As visualized in Figure 1A, the legal system consists of multiple levels: the local (e.g., municipal) level, the intermediate (e.g., state or province) level, the national (e.g., federal) level, and the supranational (including international) level. Horizontally, it is usually subdivided into the legislative, executive, and judicial branches of government. These public parts are framed by the private sector, which operates on all levels, and the research community, which studies all parts of the legal system (including itself [46–50]). In all parts of the legal system, agents of varying sizes produce different types of outputs that create, modify, delete, apply, debate, or evaluate legal rules. These agents and their typical outputs are summarized in Table 1.

As the agents interact, they consciously interconnect their outputs. For example, court decisions regularly contain references to statutes, regulations, contracts, and other court decisions. Figure 1B gives an overview of the classic dependencies between the typical outputs of agents in the legal system. It illustrates that the documented part of the legal system constitutes a multilayered document network, which is changing over time as the agents continue producing or amending their outputs. Since the connections between the legal documents are placed deliberately by the agents, they encode valuable information about the content and the context of these documents. A lot of this information cannot be inferred from the documents’ language alone (reliably or at all). Therefore, investigating the dynamic document network representation of a legal system using network analytical tools promises insights into its structure and evolution that would be hard or impossible to obtain via other methods.

To perform network analysis of a dynamic network of legal documents, we need to represent it as a series of graphs. Here, we build on a generalizable network model of statutory materials [27] and exploit the fact that the typical outputs listed in Table 1 have three common features (beyond the obvious characteristic that they all contain text):

1. They are hierarchically structured (hierarchy).

Therefore, each document at a given point in time (henceforth: snapshot) is represented as a (sub)graph, with its hierarchy modeled as a tree using hierarchy edges. We capture a document’s reference using reference edges at the level corresponding to the document’s sequence, which, inter alia, prevents the graph induced by these references from becoming too sparse (thereby eliminating some noise in the data and facilitating its analysis). The result is a directed multigraph, as illustrated in Figure 2 for documents that are statutes or regulations (whose sequence level is the section level). Depending on the analytical focus, other edge types can be included (e.g., authority edges pointing from regulations to statutes can indicate which statutes delegated the rule-making power used to create which regulations), and depending on the document types considered, different types of edit operations are possible (e.g., court decisions and scholarly articles are only seldom changed after their initial publication), but the general model applies to all outputs listed in Table 1.

To illustrate the power of the methodological framework laid out in Section 3 and produce the results presented in Section 4, we instantiate the data model described in Section 2.1 with the outputs of the legislative and executive branches of government at the national level in the United States and Germany, i.e., federal statutes and regulations, over the 22 years from 1998 to 2019 (inclusive). The rules contained in these sources are universally binding and directly enforceable through public authority (the combination of which distinguishes them from the other outputs listed in Table 1). In the United States and Germany, they are arranged into edited collections: the United States Code (USC) and the Code of Federal Regulations (CFR) in the United States, and the federal statutes and federal regulations (which have no special name) in Germany. These collections are actively maintained to reflect the latest consolidated state of the law (though, in the United States, the consolidation may lag several years behind the actual law). As such, they are a best-effort representation of all universally binding and directly enforceable rules at the federal level in their country at any point in time, commonly referred to as codified law.

For the United States, we obtain the annual versions (reflecting the state of the codified law at the end of the respective year) of the United States Code (USC) and the Code of Federal Regulations (CFR) from the United States Office of the Law Revision Council ¹ and the United States Government Publishing Office ² , respectively. For Germany, we create a parallel set of annual snapshots for all federal statutes and regulations in effect at the end of the year in question based on documents from Germany’s primary legal data provider, juris GmbH. ³ These data sources are the most complete presently available, but they may still be incomplete. They also reflect choices made by and events affecting the agents in charge of their maintenance, e.g., varying rates and orders of updates, purposeful or unintentional omissions or modifications, and changes in the agents’ composition (e.g., as a consequence of elections).

We perform several preprocessing steps on the raw input data, detailed in the Supplementary Material, to extract the hierarchy, sequence, and reference structure contained in each collection. The results are directed multigraphs, one per country and year, akin to those illustrated in Figure 2. These graphs contain all structural elements of the USC and the CFR (in the United States) or the federal statutes and regulations (in Germany) as nodes and all direct inclusion relationships (hierarchy) and atomic references (reference) as edges, where the references are resolved to the section level (sequence). Each graph represents the codified law of a particular country in a particular year, containing documents of two document types (statutes and regulations) at the federal level.

