Giesela Rühl (Humboldt University of Berlin) explains that during the past two decades, German courts have experienced a dramatic decline in cases. While the causes for the loss remain unclear, it is plausible that German courts are not an attractive means of resolving lower-value claims. Thus, these claims remain unenforced. A number of legal tech companies have entered the German legal services market to mitigate that problem. These companies enforce lower-value claims and are extremely popular with consumers. The legal profession, however, has met all this with skepticism – and at times even with hostility – as some lawyers question whether legal tech companies illegally provide legal services. These discussions have since led to various court cases, as well as the adoption of a new federal law that specifically targets legal tech companies. The chapter critically engages with these developments, outlining the regulatory environment for the provision of legal services in Germany as well as relevant case law and legislation. Overall, the chapter hypothesizes that access to justice in Germany has benefited from legal tech companies but that important problems remain to be addressed.

This appendix delves into the mathematical foundations of network representation techniques, focusing on two key areas: maximum likelihood estimation (MLE) and spectral embedding theory. It begins by exploring MLE for Erdös-Rényi (ER) and stochastic block model (SBM) networks, demonstrating the unbiasedness and consistency of estimators. The limitations of MLE for more complex models are discussed, leading to the introduction of spectral methods. The chapter then presents theoretical considerations for spectral embeddings, including the adjacency spectral embedding (ASE) and its statistical properties. It explores the concepts of consistency and asymptotic normality in the context of random dot product graphs (RDPGs). Finally, we extend these insights to multiple network models, covering graph matching for correlated networks and joint spectral embeddings like the omnibus embedding and multiple adjacency spectral embedding (MASE).

This chapter presents a unified framework for analyzing complex networks through statistical models. Starting with the Inhomogeneous Erdős-Rényi model’s concept of independent edge probabilities, we progress through increasingly sophisticated representations, including the Erdös-Rényi, Stochastic Block Model, and Random Dot Product Graph (RDPG) models. We explore how each model generalizes its predecessors, with the RDPG encompassing many earlier models under certain conditions. The crucial role of positive semidefiniteness in connecting block models to RDPGs is examined, providing insight into model interrelationships. We also introduce models addressing specific network characteristics, such as heterogeneous node degrees and edge-based clustering. The chapter extends to multiple and correlated network models, demonstrating how concepts from simpler models inform more complex scenarios. A hierarchical framework is presented, unifying these models and illustrating their relative generality, thus laying the groundwork for advanced network analysis techniques.

W. Bradley Wendel (Cornell Law) provides a useful counterpoint to a set of chapters focused on mapping the connection between the current regime of legal services regulation and access to justice. His chapter is a passionate defense of the traditional lawyer’s role as a defender of key public values and a bulwark of rule of law. His chapter elegantly reminds readers that lawyers and the legal profession sit at an important crossroads as essential defenders of rule-of-law values that are under attack and yet waning in their market and cultural power.

This chapter explores practical applications of network representation learning techniques for analyzing individual networks. It begins by addressing the community detection problem, demonstrating how to estimate community labels using network embeddings. The chapter then discusses the challenges posed by network sparsity and introduces efficient storage methods for sparse networks. The text proceeds to examine testing for differences between groups of edges, applying hypothesis testing to stochastic block models and structured independent edge models. It also covers model selection techniques for stochastic block models, helping readers choose appropriate levels of model complexity. The chapter introduces the vertex nomination problem, which aims to identify nodes similar to a set of known "seed" nodes. It presents spectral vertex nomination techniques and explores extensions to related problems. Finally, the chapter addresses out-of-sample embedding, providing efficient strategies for embedding new nodes into existing network representations. This approach is particularly valuable for large-scale, dynamic networks where frequent re-embedding would be computationally prohibitive.

Brian Libgober (Northwestern Political Science) drills down on the well-known but critically important fact that the justice gap particularly afflicts communities of color. Libgober tours new research finding that African Americans face significant barriers in finding lawyers, perhaps because of anticipated decisional bias within the legal system. The result is a bracing reminder that the justice gap is rooted in much wider structures of racial inequality and a profit-oriented legal marketplace that systematically under-serves certain segments of the population. His work shows the urgency – and difficulty – of meaningful reform.

This chapter explores techniques for analyzing and comparing pairs of networks, building on previously introduced statistical models and representation learning methods. It focuses on two-sample testing for networks, introducing methods to determine whether two network observations are sampled from the same or different random networks. The chapter covers latent position and distribution testing, addressing nonidentifiability issues in network comparisons. It then explores specialized techniques for comparing stochastic block models (SBMs), leveraging their community structure and discussing methods for testing differences in block matrices, including density adjustment approaches. A significant portion is devoted to the graph matching problem, addressing the challenge of identifying node correspondences between networks. This section introduces permutation matrices and explores optimization-based methods, including gradient descent approaches, for both exact and inexact matching scenarios. Throughout, the chapter emphasizes practical implementations with code examples, bridging the gap between theoretical concepts and real-world applications in network analysis. These techniques provide a comprehensive toolkit for comparing networks, essential for understanding evolving networks, analyzing differences across domains, and integrating multisource network data.

