A Breezing Proof of the KMW Bound

In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum degree $\Delta$ on which $\Omega(\min\{\sqrt{\log n/\log \log n},\log \Delta/\log \log \Delta\})$ (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than $15$ years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a fully self-contained and $\mathit{simple}$ proof of the KMW lower bound. The key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs.Comment: 21 pages, 6 figure

Beyond Flatland : exploring graphs in many dimensions

Societies, technologies, economies, ecosystems, organisms, . . . Our world is composed of complex networks—systems with many elements that interact in nontrivial ways. Graphs are natural models of these systems, and scientists have made tremendous progress in developing tools for their analysis. However, research has long focused on relatively simple graph representations and problem specifications, often discarding valuable real-world information in the process. In recent years, the limitations of this approach have become increasingly apparent, but we are just starting to comprehend how more intricate data representations and problem formulations might benefit our understanding of relational phenomena. Against this background, our thesis sets out to explore graphs in five dimensions: descriptivity, multiplicity, complexity, expressivity, and responsibility. Leveraging tools from graph theory, information theory, probability theory, geometry, and topology, we develop methods to (1) descriptively compare individual graphs, (2) characterize similarities and differences between groups of multiple graphs, (3) critically assess the complexity of relational data representations and their associated scientific culture, (4) extract expressive features from and for hypergraphs, and (5) responsibly mitigate the risks induced by graph-structured content recommendations. Thus, our thesis is naturally situated at the intersection of graph mining, graph learning, and network analysis.Gesellschaften, Technologien, Volkswirtschaften, Ökosysteme, Organismen, . . . Unsere Welt besteht aus komplexen Netzwerken—Systemen mit vielen Elementen, die auf nichttriviale Weise interagieren. Graphen sind natürliche Modelle dieser Systeme, und die Wissenschaft hat bei der Entwicklung von Methoden zu ihrer Analyse große Fortschritte gemacht. Allerdings hat sich die Forschung lange auf relativ einfache Graphrepräsentationen und Problemspezifikationen beschränkt, oft unter Vernachlässigung wertvoller Informationen aus der realen Welt. In den vergangenen Jahren sind die Grenzen dieser Herangehensweise zunehmend deutlich geworden, aber wir beginnen gerade erst zu erfassen, wie unser Verständnis relationaler Phänomene von intrikateren Datenrepräsentationen und Problemstellungen profitieren kann. Vor diesem Hintergrund erkundet unsere Dissertation Graphen in fünf Dimensionen: Deskriptivität, Multiplizität, Komplexität, Expressivität, und Verantwortung. Mithilfe von Graphentheorie, Informationstheorie, Wahrscheinlichkeitstheorie, Geometrie und Topologie entwickeln wir Methoden, welche (1) einzelne Graphen deskriptiv vergleichen, (2) Gemeinsamkeiten und Unterschiede zwischen Gruppen multipler Graphen charakterisieren, (3) die Komplexität relationaler Datenrepräsentationen und der mit ihnen verbundenen Wissenschaftskultur kritisch beleuchten, (4) expressive Merkmale von und für Hypergraphen extrahieren, und (5) verantwortungsvoll den Risiken begegnen, welche die Graphstruktur von Inhaltsempfehlungen mit sich bringt. Damit liegt unsere Dissertation naturgemäß an der Schnittstelle zwischen Graph Mining, Graph Learning und Netzwerkanalyse

Juristische Netzwerkforschung

Legal network science explores how legal phenomena can be represented as networks and investigates what can be gained from their quantification and visualization. Corinna Coupette introduces legal network science to the German legal discourse. Based on an original dataset of decisions by Germany's Federal Constitutional Court, she develops tools for modeling, measuring, and mapping the law. The work includes an online appendix, which is freely available at : https://doi.org/10.1628/978-3-16-157012-4-appendi

Evaluating the "Learning on Graphs" Conference Experience

With machine learning conferences growing ever larger, and reviewing processes becoming increasingly elaborate, more data-driven insights into their workings are required. In this report, we present the results of a survey accompanying the first "Learning on Graphs" (LoG) Conference. The survey was directed to evaluate the submission and review process from different perspectives, including authors, reviewers, and area chairs alike

