is the number of nodes in the opposite set. School 10 has degree centrality 0.5 as it has edges with 5 of the 10 neighborhoods, while neighborhood 2 has degree centrality of 0.357 as it has edges with 5 of the 14 schools. While there is general concordance across the three local centrality measures, there are several things to keep in mind. First, the degree centrality measure results in a larger set of tied pairs than the other two indicators. Second, there is agreement across the three measures in the least central node in the bipartite graph: h3. In contrast, there is less concordance at the top end of the centrality distributions, as h6 and h7 tie on degree centrality, while s10 has the highest closeness centrality, and h6 is the maximum for betweeness centrality. 9 Table 3: School Neighbor Cardinalities queen bpg min 1.00 1.00 25% 5.00 11.00 50% 6.00 16.00 75% 7.00 21.00 max 31.00 59.00 Similar patterns in the comparison between school contiguity and projected graphs shown in Figure 6b and Table 3. Again, it is clear that the projection has the anticipated densifying effect, creating a greater number of connections and, here the mean neighbor set from about 6 neighbors to about 16 while also increasing the spread of the distribution. Again, this has the effect of greater network influences, as schools can be affected not just by their neighboring schools, but by also by schools who draw from the same neighborhoods of students. An intervention in a given neighborhood may affect all schools that draw from this neighborhood. Exploratory Spatial Data Analysis Neighborhoods To examine the role of the choice of W in a local analysis of the neighborhood context, we focus on the pattern of parental education levels as measured by the percentage of adults with a college education displayed in Figure 7. There is a clear gradient to educational levels with attainment falling with distance away from the neighborhoods in the central portion of the coast. The gradient is steeper to the southeast relative to the northeast. A global Moran’s I test applied to this attribute using the spatial weights obtained from Queen contiguity and the bipartite projection for the neighborhoods yielded 0.67 and 0.74, respectively, both of which have pseudo p-values of 0.001, pointing to spatial clustering in the distribution of this important aspect of human capital. Turning to a local autocorrelation analysis using the same pair of spatial weights, the cluster maps for the local Moran’s I statistic are displayed in Figure 8a and Figure 8b. As expected from the previous discussion of the underlying graph structures, the greater density in the weights from the projection results in a larger number significant local statistics relative to the contiguity weights. Upon further consideration of the results from the two local analyses, the two approaches concur in 156 out of 198 cases. In these cases, 76 neighborhoods have non-significant local statistics in both methods, and 80 neighborhoods receive a classification of significant by both. The most substantial disagreements occur in the 38 neighborhoods classified as insignificant using contiguity weights but significant with projected weights. This type of disagreement reflects the higher density in the second connectivity graph. There are, however, 4 instances where a neighborhood shows a significant LISA with contiguity but becomes insignificant with 16 Figure 7: Percent of adults with a college education by neighborhood. 17 projected weights. Overall, the two approaches disagree in close to 1 out of 5 cases. (a) Contiguity (b) Projection Figure 8: Cluster Maps for Neighborhood Adult Education Levels Schools We carry out a second set of autocorrelation analyses for the catchments with a focus on the spatial distribution of average test scores by catchment displayed in Figure 9. Global Moran’s I values for the two school weights (contiguity and projection) were significant in both cases. Figure 10a and Figure 10b display the local autocorrelation for these two definitions of spatial weights. In contrast to what held for the local analysis of neighborhoods, here the differences in the patterns between the two sets of weights are more complex. There is less concordance between the classification of each local statistic for the two weight structures: agreement for 180 catchments as insignificant and 98 as significant out of 370 catchments. A clear difference from the neighborhoods case is that the areas of disagreement for the contiguity and projection weights show less asymmetry in LISA classifications. For 57 catchments the LISA is not significant when using contiguity but turns significant for the projection based weights, while there are 35 catchments in the reverse situation. These differences result in the weakening of two hot-spots, one in the northern part of the county, which disappears, and one in the southern part of the county, which shrinks. At the same time, the more central hot-spot becomes more solidified when moving to the projection weights, as do two cold-spots. Bivariate Local Autocorrelation We also leverage the bipartite graph to operationalize new approaches towards measuring bivariate local autocorrelation. A bivariate local Moran’s I captures the relationship between the value for one variable at location i , xi and the average of the neighboring values for the spatial lag of a second variable yli given as Pjwi,jyj. Both xand yare standardized to have 0 means and unit variances. 18 Figure 9: Average test scores by school catchment. (a) Contiguity (b) Projection Figure 10: Cluster Maps for Average Test Scores by Catchment 19 In the case of two variables subject to the ISP, the difference is that each location doesn’t have an in-situ pair of observations. For example, consider the bivariate Moran for school performance and adult education levels, the former is measured for each catchment while the latter is observed for neighborhoods. To address the incomensurability of these two partitions we implement the bivariate LISA as: Is=scoresB′ .,sed where scores is the test score for school s , B.,s is column s of the bipartite adjacency matrix B and ed is a vector of neighborhood education levels. For inference on the bivariate LISA we rely on random permutations of the ed vector. Figure 11 reports the results for the bivariate LISA of the test score and neighborhood education relationship. Based on the random permutations, there are significant local statistics in each of the four quadrants of the scatter plot, with the High-High and Low-Low dominating. The former represents schools