2 Application to Redistricting

To illustrate the method, we imagine ourselves as a researcher new to studying redistricting and conducting a literature review. Figure 1 summarizes steps we take as an example for researchers interested in this approach. Redistricting following the 2020 U.S. Census attracted the attention of courts, politicians, and the public, to the prior decade of academic work, highlighting the importance of district boundaries to political outcomes. Understanding this literature poses a challenge for new scholars. We focus on work over the last 10 years as a way to demarcate the “latest research” (Dion et al. Reference Dion, Sumner and Mitchell2018).

Figure 1 Steps to creating a literature network. Application to redistricting demonstrated in italics.

What constitutes the relevant literature? We recommend selection criteria based on predefined and replicable rules. Our criteria prioritize recent and impactful work on redistricting, indicated by journal rankings and citations. We select six highly ranked political science journals broad enough in scope to cover the topic of redistricting (Scimago 2020), two journals specifically from the American politics subfield, and finally, Election Law Journal, which has systematically published redistricting research cited by courts, expert witnesses, and government entities. Within these journals, we conduct keyword searches among articles published since 2010 containing any of the following phrases in either title or abstract: efficiency gap, gerrymander, partisan symmetry, and redistrict. To capture relevant work outside these journals, we search Google Scholar for any post-2010 peer-reviewed article with 50 or more citations that included a key phrase in the title/abstract. One hundred fifteen articles matched these criteria, constituting the corpus of studies for this review.Footnote ¹ Keyword selection necessarily affects article selection. For a comprehensive search that balances exploration and “starting values,” we recommend an iterative process of selecting seeding keywords to survey the literature and snowball-sample highly related keywords (e.g., searching articles with keyword redistricting returns articles that regularly speak of gerrymandering and efficiency gap) and soliciting keyword suggestions from domain area experts.

For each article, network data is input through familiar steps: reading the work and identifying the main concepts and connections posited between concepts. We select concepts representing the main causes and effects investigated, often concepts discussed in the abstract or theory sections. We enter this information into an edgelist spreadsheet—such that concepts constitute nodes, and their connections are edges, often hypothesized with directions so the edge can be drawn as an arrow from cause to effect.Footnote ² For example, the two main concepts in Cain et al. (Reference Cain, Tam Cho, Liu and Zhang2017) are “independent redistricting commissions” and “partisan advantage.” The authors further posit that such commissions are unlikely to produce extremely partisan maps, which we record as a directed edge. Information pertinent to this edge—such as whether the effect is positive or negative—and the number and identity of works that address this same connection are edge attributes. Attention must be paid to the important process of defining nodes and edges—no surprise to regular users of network analysis or analyses that rest on well-measured concepts—how and what constitutes a node that represents a concept and whether it relates to another is ultimately an interpretive process by the researcher from research piece to row in an edgelist.Footnote ³ Authors may use different language in referencing the same concept (i.e., partisan bias and partisan advantage); in these cases, we employ an iterative concept-naming process. We add terms used by each author to the spreadsheet, then visualize the draft network to identify similar terms. After consulting the relevant articles, we consolidate terms referring to the same concept under an umbrella term in the spreadsheet.

In this example, the final spreadsheet contains 57 concept nodes and 69 edges describing relationships among concepts.

Figure 2 shows the resulting redistricting literature network. A global summary of the literature begins with describing the network itself—57 theoretical concepts studied, as causes or effects, shaded by the node’s total degree centrality.Footnote ⁴ Sixty-nine edge arrows show each directional relationship explicitly theorized in our corpus, colored by number of publications addressing that relationship. Edge colors can illustrate attributes of relationships supplied as an additional column in the input spreadsheet (here representing the number of citations that theorized the edge relationship). The $\textit{netlit}$ vignette presents attributes that one may wish to highlight, including edge statistics (e.g., edge betweenness) produced by $\textit{netlit::review()}$ . Literature discussing the measurement of a single concept appears as self-ties. For example, measuring the concept of compactness (right-hand side of Figure 2) has inspired a series of works (Barnes and Solomon Reference Barnes and Solomon2021; Chen and Rodden Reference Chen and Rodden2015; De Assis, Franca, and Usberti Reference De Assis, Franca and Usberti2014; Magleby and Mosesson Reference Magleby and Mosesson2018; Saxon Reference Saxon2020; Tam Cho and Liu Reference Tam Cho and Liu2016).

Figure 2 Redistricting literature network. Nodes represent theoretical concepts, shaded by total degree centrality. Arrows connect concepts theorized as directional relationships in works, colored by number of works (and can be both directions—such as might indicate an endogenous relationship). Solid edges indicate empirically studied connections; dashed are relationships that have been theorized but not studied empirically. The $\textit{netlit}$ vignette walks through production of network graphs, using the $\textit{graph}$ object returned by the $\textit{netlit::review()}$ function as the input to network graphing functions from packages like $\textit{ggnetwork}$ . The vignette illustrates how $\textit{nodelist}$ and $\textit{edgelist}$ objects provide required inputs for other network visualization packages, for example, $\textit{ggraph}$ or $\textit{visNetwork}$ .

