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Fiber Graphs Fiber graphs Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) von Tobias Windisch, M.Sc. (hons) geb. am 12.10.1987 in Bamberg genehmigt durch die Fakultät für Mathematik der Otto-von-Guericke-Universität Magdeburg Gutachter: Jun.-Prof. Dr. Thomas Kahle Otto-von-Guericke-Universität Magdeburg Prof. Seth Sullivant, PhD North Carolina State University eingereicht am: 22.07.2016 Verteidigung am: 09.12.2016 Abstract A fiber graph is a graph on the integer points of a polytope whose edges come from a set of allowed moves. Fiber graphs are given implicitly which makes them a useful tool in many applications of statistics and discrete optimization whenever an exploration of vast discrete structures is needed. The first part of this thesis discusses the graph-theoretic structure of fiber graphs with a particular focus on their diameter and edge-expansion. We define the fiber dimension of a simple graph as the smallest dimension where it can be represented as a fiber graph and prove an upper bound on the fiber dimension that only depends on the chromatic number of the graph. In the second part, random walks on fiber graphs are studied and it is shown that, when a fixed set of moves is used, rapid mixing is impossible. In order to improve mixing rates for fiber walks in fixed dimension, we evaluate possible adaptions of the set of moves, one that adds a growing number of linear combinations of moves to the set of allowed moves and one that allows arbitrary lengths of single moves. We show that both methods lead to spectral expanders in fixed dimension. Finally, the parity binomial edge ideal of a graph is introduced. Unlike the binomial edge ideal, it does not have a square-free Gröbner bases and is radical if only if the graph is bipartite or the characteristic of the ground field is not two. We compute the universal Gröbner basis and the minimal primes and show that both encode combinatorics of even and odd walks. Zusammenfassung Ein Fasergraph ist ein Graph auf den ganzzahligen Vektoren eines Polytops, dessen Kanten aus einer Menge zugelassener Richtungsvektoren entstehen. Fasergraphen sind implizit gegeben und deshalb ein wichtiges Werkzeug in vielen Anwendungen der Statistik und Optimierung, wann immer riesige diskrete Strukturen untersucht werden. Der erste Teil dieser Arbeit beschäftigt sich mit der graphen-theoretischen Struktur von Fasergraphen mit besonderem Augenmerk auf deren Durchmesser und Kanten-Expansion. Wir definieren die Faserdimension eines einfachen Graphen als die kleinste Dimension, in der er als Fasergraph dargestellt werden kann, und beweisen eine obere Schranke der Faserdimension, die nur von der chromatischen Zahl des Graphen abhängt. Im zweiten Teil werden Zufallsbewegungen auf Fasergraphen untersucht und es wird gezeigt, dass mit fixierten Richtungsvektoren eine schnelle Konvergenz nicht möglich ist. Um die Konvergenzrate zu verbessern, untersuchen wir mögliche Anpassungen der zugelassenen Richtungsvektoren: eine, die eine wachsende Zahl an Linearkombinationen der Richtungen hinzufügt und eine zweite, die fixierte Richtungen beliebiger Länge zulässt. Wir zeigen, dass beide Methoden spektrale Expander in fixierter Dimension liefern. Zum Schluss wird das paritäre binomische Kantenideal eines Graphen vorgestellt. Dieses hat, anders als das binomische Kantenideal, keine quadratfreie Gröbnerbasis und ist radikal genau dann, wenn der Graph bipartit oder die Charakteristik des Grundkörpers ungleich zwei ist. Wir berechnen die universelle Gröbnerbasis sowie die minimalen Primideale und zeigen, dass beide eine Kombinatorik von geraden und ungeraden Pfaden kodieren. ii Acknowledgements First and foremost, I am indebted to my advisor, Thomas Kahle, for his excellent guidance and enduring support. During all these years, his door was open for me at any time and he never failed to be reachable. He taught me so extremely many things about mathematics and beyond, not alone because he was always prepared to give an unprepared blackboard talk about almost any topic I came up with. All my knowledge about binomial ideals, Markov bases, 3D-printing, handball, and crypto currencies is due to him. Research is impossible without money, and I would like to thank all the institutions that where willing to support me with theirs: the German National Academic Foundation (Studienstiftung des deutschen Volkes), the TopMath program at TU Munich, and the German Academic Exchange Service (Deutscher Akademischer Austauschdienst). I also would like to thank all the current and former members of the Institute of Algebra and Geometry at OvGU Magdeburg and the research group on Applied Geometry and Discrete Mathematics at TU Munich for the very inspiring working atmosphere and the many great talks, discussions, and