performing above average that draw from neighborhoods with above average adult education levels, the latter being under-performing schools drawing from neighborhoods with below average adult education levels. These two sets of observations be viewed as cases where there is a strong positive association between educational levels of parents and school performance. Quadrants two and four identify where this linkage is disrupted. For the schools with LISAs in the High-Low quadrant (4), test scores are above average yet these schools are pulling from neighborhoods with below average levels of adult education. Not only are these high-performing schools, they are also able to escape the downward pull of low adult education levels that we see in the schools in quadrant 3. Significant bivariate LISAs in the Low-High quadrant (2) identify schools that under-perform yet pull from neighborhoods with above average adult education levels. The spatial locations associated with these LISA values are shown in Figure 11b. By definition, the location of the significant High-Low observations will be in areas with Low adult education neighborhoods (seen earlier in Figure 8b). However, the Low-Low LISA values will also be constrained to these regions as well, and the two sets of bivariate LISAs are often in close proximity to one another. This suggests that neighborhood education levels may not be destiny for school performance. A particularly prominent example of this is the large polygon in the south-eastern portion of the study area. This is the Mountain Unified School District that contains two elementary schools with significant LISAS, however one, Descanso Elementary has a High-Low value with the second, Potrero Elementary, has a Low-Low LISA. A second bivariate LISA considers the association between neighborhood adult education levels and the spatial lag of school test scores. Here the LISA is Ih=edhBh,.score where edh is education levels in neighborhood h , Bh,. is row h of the bipartite graph adjacency matrix and 20 (a) Moran Scatterplot (b) LISA Cluster Map Figure 11: Bivariate Local Autocorrelation Average Test Score and Lag Parental Education (by school) score is a vector of school catchment average test scores. Inference here is obtained using random spatial permutations of the catchment test scores. The results of the local analysis are shown in Figure 12. As in the case of the previous bivariate LISA, the High-High and Low-Low quadrants contain the majority of the significant local measures of bivariate spatial association. This dominance is more pronounced in this case as there are relatively few significant spatial outliers in the High-Low or Low-High quadrants. In this case, the focal neighborhoods that have significant bivariate LISAs in quadrant 4 are neighborhoods with high levels of adult education that feed under-performing schools, while those in quadrant 2 are neighborhoods with lower levels of adult education that feed higher performing schools. (a) Moran Scatterplot (b) LISA Cluster Map Figure 12: Bivariate Local Autocorrelation Parental Education and Lag Average Test Score (by neighborhood) Implementing the two sets of bivariate LISAs using the rectangular B matrix offers several advantages. Firstly, 21 interpreting the original bivariate LISA requires caution, as disentangling whether any spatial association in the bivariate case stems from the in-situ bivariate correlation of the original variables or from the similarity due to being spatial neighbors can be challenging. However, the ISP “problem” eliminates the in-situ pair of observations, thus avoiding this complication. The second advantage of a bivariate LISA based on the bipartite graph is that conditional randomization is no longer needed to develop the reference distribution for the local test statistics at each location. Again this is due to the absence of in-situ pairs of observations for the two variable which means conditional randomization to ensure the observation for the same unit on the second variable is not selected in constructing the lag is no longer required. Segregation To compare the difference between a standard measure of spatial proximity and the BPG representation of the neighborhood, we compute three multi-group spatial segregation indices: the Gini index, the Information Theory Index, and the Isolation/Exposure Index, all of which are based on four mutually-exclusive racial groups (white non-Hispanic, Black non-Hispanic, Hispanic, and Asian 1 ) as commonly found in the literature. As Knaap & Rey (2023) describe, a fundamental element in any spatial segregation index is the definition of the neighborhood that describes the expected interaction between different population groups. As such, they propose the use of a spatial weights graph W as a flexible definition that can be incorporated into an abstract segregation index following the logic of Reardon & O’Sullivan (2004) and Reardon et al. (2008). In the context of segregation research, the concept of a projected bipartite graph represents the ways that schools serve as mixing sites for students from different neighborhoods. More specifically, by using a W graph based on the school reprojection described above, the “neighborhood” of expected interaction represents the area-weighted environment of neighborhoods that other students in the same school live in. Put differently, rather than weighting a student’s nearby population of adults, the graph represents the weighted population of adults nearby other students, and represents the ways that students may bring their own neighborhood influences to bear during social interactions at school, or the ways that a given student may be expected to interact with other adults when visiting the home of a friend from school. For each of the segregation indices, we compute a spatial measure based on each concept of the W graph; in all cases the contiguity-based graph results in a higher index (i.e. greater levels of segregation) than the projected graph. Conceptually, this suggests that schools may serve as a medium for students to interact with a more diverse population group, by interacting not just with their own neighbors, but also with their classmates neighbors. In the case of San Diego, schools appear to diversify rather than insulate the home environment, supporting earlier evidence by Rey et al. (2023). To test the robustness of this difference, we apply the method described in Cortes, Rey, et al. (2019) and Rey et al. (2021) that provides for comparative inference between two segregation measures using the segregation 1 All calculations are based