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Influences in Voting and Growing Networks UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Influences in Voting and Growing Networks Permalink https://escholarship.org/uc/item/0fk1x7zx Author Racz, Miklos Zoltan Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Influences in Voting and Growing Networks by Mikl´osZolt´anR´acz A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Statistics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Elchanan Mossel, Chair Professor James W. Pitman Professor Allan M. Sly Professor David S. Ahn Spring 2015 Influences in Voting and Growing Networks Copyright 2015 by Mikl´osZolt´anR´acz 1 Abstract Influences in Voting and Growing Networks by Mikl´osZolt´anR´acz Doctor of Philosophy in Statistics University of California, Berkeley Professor Elchanan Mossel, Chair This thesis studies problems in applied probability using combinatorial techniques. The first part of the thesis focuses on voting, and studies the average-case behavior of voting systems with respect to manipulation of their outcome by voters. Many results in the field of voting are negative; in particular, Gibbard and Satterthwaite showed that no reasonable voting system can be strategyproof (a.k.a. nonmanipulable). We prove a quantitative version of this result, showing that the probability of manipulation is nonnegligible, unless the voting system is close to being a dictatorship. We also study manipulation by a coalition of voters, and show that the transition from being powerless to having absolute power is smooth. These results suggest that manipulation is easy on average for reasonable voting systems, and thus computational complexity cannot hide manipulations completely. The second part of the thesis focuses on statistical inference questions in growing ran- dom graph models. In particular, we study the influence of the seed in random trees grown according to preferential attachment and uniform attachment. While the seed has no effect from a weak local limit point of view in either model, different seeds lead to different distri- butions of limiting trees from a total variation point of view in both models. These results open up a host of new statistical inference questions regarding the temporal dynamics of growing networks. i Mam´anak´esPap´anak ii Contents 1 Introduction 1 1.1 Quantitative social choice . 1 1.2 Statistical inference in networks . 2 1.3 Overview . 3 I Influences in Voting 4 2 A quantitative Gibbard-Satterthwaite theorem 5 2.1 Introduction . 5 2.1.1 Basic setup . 7 2.1.2 Our main result . 8 2.1.3 Discussion . 9 2.1.4 Proof techniques and ideas . 11 2.1.5 Organization of the chapter . 13 2.2 Preliminaries: definitions and previous technical results . 14 2.2.1 Boundaries and influences . 14 2.2.2 Large boundaries . 15 2.2.3 General isoperimetric results . 15 2.2.4 Fibers . 16 2.2.5 Boundaries of boundaries . 17 2.2.6 Reverse hypercontractivity . 17 2.2.7 Dictators and miscellaneous definitions . 17 2.3 Inverse polynomial manipulability for a fixed number of alternatives . 18 2.3.1 Overview of proof . 19 2.3.2 Division into cases . 20 2.3.3 Big mass on large fibers . 21 2.3.4 Big mass on small fibers . 23 2.3.5 Proof of Theorem 2.9 concluded . 27 2.4 An overview of the refined proof . 27 2.5 Refined rankings graph|introduction and preliminaries . 29 2.5.1 Transpositions, boundaries, and influences . 29 iii 2.5.2 Manipulation points on refined boundaries . 30 2.5.3 Large refined boundaries . 30 2.5.4 Fibers . 31 2.5.5 Boundaries of boundaries . 32 2.5.6 Local dictators, conditioning and miscellaneous definitions . 33 2.6 Quantitative Gibbard-Satterthwaite theorem for one voter . 34 2.6.1 Large boundary between two alternatives . 34 2.6.2 Division into cases . 35 2.6.3 Small fiber case . 36 2.6.4 Large fiber case . 44 2.6.5 Proof of Theorem 2.4 concluded . 46 2.7 Inverse polynomial manipulability for any number of alternatives . 46 2.7.1 Division into cases . 47 2.7.2 Small fiber case . 48 2.7.3 Large fiber case . 57 2.7.4 Proof of Theorem 2.35 concluded . 61 2.8 Reduction to distance from truly nonmanipulable SCFs . 62 2.9 Open problems . 65 3 Coalitional manipulation 66 3.1 Introduction . 66 3.1.1 Our results . 67 3.1.2 Additional related work . 70 3.2 Large coalitions . 71 3.3 Small coalitions and the phase transition . 72 3.3.1 Generalized scoring rules, hyperplane rules, and their equivalence . 73 3.3.2 Small coalitions for generalized scoring rules . 76 3.3.3 Smoothness of the phase transition . 