coffee breaks. I thank my co-authors Raymond Hemmecke, Thomas Kahle, Camilo Sarmiento, and Caprice Stanley for their work and effort, and for the great time I had while working on our projects. My interests in algebraic statistics and fiber graphs in particular was triggered by Raymond Hemmecke, and I thank him for all his advice and support during my time at TU Munich. I am indebted to Seth Sullivant for pointing me to the hit-and-run algorithm and for initiating the joint work with Caprice Stanley on compressed fiber graphs. I owe much to Alexander Engström who had always a friendly ear for my questions and who encouraged me to pursue them further. Mathematical thanks go also to Gennadiy Averkov, Johannes Hofscheier, and Benjamin Nill for helping me with my various questions on polytopes and to the innumerably many people I talked to on all the conferences I visited over the last years. I also want to express my gratitude to Sonja Petrović, Seth Sullivant, and Ruriko Yoshida for their great hospitality during my US visit. This thesis could not have been created without the many magnificent open-source software packages that helped me finding my theorems. Thus, I am grateful to the developers of 4ti2 [1], LattE [9], Macaulay2 [58], Normaliz [20], polymake [57], R [98], sage [34], and many more for their effort to make mathematics available to computers. I thank all my friends, particularly Linus and Markus, for their tireless persistence to bring me back to life whenever I was buried in mathematics. Finally, I thank Katharina for her unconditional love and her support. In my life as a PhD student, geographically teared between Magdeburg, Immenstadt, Bamberg, and Munich, mentally spread among algebra, combinatorics, and statistics, in planes and trains, up hill and down dale, she was always my universal constant and the source of my motivation. iii Contents 1 Introduction 1 1.1 Random walks.....................................2 1.2 Fiber graphs......................................6 1.3 Running examples...................................9 1.4 Fiber walks in statistics................................ 10 2 Graph properties of fiber graphs 15 2.1 Connectedness and connectivity............................ 15 2.2 Bounds on the diameter................................ 18 2.3 Graph degrees...................................... 22 2.4 Edge-expansions.................................... 24 3 The fiber dimension of a graph 30 3.1 Embeddings....................................... 30 3.2 Fiber dimension one.................................. 33 3.3 Complete graphs.................................... 35 3.4 Distinct pair-sum polytopes.............................. 36 3.5 Computational aspects................................. 38 4 Symmetric fiber walks 40 4.1 Fixed Markov bases.................................. 40 4.2 Adapted Markov bases................................. 44 4.3 Varying constraint matrices.............................. 47 5 Heat-bath walks 49 5.1 Spectral analysis.................................... 50 5.2 Augmenting Markov bases............................... 56 5.3 Best move selection................................... 61 6 Parity binomial edge ideals 63 6.1 Markov bases...................................... 64 6.2 Universal Gröbner basis................................ 66 6.3 Minimal primes..................................... 72 6.4 Radicality........................................ 76 Bibliography 79 Index 87 iv 1 Introduction The assessment of statistical models, based on expectation and experience, tries to reduce the complexity of our ambient world and makes questions about nature amendable to algorithms and mathematics. The prevalent working scheme in inferential statistics is to draw conclusions based on finitely many independent observations about the whole set of interests. Frequently, observations are represented as multi-way contingency tables and the class of log-linear statistical models describes how their attributes relate among each other [28]. Given an observed contingency table on the one and a statistical model on the other hand, an intruding question is how well the model explains the observed data, that is determining its goodness-of-fit. The running engine that many exact goodness-of-fit tests for log-linear models have under their hood is a random walk on a fiber graph. Essentially, a fiber graph is a graph on the lattice points of a polytope where two nodes are adjacent if their difference lies in a set of allowed moves. Being graphs on lattice points, the combinatorial outreach of fiber graphs goes far beyond statistics as they appear naturally in discrete optimization [31] and commutative algebra, particularly in the context of toric ideals [109] and matrix gradings [79]. Regardless of the
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