What are key concepts in redistricting literature? As posed in Table 1 “Assess and Identify,” a natural translation of this question to a network is “what are the central nodes?” The concept of partisan advantage is most central with 14 total edges. Efficiency gap, partisan gerrymandering, and preserve communities of interest each have degree centrality of eight. We define preserve communities of interest as an umbrella term covering the broad goals of preservation of minority areas and political subdivisions within districts, and core district retention (Figure 3a). It is unsurprising that this predominantly legal concept scores high in total degree (five out edges, one in-edge, and one self-tie) as it is both a traditional redistricting criterion and widely studied.

Figure 3 (a) Redistricting literature directly related to the concept of preserving communities of interest. Nodes represent theoretical concepts (shaded by total-degree centrality). Arrows connect concepts explicitly theorized as directional relationships, colored by number of works. Solid edges indicate empirically studied connections. Dashed edges indicate theorized relationships that aren’t studied in the empirical literature reviewed. Preserving communities of interest is often studied as a cause to rolloff and partisan gerrymandering, and has been theorized to affect voters’ information and fellow constituents; ways to measure it (self-ties) have also been explored. (b) Confounding concepts for the effect of preserving communities of interest on rolloff.

Are there communities of work that have developed recently? In network terms, we can ask: “what are communities in the network?” A distinct community (Figure 3a) of scholars has investigated how changes in the electorate’s composition (change in constituency boundaries) can affect downstream campaign resource allocations and vote power, highlighting the effects that redrawing of maps might have on political environments for individual candidates.

Beyond assessing the state of the literature, two defining tasks in research are finding areas for theory building and identifying where theory has yet to be empirically tested; that is, finding the “gaps” in knowledge. A visual approach to the first question looks for areas in the literature network where two concepts are discussed separately and where a researcher might posit a connection. For example, recent work (Ansolabehere and Snyder Jr. Reference Ansolabehere and Snyder2012; Carsey, Winburn, and Berry Reference Carsey, Winburn and Berry2017; Hood and McKee Reference Hood and McKee2013; Limbocker and You Reference Limbocker and You2020) has shown that redrawn lines that change the composition of the electorate exert an exogenous effect on the vote, but less attention has been paid to how such changes affect minority representation; there exists an opportunity for research concerned with political consequences of constituency boundaries to engage more directly with scholarship on minority representation.

Answering questions about empirical gaps is a simple matter of analyzing edge characteristics in the network—whether concepts that are linked in theory are also linked in empirical work. In Figure 2 edges drawn as dashed lines indicate a theorized but not empirically validated relationships. Solid lines represent empirically demonstrated connections between concepts. Partisan advantage is hypothesized to affect whether floor votes align with district/state preferences (bottom of Figure 2)—both through connections that remained untested empirically until recently (Caughey, Xu, and Warshaw Reference Carsey, Winburn and Berry2017). Likewise, Figure 2 suggests equal population and partisan dislocation are concepts that are important to measure (visually verified with self-ties). Measuring equal population has been discussed more often than partisan dislocation (Gatesman and Unwin Reference Gatesman and Unwin2021; Magleby and Mosesson Reference Magleby and Mosesson2018), which is reasonable given that equipopulation is a long-standing and firm legal rule in redistricting; whereas the latter concept is relatively new (DeFord, Eubank, and Rodden Reference DeFord, Eubank and Rodden2021).

For researchers interested in causal relationships, a network approach offers two tools for isolating and controlling, potentially the first step toward a more complete directed analytic graph. To answer the question “what causal pathways are related to a theorized concept?”, we inspect the neighborhood of nodes and edges. Consider the node preserve communities of interest. Exploring its neighborhood (Figure 3a) reveals hypothesized downstream effects on preserving community interests, including changes at the voter (voter information about their district and stability in voters’ fellow constituents) and district levels (partisan gerrymandering and rolloff). It also suggests that preserving communities of interest is a prominent confounding concept that affects how voter information about their district contributes to rolloff.

How does a network approach differ from a traditional expert-guided review? As a thought exercise, one of our team members (Mayer)—a redistricting scholar and experienced expert witness in gerrymandering litigation—prepared a traditional review. We compare our approach against his and McGhee (Reference McGhee2020)’s recent redistricting literature review. Mayer and McGhee separately identify three key themes in the recent redistricting literature that parallel our network findings: developing metrics, automation of redistricting methods, and exploring downstream effects of gerrymandering. The network approach brings some nuance to each of these themes, however, by allowing quick identification of metric-oriented works, avoiding over-inflating the importance of growing communities of work, and allowing us to develop more complex directed acyclic graphs from the literature around the effects of gerrymandering.