Natalie Byrom explains how the Legal Services Act 2007 (LSA 2007) aimed to reform legal services in England and Wales to enhance consumer protection and access to justice. However, its focus on professional titles and reserved activities created complexity and hindered innovation, especially for low-income individuals. Public funding cuts in 2013 worsened the situation, leading to increased self-representation and strain on the judiciary. In response, the Ministry of Justice and Senior Judiciary launched a £1.3bn digital reform in 2014 to modernize court operations. However, by 2023, only twenty-four out of forty-four projects were completed, with key initiatives like the Online Solutions Court abandoned due to delays and COVID-19 disruptions. In November 2023, a new vision proposed a public–private partnership for digital justice, leveraging technology to streamline processes and support from private sector services. This raises questions about market readiness, incentives for data sharing, and necessary regulatory adjustments to ensure fair access to justice. Addressing these challenges is crucial for improving legal service delivery and access to justice.

This appendix provides a comprehensive overview of statistical network models, building from fundamental concepts to advanced frameworks. It begins with essential mathematical background and probability theory, then introduces the foundations of random network models. The appendix covers a range of models, including Erdös-Rényi, stochastic block models (both a priori and a posteriori), random dot product graphs, and their generalizations. Each model is presented with its parameters, generative process, probability calculations, and equivalence classes. The appendix also explores degree-corrected variants and the Inhomogeneous Erdös-Rényi model. Throughout, we emphasize the relationships between models and their increasing complexity, providing a solid theoretical foundation for understanding network structures and dynamics.

David Freeman Engstrom (Stanford) and Daniel B. Rodriguez (Northwestern) argue that current structure of American legal services regulation, known as “Our Bar Federalism,” is outdated. Fifty states maintain their own rules and regulatory apparatus for a legal profession and industry that are now national and multinational. This fragmented system is a key factor in the American civil justice system’s access-to-justice crisis, where restrictive state rules support the lawyers’ monopoly. With new legal services delivery models and AI, this scheme will seem increasingly provincial and retrograde. This chapter argues it’s time to rethink "Our Bar Federalism," and explore hybrid state-federal regulatory system.

David Engstrom and Jess Lu (both Stanford Law) first show that an otherwise fast-growing and dynamic “legal tech” industry has not generated significant “direct-to-consumer” technologies designed to help self-represented litigants navigate a complex legal system. They then interrogate that puzzle: Why is it that better consumer legal tech hasn’t flourished? They ultimately settle on the idea that rule reforms alone may not stimulate high-scale, direct-to-consumer technology. Instead, other policy interventions may be necessary, including standardizing what is currently a checkerboard of court technology and data infrastructures. Perhaps more importantly, direct-to-consumer legal tech may have trouble overcoming some of the problems that are inherent to markets that are attempting to serve individuals with episodic attachment to the civil justice system and limited ability to pay. The result is an important meditation on whether reforms to UPL, Rule 5.4, or something else entirely are necessary to unlock the potential of potent new technologies in order to narrow the justice gap.

This chapter presents a comprehensive workflow for applying network machine learning to functional MRI connectomes. We demonstrate data preprocessing, edge weight transformations, and spectral embedding techniques to analyze multiple brain networks simultaneously. Using multiple adjacency spectral embedding (MASE) and unsupervised clustering, we identify functionally similar brain regions across subjects. Results are visualized through abstract representations and brain-space projections, and compared with established brain parcellations. Our findings reveal that MASE-derived communities often align with known functional and spatial organization of the brain, particularly in occipital and parietal areas, while also identifying regions where functional similarity doesn’t imply spatial proximity. We illustrate how network machine learning can uncover meaningful patterns in complex neuroimaging data, emphasizing the importance of combining algorithmic approaches with domain expertise to motivate the remainder of the book.

Rebecca Sandefur (Arizona State) and Mathew Burnett (American Bar Foundation) – one a MacArthur Genius Award-winning sociologist, the other a longtime leader on access-to-justice issues – explore ways to reform legal services regulation, from relaxing UPL rules (to welcome new providers into the system) to relaxing Rule 5.4’s bar on nonlawyer ownership of law firms (to make available new sources of capital investment). After reviewing existing empirical evidence, they argue in favor of the former, in order to spur new human-centered service models, as against longer-term and less proven reforms altering law firm ownership.