When modeling codified law as just described, we take a couple of design decisions that limit the scope of the results presented in Section 4. First, we focus on codified law, i.e., law in books, excluding other legal materials listed in Table 1, especially those representing law in action (in the sense of [51]), or even other representations of legislative materials such as the United States Statutes at Large or the German Federal Law Gazette (Bundesgesetzblatt). These materials all merit investigation, and they need to be included in an all-encompassing assessment of the legal system. Our current work also serves as a preparatory step toward realizing this larger vision.

Second, we extract atomic and explicit references that follow a specified set of common patterns only, i.e., references including—in a typical format—a particular section (called “Paragraph” or “Artikel” in German law), a list of sections, or a range of sections. With this procedure, we exclude container references (e.g., references to an entire chapter of the USC), pinpoint references (e.g., references to a codified Act of the United States Congress by its popular name), implicit references (e.g., the use of a certain term implying its definition), and explicit references following uncommon patterns. As sketched in Section 3.3 of the Supplementary Material, there are plenty of such references, especially in the CFR, and including them would produce results different from those presented in Section 4. However, the graph representation of such references is inherently ambiguous, and their extraction is inherently more challenging than the extraction of atomic citations. Solving these problems falls outside the scope of this paper but presents an interesting opportunity for future work.

Third, we resolve the atomic references we extract to the level of sections, rather than the smallest referenced unit (which might be a subsection or even an item in an enumeration), thereby effectively discarding potentially valuable information. Since for statutes and regulations, the section level corresponds to the documents’ sequence level, this is consistent with our data model. It also reflects a focus on the perspective of the user, who tends to navigate the law on the section level because it is the only level at which the individual German laws or their United States counterparts, the chapters of the USC and the CFR, are uniquely sequentially labeled. Finally, it ensures a certain degree of comparability because sections are the only structural elements in which text is (with very few exceptions) guaranteed to be contained (albeit the amount of text varies widely across sections). Therefore, resolving references to the section level is reasonable for our purposes, but further research is needed on how the choice of the resolution level impacts the analysis of legal networks.

As detailed in Section 2.1, despite their diversity, the typical outputs of the legal system have hierarchy, sequence, and reference as common structural features, and they also all contain text. Therefore, to assess the growth of the legal system, we track the number of tokens (roughly corresponding to words), the number of structural elements (i.e., hierarchical structures), and the number of references between documents of the same type at the documents’ sequence level (e.g., the section level for statutes and regulations) across all documents in the collection, separately for each document type and across all temporal snapshots (e.g., all years). For the token counts, we concatenate the text of all materials for one snapshot and document type and split on whitespace characters. For a given document type, the structural element counts reflect the number of nodes of that type, and the reference counts reflect the number of reference edges between nodes of that type in our graphs. These measures give a first, high-level idea of how the legal system evolves, and the aggregation by document type allows us to uncover, e.g., differences in growth rates.

To explore the relevance of references between documents of different types, if the document collection contains x types of documents, we distinguish between x ² types of references. Figure 3 illustrates the idea for x = 2 with statutes and regulations as document types. We count the number of references of each type for each temporal snapshot.

Raw reference counts do not show how the incoming and outgoing references are distributed across the individual sequence-level items. The user experience of a legal system, however, depends crucially on these distributions: If the typical item on the sequence level has very few outgoing references, the expected cost of navigating the law is much smaller than if the outgoing references are uniformly distributed, and if the distribution of incoming references is very skewed, when reading the law, we are much more likely to encounter a few prominent sequence-level items than a large number of less prominent ones (of course, the actual cost of the user also depends on the size of the items to be navigated, which can vary widely, e.g., among the sections in the USC). Therefore, we inspect the evolution of the in-degree distribution and the out-degree distribution of the subgraph induced by the reference edges. We compute these distributions separately for each combination of reference edge types (e.g., considering any combination of the reference types depicted in Figure 3 for a document collection containing statutes and regulations) and across all snapshots. This allows us to evaluate whether growth in the number of references further amplifies differences in the prominence of different parts of the law, which would be reflected in a lengthening or thickening of the distributions’ tails, and to assess how this affects the navigability of a legal system.

When exploring the connectivity of the legal system over time, we distinguish between macro-level connectivity (3.2.1), meso-level connectivity (3.2.2), and micro-level connectivity (3.2.3).