on blockgroup-level data from the U.S. Census American Community Survey 5-year estimates from the 2017-2021 release (which adopt the 2020 census boundary delineations) 22 (a) Information Theory Index (b) Exposure Index (c) Gini Index Figure 13: Comparative Segregation Analytics module from the PySAL software ecosystem (Cortes, Knaap, et al., 2019; Rey et al., 2022). Specifically, the test uses a computational inference to develop a distribution of differences between segregation measures under the null hypothesis of no difference. The observed difference (in this case the difference between a measure computed with two different Wspecifications) is then tested against this reference distribution. The results from our analysis are shown in Figure 13, where the reference distribution is shown as a blue histogram, and the observed difference is shown as a red line. In all three cases, the difference is significant, showing that the greater level of segregation observed using the contiguity graph is not due to random chance. In other words, there is strong evidence that the interactions posited by the projected graph serve to reduce segregation by providing a more diverse environment with which to interact. Conclusion In this paper, we introduced a novel methodological framework for analyzing incommensurate spatial partitions (ISPs) through the lens of bipartite graphs, focusing on the intricate interplay between school catchments and neighborhoods. This approach reframes the relationship between different spatial units, allowing for the analysis of multi-process spatial contexts without needing harmonization of spatial supports. Doing so allowed us to move beyond traditional models, offering a more nuanced understanding of the complex interactions and spillovers between overlapping yet distinct spatial units. Our findings from the empirical case study in San Diego, California, underscore the significance of considering the multidimensional nature of space in applied work. Applying our bipartite graph approach, we uncovered intricate patterns of association between schools and neighborhoods, revealing the intertwined nature of educational outcomes and neighborhood characteristics within their spatial context. This not only highlighted the potential of our methodological contributions but also provided valuable insights regarding the identification of schools that are over-performing (or under-performing) given their neighborhood contexts. The development of new spatial weights from bipartite graphs emerged as a key innovation, facilitating both exploratory and confirmatory spatial analyses. The weights obtained from projections of the bipartite graph result in much wider spatial linkages between different schools, and between different neighborhoods, 23 than those afforded by traditional contiguity-based weights. Direct use of the bipartite adjacency graph results in rectangular spatial weights matrices which, in turn, opens up a rich set of model specifications to consider different forms of spatial spillovers in the school-neighborhood nexus. This advancement creates new possibilities for researchers and policymakers to investigate and understand the spatial underpinnings of socio-economic phenomena more deeply. More specifically, the bipartite adjacency graph allows researchers to consider complex albeit natural patterns of social interaction that are challenging to represent in classic quantitative models. As a concrete example, the bipartite projection allows a method for considering how social influences may be transmitted via institutions with a geographic footprint. Schools are one such institution whose boundaries define a container of social interaction (i.e., students are more likely to interact with one another if they attend the same school); neighborhoods are another, where interaction is more likely among neighbors and family members who are closer nearby. To date, a major methodological workhorse of the neighborhood effects and school effects literature is the multilevel model, which typically treats each of these contexts in isolation. The bipartite graph approach introduced here, however, offers an avenue for considering the ways these contexts may interact and provides a formal mechanism for studying, for example the transmission of behavior from a neighborhood to a classmate– even when those classmates live in different neighborhoods. These kinds of multi-contextual spillovers are conceiveable, even expected for a wide variety of additional research questions (information transmission between colleagues via social media, innovation transmission among small businesses through conference attendance, or disease transmission through superspreader events, etc.) We recognize that this work is not without limitations. While the San Diego case study provided a robust test bed for our methodologies, we need further research to validate and refine these approaches across different contexts and scales. The variance in the node degrees allows for a consideration of weighted edges in the original bipartite graph as well as in the projections. Consideration of weighted edges that take into account distance decay relationships seems particularly important given the higher density of the bipartite graphs relative to the traditional contiguity based spatial weights. These extensions will require addressing a number of challenges with weighted bipartite graphs. Related to this is the extraction of the backbone of the graph (Domagalski et al., 2021), which would identify the most significant edges. Finally, extension of the single equation specification presented in this paper to the case of simultaneous equations systems (Rey & Boarnet, 2004) would allow for bidirectional feedback between the multiple processes. In conclusion, our research contributes an important step forward in the understanding and application of ISPs in urban science. By providing a novel set of tools and perspectives, we aim to inspire further research and practical applications that embrace the complexity of spatial interdependencies. 24 Declarations Funding This research was supported by NSF Grant TI-2345820. References Anderson, T. K. (2010). Using geodemographics to measure and explain social and environment differences in road traffic accident risk. Environment and Planning A,42 (9), 2186–2200. https://doi.org/10.1068/a43157 Anselin, L., & Rey, S. J. (2014). Modern spatial econometrics in practice: A guide to GeoDa, GeoDaSpace and PySAL. GeoDa Press LLC. Anselin, L., Rey, S., & Deichmann, U. (1990). 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