81 d 3.4 Decomposing R as the disjoint union of finitely many convex cones: only via hyperplanes . 84 3.5 Most voting rules are hyperplane rules: examples . 88 II Influences in Growing Networks 94 4 The influence of the seed in growing networks 95 4.1 Introduction . 95 4.2 Notions of influence and main results . 96 4.3 Discussion . 98 4.3.1 Chronological history of main results . 98 4.3.2 Comparison of preferential attachment and uniform attachment . 99 4.3.3 Further related work . 100 iv 4.4 Future directions and open problems . 101 5 Preferential attachment trees 103 5.1 Overview . 103 5.2 Useful results on the maximum degree . 103 5.2.1 Previous results . 104 5.2.2 Tail behavior . 106 5.3 Distinguishing trees using the maximum degree . 107 5.3.1 Proofs . 107 5.3.2 Towards an alternative proof of Theorem 4.2 . 110 5.4 The weak limit of PA(n; T ) ........................... 112 5.5 Proof of Lemma 5.1 . 113 6 Uniform attachment trees 119 6.1 Overview . 119 6.2 Partitions and their balancedness: simple examples . 119 6.2.1 Preliminary facts . 120 6.2.2 A simple example . 121 6.2.3 Distinguishing large stars: a proof of Theorem 4.5 . 122 6.3 Proof of Theorem 4.3 . 124 6.3.1 Decorated trees . 124 6.3.2 Statistics and distinguishing martingales . 124 6.3.3 Recurrence relation . 127 6.3.4 Moment estimates . 129 6.3.5 Constructing the martingales . 137 6.4 Comparison to the work of Curien et al. 141 6.5 Technical results used in the proofs . 142 6.5.1 Facts about the beta-binomial distribution . 142 6.5.2 Estimates on sequences . 143 6.5.3 Tail behavior of the diameter . 144 Bibliography 146 v Acknowledgments First and foremost, I thank Elchanan Mossel, my advisor, for being an outstanding mentor, teacher, collaborator, and friend. I feel very lucky to have been in the right place at the right time: during my first semester at Berkeley, Elchanan taught a course called \Social Choice and Networks" which served as a springboard for my PhD. This was a fascinating class with lots of exciting topics and interesting discussions among a great group of students. I am thankful to Elchanan for letting me continue in the same spirit and for accepting me as his student. I could not have wished for a better advisor. I have grown tremendously by learning from Elchanan's advice and from his general approach to life. I have also enjoyed going hiking, climbing, playing frisbee, and just hanging out and having a good meal. Thank you, Elchanan, for everything, and I look forward to more interesting problems and fun in the future. I am grateful to Jim Pitman, Allan Sly, and David Ahn for being on my dissertation committee. I especially thank Jim for his careful reading of my work, for his numerous comments that improved my dissertation, and for his insights and ideas that are leading to further explorations. This thesis would not have been possible without my collaborators. I thank them for all their efforts regarding our projects and for their company which made working on these problems so much more enjoyable. I am thankful to M´artonBal´azsand B´alint T´othfor advising me on my first research problem, a project which we finished during the first year of my PhD; I also thank them for teaching me the fundamentals of probability theory. Another project that was finished during the beginning of my PhD was started during an unforgettable summer at IPAM with Jasmine Nirody and Susana Serna|I will always remember all our laughter. Working with Elchanan is wonderful, and I am constantly amazed at the clarity with which he sees the key issues of the problem at hand. I am very happy that I got to work on a wide variety of problems during my PhD. It was a pleasure to work with Ariel Procaccia, and I also greatly enjoyed and learned from the summer school he organized at CMU. Ton´ciAntunovi´ctaught me many things about P´olya urns and I also thank him for his hospitality during my visits to LA. Misha Shkolnikov's drive to know all things about interacting particle systems is infectious and I am glad that our paths intersected at Berkeley. I am thankful to Eric Friedman for taking me on board his project and to Scott Shenker for his input. Thanks to Nathan Ross for all the great times, I am happy we got to work together; thanks also to Junhyong Kim for doing his best to ground our project in biological reality. I feel very fortunate to have started working with S´ebastienBubeck when he was visiting the Simons Institute and I am thankful for the opportunity to intern with him at Microsoft Research last summer.
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