To assess the evolution of individual units of law (e.g., individual court decisions or chapters of a regulation) in the legal system, we create profiles of these units covering all temporal snapshots in the document collection under study. More specifically, based on the quotient graphs that are created on the level of our unit of interest and contain only reference edges (like the preprocessed graphs described in Step 1, Section 3.2.2), we track ten statistics in five groups (note that not all of these statistics can change over time for units of all legal document types):

1. the number of tokens and the number of unique tokens,

These statistics capture how the unit of law in focus evolves in size (number of tokens), topical breadth (number of unique tokens), structure (number of items above, on, and below the sequence level), self-referentiality (number of self-loops), scope of interdependence within the legal system (weighted in-degree and weighted out-degree), and diversity of interdependence within the legal system (binary in-degree and binary out-degree). Finally, by constructing the ego graphs of the profiled unit for its out-neighborhood and its in-neighborhood and following the evolution of these ego graphs across snapshots, we assess to what extent a profiled unit references which other units (reliance) and to what extent it is referenced by which other units (responsibility).

Figure 5 summarizes the growth of the United States legal system and the German legal system as measured by the tokens, structural elements, and lateral references contained in their codified law. Each row of the figure corresponds to a country, and each column corresponds to a document type. All counts are divided by their value in 1998, i.e., Figure 5 depicts growth relative to the 1998 baseline. Supplementing the time series data, Table 2 provides the absolute counts for 1998 and 2019 and the total percentage change between these years (Δ).

Figure 5 and Table 2 show that over the last two decades, the legal systems of both countries have grown substantially. In the United States, the USC (containing codified statutes) has over 60 new structural elements (e.g., chapters, parts, or sections) in 2019 for every 100 such elements it had in 1998. Notably, as evident from the upper left panel of Figure 5, the growth rate of the USC appears to have experienced two distinct periods when measured by its structural elements: one period with a monotonic growth rate of approximately 4% per year (1998–2010), followed by another period with a decelerated monotonic growth rate of approximately 2% per year (2010–2019). At a slightly lower level, this trend also occurs for both the number of tokens and the number of intra-USC references. For example, there are approximately 40 new tokens or references in 2019 for every 100 tokens or references that existed in 1998. Shifting the focus for the United States to the CFR (containing codified regulations), as observable from the upper right panel of Figure 5, the quantity of regulations has increased by an even greater factor. For every 100 structural elements or tokens that were present in 1998, approximately 90 additional elements or tokens exist in 2019. This increase is even more extreme for intra-CFR references, where there are almost 160 new references in 2019 for every 100 that existed in 1998. Apart from brief intervals of stagnation or slight decrease (2009–2010, 2013–2014), these increases have been monotonic.

Corresponding trends for German statutes and regulations are presented in the bottom row of Figure 5. Growth in the German legal system has been qualitatively similar to that in the United States legal system but quantitatively less pronounced and of different functional shape. For both German statutes and German regulations, there are approximately 30 new structural elements in 2019 for every 100 that existed in 1998. Unlike in the United States, however, this growth has been non-monotonic: When measured through structural elements, both statutes and regulations experienced some periods of shallow decline between 2005 and 2010. These shrinking periods are generally not mirrored by the token and lateral reference counts, with one notable exception: In the period from 2005 to 2006, all German statistics decreased. This is likely due to statutes aiming to cleanse the law (Rechtsbereinigungsgesetze), eight of which were introduced in 2006 (recall that our 2006 snapshot represents the law at the end of 2006). ⁴ In total, there are approximately 55 new statute tokens in Germany in 2019 for every 100 such tokens that existed in 1998. Like regulations in the United States, German statutes experienced a greater increase in the quantity of lateral references than in other metrics: For every 100 references in 1998, there are approximately 70 new references in 2019. For German regulations, as for statutes in the United States, the rate of change has been more similar across metrics, with growth varying roughly between 30% and 40% (as noted in Table 2). At a high level, the growth of the German legal system thus seems to be driven by statutes, whereas the growth of the United States legal system appears to be driven by regulations.

Figure 5 and Table 2 only account for lateral references, excluding references between documents of different types. Therefore, Figure 6 shows growth relative to the 1998 baseline for lateral and upward (i.e., regulation-to-statute) references. We exclude downward references because they are very few in number (which means that even a small absolute increase results in a large relative increase) but note that, contrary to the legal theory intuition, they do occur. ⁵ As evident from Figure 6, upward references have grown at similar rates in both countries, with approximately 80 new upward references existing in 2019 for every 100 upward references that existed in 1998. This relative increase is larger than that of the lateral references in both countries, with the exception of lateral regulation references in the United States, whose growth rate dwarfs all others. Since the token and structural element growth rates of German regulations are lower than or similar to those of German statutes, this means that over the period under study, connectivity between statutes and regulations in Germany has grown faster than connectivity within statutes or within regulations.