Specifically, Mayer and McGhee note that recent work has focused on developing metrics to propose a legal standard for federal courts to place limits on partisan plans (and which Justice Anthony Kennedy appeared to request in LULAC v. Perry 584 U.S.399 (2004)).Footnote ⁵ Our network approach also identifies scholarship on metrics, represented as nodes with self-ties. Further, it parses out where scholarship on metrics is more or less likely to contribute to theories of redistricting. For example, measures of compactness versus equal population both have self-ties, but the network shows that only the former has been recently theorized to affect political outcomes such as voter turnout.

Similarly, Mayer notes that automated redistricting methods have captured substantial attention recently (Chen and Rodden Reference Chen and Rodden2013; Cho and Liu Reference Cho and Liu2018; Liu, Cho, and Wang Reference Liu, Cho and Wang2016; Magleby and Mosesson Reference Magleby and Mosesson2018; Vanneschi, Henriques, and Castelli Reference Vanneschi, Henriques and Castelli2017). One method draws large numbers of maps with different decisions rules and initial conditions, with the resulting maps used to identify outliers that indicate partisan gerrymanders or possible “natural” gerrymanders (Cain et al. Reference Cain, Tam Cho, Liu and Zhang2017; Chen Reference Chen2017; Chen and Cottrell Reference Chen and Cottrell2016; Chen and Rodden Reference Chen and Rodden2013, Reference Chen and Rodden2015; Fifield et al. Reference Fifield, Higgins, Imai and Tarr2020; Ramachandran and Gold Reference Ramachandran and Gold2018; Tam Cho and Liu Reference Tam Cho and Liu2016). The network figure shows these studies with self-tying nodes and node connections, including the relationship between geographic partisan distribution and partisan advantage. Both expert reviews emphasized methodological advancements. While this strand of work is prominent in the full network graph, it is a minority of scholarship that is still primarily concerned with political science theories related to redistricting. Thus, we see our approach as avoiding the conflation of overall patterns of scholarship with popular and highly discussed work. The latter would be better captured with a citation network than a causal graph.

The third insight of a traditional literature review is that recent work has continued exploring the effects of gerrymandering on various outcomes including incumbency advantage (Henderson, Hamel, and Goldzimer Reference Henderson, Hamel and Goldzimer2018); electoral competition (Cottrell Reference Cottrell2019); candidate quality and emergence (Williamson Reference Williamson2019); roll-call voting and state policy (Caughey, Tausanovitch, and Warshaw Reference Caughey, Xu and Warshaw2017); political parties (Stephanopoulos and Warshaw Reference Stephanopoulos and Warshaw2020); campaign contributions (Crespin and Edwards Reference Crespin and Edwards2016); and constituent access (Niven, Cover, and Solimine Reference Niven, Cover and Solimine2021). Our network also captures these relationships as dyadic connections, but can further illuminate downstream causal chains, confounding concepts, and multiple causal paths.

3 Discussion and Conclusion

We present an organizing framework based on network representations to conduct literature reviews. Our application focused on redistricting, but the approach is general; where research builds on complex combinations of prior work, a network approach might prove especially fruitful.

We highlight several helpful features of networks as a way of uncovering patterns in scholarship. Beyond assessing prominent themes and communities of work, the network representation most importantly lends itself to theoretical exploration and identification of relationships that have yet to be studied empirically. Finally, the directed graph representations in this framework can be used to inspect causal pathways related to a concept or to identify confounding relationships (see, for instance, discussion of causal interpretations in regression models in (Keele, Stevenson, and Elwert Reference Keele, Stevenson and Elwert2020)).

Our approach may also lower barriers to entry: while substantive expertise always improves exercises like these, the input units to the network require summarizing concepts and identifying posited relationships between them within single research works, repeating this over the list of works, and submitting this information into a spreadsheet. This process is accessible to newcomers to a literature. Illustrated in the $netlit$ vignette is another pattern-discovering tool for reviewing literature evolution—by subsetting the input data to prior periods and comparing the generated literature network to the most complete and up-to-date network.

In emphasizing the importance of clearly delineating inclusion criteria for work included in a literature review, our approach may also limit unintentional biases in the process of assembling “relevant” works for literature reviews (e.g., favoring personal or institutional social networks, running the risk of over-representation of well-connected works at the expense of research from underrepresented scholars (Lalanne and Seabright Reference Lalanne and Seabright2022)). While our approach does not eliminate systemic under-representation, we hope that clear criteria and full visualization of included work can sidestep under-representation and under-inclusion.

Ultimately, the proposed framework still relies on researcher choices—including the universe of sources, selection criteria, and identification of main concepts—choices that are still undertaken in traditional reviews of literature. By clarifying choices and utilizing our network framework to visualize the resulting network, we expect it to be easier to evaluate such choices and how sensitive assertions about gaps in the literature are to these choices.