Complex Societies and the Growth of the Law

While a large number of informal factors influence how people interact, modern societies rely upon law as a primary mechanism to formally control human behaviour. How legal rules impact societal development depends on the interplay between two types of actors: the people who create the rules and the people to which the rules potentially apply. We hypothesise that an increasingly diverse and interconnected society might create increasingly diverse and interconnected rules, and assert that legal networks provide a useful lens through which to observe the interaction between law and society. To evaluate these propositions, we present a novel and generalizable model of statutory materials as multidimensional, time-evolving document networks. Applying this model to the federal legislation of the United States and Germany, we find impressive expansion in the size and complexity of laws over the past two and a half decades. We investigate the sources of this development using methods from network science and natural language processing. To allow for cross-country comparisons over time, we algorithmically reorganise the legislative materials of the United States and Germany into cluster families that reflect legal topics. This reorganisation reveals that the main driver behind the growth of the law in both jurisdictions is the expansion of the welfare state, backed by an expansion of the tax state.Comment: 22 pages, 6 figures (main paper); 28 pages, 11 figures (supplementary information

Measuring Law Over Time: A Network Analytical Framework with an Application to Statutes and Regulations in the United States and Germany

How do complex social systems evolve in the modern world? This question lies at the heart of social physics, and network analysis has proven critical in providing answers to it. In recent years, network analysis has also been used to gain a quantitative understanding of law as a complex adaptive system, but most research has focused on legal documents of a single type, and there exists no unified framework for quantitative legal document analysis using network analytical tools. Against this background, we present a comprehensive framework for analyzing legal documents as multi-dimensional, dynamic document networks. We demonstrate the utility of this framework by applying it to an original dataset of statutes and regulations from two different countries, the United States and Germany, spanning more than twenty years (1998-2019). Our framework provides tools for assessing the size and connectivity of the legal system as viewed through the lens of specific document collections as well as for tracking the evolution of individual legal documents over time. Implementing the framework for our dataset, we find that at the federal level, the United States legal system is increasingly dominated by regulations, whereas the German legal system remains governed by statutes. This holds regardless of whether we measure the systems at the macro, the meso, or the micro level.Comment: 32 pages, 13 figures (main paper); 32 pages, 14 figures (supplementary information

Graph Similarity Description: How Are These Graphs Similar?

How do social networks differ across platforms? How do information networks change over time? Answering questions like these requires us to compare two or more graphs. This task is commonly treated as a measurement problem, but numerical answers give limited insight. Here, we argue that if the goal is to gain understanding, we should treat graph similarity assessment as a description problem instead. We formalize this problem as a model selection task using the Minimum Description Length principle, capturing the similarity of the input graphs in a common model and the differences between them in transformations to individual models. To discover good models, we propose Momo, which breaks the problem into two parts and introduces efficient algorithms for each. Through an extensive set of experiments on a wide range of synthetic and real-world graphs, we confirm that Momo works well in practic

Simplify Your Law: Using Information Theory to Deduplicate Legal Documents

Textual redundancy is one of the main challenges to ensuring that legal texts remain comprehensible and maintainable. Drawing inspiration from the refactoring literature in software engineering, which has developed methods to expose and eliminate duplicated code, we introduce the duplicated phrase detection problem for legal texts and propose the Dupex algorithm to solve it. Leveraging the Minimum Description Length principle from information theory, Dupex identifies a set of duplicated phrases, called patterns, that together best compress a given input text. Through an extensive set of experiments on the Titles of the United States Code, we confirm that our algorithm works well in practice: Dupex will help you simplify your law.Comment: 8 pages, 3 figures; to appear in ICDMW 202

Differentially Describing Groups of Graphs

How does neural connectivity in autistic children differ from neural connectivity in healthy children or autistic youths? What patterns in global trade networks are shared across classes of goods, and how do these patterns change over time? Answering questions like these requires us to differentially describe groups of graphs: Given a set of graphs and a parti- tion of these graphs into groups, discover what graphs in one group have in common, how they systematically differ from graphs in other groups, and how multiple groups of graphs are related. We refer to this task as graph group analysis, which seeks to describe similarities and differences between graph groups by means of statistically significant subgraphs. To per- form graph group analysis, we introduce GRAGRA, which uses maximum entropy modeling to identify a non-redundant set of subgraphs with statistically significant associations to one or more graph groups. Through an extensive set of ex- periments on a wide range of synthetic and real-world graph groups, we confirm that GRAGRA works well in practice