To evaluate how the growth in the number of references affects the differences in the prominence of individual sections of codified law in the legal systems under study, in Figure 7, we examine the in-degree distribution and the out-degree distribution of the graphs induced by the reference edges in 1998 and 2019 (an analogous figure normalizing section degrees by section size in tokens can be found in Section 4.1 of the Supplementary Material). Since these distributions are highly skewed (as in many graphs arising from complex systems), we plot them on a log-log scale.

All distributions plotted in Figure 7 demonstrate features common among graphs arising from bibliometric dynamics. For example, most sections of statutes and regulations in the United States and Germany are referenced very few times (if at all). The detailed characteristics of the distributions, however, differ between distribution types, document types, and countries: For the United States, the out-degree distributions exhibit less right skew than their in-degree counterparts, while in Germany, we observe the opposite: All out-degree distributions are either within an order of magnitude of or have a longer and thicker right tail than their in-degree counterparts. Similarly, the sections contained in United States regulations exhibit a higher degree of skew in their in-degree distributions than the sections contained in United States statutes (e.g., a higher fraction of these sections has more than 500 ingoing references), but in Germany, the opposite phenomenon occurs at a lower absolute level: There are many statute sections with more than 100 ingoing references but hardly any regulation sections clearing that threshold. These national divergences might be partly due to the differing ratio between statutes and regulations, but they could also point to peculiarities in United States and German drafting style.

Comparing all distributions across countries, we observe that the United States legal system exhibits more extreme statistics than the German legal system (which might, at least in part, be due to its larger size). Finally, we see that from 1998 to 2019, most distributions shift to the right, i.e., the tails become both longer and thicker, which is in line with bibliometric preferential attachment dynamics [59, 60]. This indicates that reference growth has disparate impact, amplifying the differences in relevance between the individual sections contained in United States and German statutes and regulations. As a consequence, the difficulty of navigating the law increases more slowly than the growth of the reference count may suggest.

When exploring the connectivity of the national legal systems of the United States and Germany over time, we distinguish between the macro level, the meso level, and the micro level as suggested in Section 3.2.

In a step toward developing a dashboard for measuring and monitoring the law, we demonstrate the utility of the profiling procedure described in Section 3.3 by applying it to selected statutes and regulations from the United States and Germany in a case study focusing on financial regulation. We profile a total of four statutes (two from each country) that constitute landmark legislation in this domain and trace their statistics over time in Figure 12, along with those of two additional regulations from the same area.

12 USC Ch. 16, popularly known as the Gramm-Leach-Bliley Act or Financial Services Modernization Act of 1999 (GLBA), liberalized the United States financial market by allowing the combination of investment banks, commercial banks, and insurance companies in one institution. It has been in effect for nearly our entire investigation period (1999–2019) and, as indicated by nearly flat lines in all but the panels related to in-degree, has not materially changed. However, the interaction of the GLBA with other parts of the legal system has been anything but static, with its initial weighted in-degree of 1000 increasing by 60% between 1999 and 2019 due to incoming references from other statutes and regulations. Unlike the growth trend in the weighted in-degree, the growth trend in the binary in-degree is nearly monotonic. This indicates that most of the fluctuations in the GLBA’s regulatory environment occur within individual chapters of the USC or the CFR. In summary, the GLBA can therefore be rightfully regarded as a landmark statute, which has required little engineering but has remained an important reference throughout the period under study.

The profile of 12 USC Ch. 53, popularly known as the Dodd–Frank Act or the Wall Street Reform and Consumer Protection Act (DFA), shows similarities with the GLBA in most statistics we track. Introduced in response to the Great Recession in 2010, it is approximately half as old as the GLBA, and like the GLBA, it has barely changed in size, breadth, or structure. But although the DFA is comparable to the GLBA in size, its interaction with the environment seems much more dynamic, with its weighted in-degree increasing by a factor of almost ten over its lifetime. In absolute terms, however, the references increase by less than 500, i.e., in the same order of magnitude as for the GLBA. This is in line with the fact that both statutes are part of the same legal domain, and it highlights how much the evolution of individual pieces of legislation is influenced by their initial conditions, e.g., whether they are strongly connected with their environment already at birth. For the DFA, the growth of the weighted in-degree again is not monotonic, but it is visibly steeper before 2017 than afterwards, leveling off in the last years of the period under study. The binary in-degree, whose gradient is almost monotonically decreasing from the start, anticipates this deceleration by several years. This suggests the existence of an onboarding period, in which the DFA is integrated into the regulatory environment before finding its place in the United States legal system (see the related discussion in [61]).

The profiles of the two German statutes we examine are starkly different from those of the United States statutes. Statutes under the names of both the Stock Exchange Act (SEA) and the Securities Trading Act (STA) have been in effect for the entire observation period. As indicated by their unique token count, they are both constantly narrow in thematic scope (with the STA an order of magnitude more narrow than the SEA to start with), and their token count increases over time. While the SEA and the STA start at comparable sizes in 1998, the STA grows by a factor of seven, while the SEA merely doubles. Possibly as a result, the SEA largely maintains its original number of structures, while the number of structural elements in the STA triples. This is accompanied by an expected, nearly parallel increase in the number of STA sections, and even a decrease of about 25% in the number of SEA sections. Beyond the general growth trends present in almost all STA statistics, the period from 2006 to 2007 stands out, as most of its statistics experience a relatively steep increase between these years. The doctrinal investigation prompted by this observation reveals that the source of the increases is a legislative project translating extensive transparency requirements mandated by the European Union into German law (Transparenzrichtlinie-Umsetzungsgesetz), which came into effect in January 2007. This finding also shows how our methods can complement doctrinal legal scholarship, as has been demonstrated, e.g., for the development of the STA over its entire lifetime [28].

Our statistics produce interesting insights not only for statutes but also for regulations. For example, there is a noticeable increase in the self-referentiality of the CFR chapter about the Commodity Futures Trading Commission (CFTCR), and the German Reports and Documents Regulation (RDR) shows structural changes between 2005 and 2006 as well as between 2015 and 2016. As our framework enables the joint modeling of data from different document types, its application can surface characteristic differences between these types, too. Examining our exemplary regulations and statutes in Figure 12, we find that the CFTCR experiences noticeable growth (about 200%), while the size of the featured statutes barely changes. At the same time, the regulation’s weighted in-degree is several orders of magnitude smaller than that of the DFA or the GLBA, supporting the intuition that statutes should be referenced more frequently than regulations for this particular case. In Germany, the RDR is smaller than the featured German statutes, and it covers less diverse content (as would be expected for a regulation from a legal theory perspective). Its structural organization is minimal, as is its self-referentiality. This confirms that smaller units of law require less internal organization by both structure and references. The RDR references, and is referenced by, a small number of different documents, indicating homogeneity in its regulatory environment. Its weighted out-degree falls between the SEA and STA, i.e., it has a non-negligible number of references to a limited number of targets. In summary, the RDR has most characteristics expected from a German regulation, and its overall profile can clearly be distinguished from that of the featured German statutes.

Our framework enables comparisons not only across document types but also across countries. When examining the DFA and the STA, we see that the STA starts at a size of roughly a third of the DFA but grows to two thirds of the DFA over time. Both statutes have similar degrees of structure at the section level and above, but the DFA contains ten to fifteen times the amount of items below section level, indicating a vastly more granular hierarchical organization. Conversely, the STA contains three to four times more self-loops than the DFA, with its weighted in-degree about 1.5 times and its binary in-degree between 1.5 and 2 times higher than those of the DFA after the first couple of years. This mirrors the more general finding that rule-making agents in the United States and Germany favor different mechanisms to handle the token growth of their statutory corpora: Americans like adding hierarchical structure, while Germans prefer adding references [27].

Figure 13 combines profile statistics concerning size and interdependence to enable direct visual comparisons. Here, we compare the ego graphs of the DFA and the STA for every quarter of their existence during our investigation period. Note that the distance between the snapshots is different for both statutes, as the DFA was adopted only in the middle of our study period, but both series end in 2019. Complementing the statistics presented in Figure 12, the visualization attributes the references to their actual sources and targets, indicating their number by the weight of the edges. For the DFA, its reliance (i.e., how much and how diversely it references other statutes and regulations) barely changes, while its responsibility (i.e., how much and how diversely it is referenced by other statutes and regulations) discernibly increases, as the DFA becomes more and more integrated with its environment. In contrast, both the reliance and the responsibility of the STA increase over time, with its responsibility starting nearly twice as high and increasing at a much faster rate than its reliance. As shown by the edge colors, the diverse responsibility of the STA concerns both statutes and regulations, and the DFA is most intensively responsible for regulations. In line with the expectation derived from legal theory, both the DFA and the STA rely mostly on statutes.