Sebastian Rosengren Networks and processes that live on them are everywhere. We find them in our social structures, in our brain, and in the way a disease Random Graph and Growth spreads amongst us. There is a need to understand these structures and processes. Paper I deals with a dynamic extension to the famous Erdös- Models Rényi graph. Paper II deals with a multi-type extension of the preferential attachment tree. Paper III is concerned with how the distribution over small degrees affects the size of the giant component Sebastian Rosengren in the configuration model. In Paper IV, we consider the frog model Models and Growth Random Graph and a two-type extension of it, showing that the shape of the asymptotic set does not depend on the initial starting set(s) and particle configuration(s) there. Paper V is concerned with the predictability of the set of discovered sites generated by the first passage percolation model. We show that it is possible to predict the shape of this set using a neural network. Sebastian Rosengren is motivated by using probability models to describe complexity. He hopes his research has helped explain a little of the complexity in the world. ISBN 978-91-7911-256-1 Department of Mathematics Doctoral Thesis in Mathematical Statistics at Stockholm University, Sweden 2020 Random Graph and Growth Models Sebastian Rosengren Academic dissertation for the Degree of Doctor of Philosophy in Mathematical Statistics at Stockholm University to be publicly defended on Friday 25 September 2020 at 13.00 in sal 15, hus 5, Kräftriket, Roslagsvägen 101. Abstract Random graphs is a well-studied field of probability theory, and have proven very useful in a range of applications — modeling social networks, epidemics, and structures on the Internet to name a few. However, most random graphs are static in the sense that the network structure does not change over time. Furthermore, standard models also tend to consist of single-type objects. This puts restrictions on possible applications. The first part of this thesis concerns random graphs with a focus on dynamic and multi-type extensions of standard models. The second part of the thesis deals with random growth models. Random growth models are important objects in probability theory and, as the name suggests, models the random growth of some entity. Typical examples include infectious disease spread; how a liquid flows through a random medium; and tumor growth. The growth of these models, properly scaled by time, tends to be deterministic. The second theme of the thesis concerns the final shape of the growing entity for two standard random growth models. In Paper I, we study a dynamic version of the famous Erdős-Rényi graph. The graph changes dynamically over time but still has the static Erdős-Rényi graph as its stationary distribution. In studying the dynamic graph we present two results. The first result concerns the time to stationarity, and the second concerns the time it takes for the graph to reach a certain number of edges. We also study the time until a large component emerges, as well as how it emerges. In Paper II, we introduce and study an extension of the preferential attachment tree. The standard version is already dynamic, but its vertices are only allowed to be of one type. We introduce a multi-type analog of the preferential attachment tree and study its asymptotic degree distributions as well as its asymptotic composition. Paper III concerns the configuration model — a random graph neither dynamic nor multi-type — and we break with the first theme of the thesis since no extensions are made to the model. Instead, we argue that the size of the largest component in the model does not depend on the tail of the degree distribution, but rather on the distribution over small degrees. This is quantified in some detail. In Paper IV, we consider the frog model on ℤd and a two-type extension of it. For the one-type model, we show that the asymptotic shape does not depend on the initial set and the particle configuration there. For the two-type model, we show that the possibility of both types to coexist also does not depend on the initial sets and the particle configurations there. Paper V is concerned with the predictability of the set of discovered sites generated by the first passage percolation model. First passage percolation has the property that the set of discovered sites, scaled properly by time, converges to some deterministic set as time grows. Typically, not much is known about this set, and to get an impression of it simulations are needed. Using simulated data we show that it is possible to use a neural network to adequately predict the shape, on this dataset, from some easily calculable properties of the passage times. The purpose of the paper is to give researchers a proof of concept of this method as wells as a new tool for quickly getting an impression of the shape. Stockholm 2020 http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-184028 ISBN 978-91-7911-256-1 ISBN 978-91-7911-257-8 Department of Mathematics Stockholm University, 106 91 Stockholm RANDOM GRAPH AND GROWTH MODELS Sebastian Rosengren Random Graph and Growth Models Sebastian Rosengren ©Sebastian Rosengren, Stockholm University 2020 ISBN print 978-91-7911-256-1 ISBN PDF 978-91-7911-257-8 Printed in Sweden by Universitetsservice US-AB, Stockholm 2020 It was the best of times, it was the worst of times — Charles Dickens List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I: A dynamic Erdős-Rényi graph model. Rosengren S., Trapman P. (2019). Markov Processes and Related Fields, 25(2):275-301. II: A multi-type preferential attachment tree. Rosengren, S. (2018). Internet Mathematics, 1(1). DOI:10.24166/im.05.2018 III: The tail does not determine the size of the giant. Deijfen, M., Rosengren, S., Trapman, P. (2018). Journal of Statistical Physics, 173:736–745. DOI:10.1007/s10955-018-2071-4. IV: The initial set in the frog model is irrelevant. Deijfen, M., Rosen- gren, S. (2020). Electronic Communications in Probability, 25(50):1-7 DOI:10.1214/20-ECP329 V: Predicting first passage percolation shapes using neural net- works. Rosengren, S. (2020). arXiv:2006.14004. Reprints were made with permission from the publishers. Author’s contributions: S. Rosengren has taken an active part in develop- ing the content of all papers. Paper I was based on the ideas of P. Trapman. The results in the first part of the paper was derived by S. Rosengren, and the second part was done jointly. The writing of the manuscript was done mainly by S. Rosengren. S. Rosengren is the sole author of Paper II, which is based on the ideas of M. Deijfen. Paper III was based on the ideas of M. Deijfen and P. Trapman. S. Rosengren derived all main results, and the simulations were done by S. Rosengren. The writing of the manuscript was mainly done by M. Deijfen. Paper IV was based on the ideas of M. Deijfen. The coupling argument, important for the main results, was provided by M. Deijfen but details were carried out by S. Rosengren. The writing of the manuscript was done jointly. Finally, S. Rosengren is the only author of Paper V, which is based on his ideas. Acknowledgments Here we are then, at the end of this journey. I know what you are thinking. A lazy person of average intelligence, now a PhD in Mathematical Statistics, how the hell did he pull this off? Turns out, you can get quite far with witty banter and wry observations. Joking aside, the answer is of course that I have had a lot of help, and I would like to thank all the people who made this thesis possible. First and foremost, I would like to thank my supervisor Professor Mia Deijfen. For her continuous support and guidance, her wisdom, intelligence, and endless patience. Thank you for being so generous with your ideas, and your time. My co-supervisor Pieter Trapman for all our fruitful discussions, collaborations, and for encouraging me to pursue a PhD. I want to thank all my colleagues at the Department of Mathematics at Stock- holm University for providing a fun and friendly work environment. To all the PhD students at the department — thank you for all the fun we have had. A special thank you to Felix, Måns, Hampus, and Erik for being very good friends and for making the quarantine quite enjoyable. Thank you Carl and Gabriel for our ventures outside academia. Finally, I want to thank my family, for all their love and support. My father for interesting me in the more difficult things of my life — mathematics and running — and for teaching me that the best tastes are acquired. My mother for taking such good care of me and my brother, and for always cheering us on. Contents List of Papers i Acknowledgments iii I Introduction 3 1 Random Graphs 5 1.1 Motivation . .5 1.1.1 Graph Properties . .6 1.2 Random Graph Models . .9 1.2.1 Erdős-Rényi Graph . .9 1.2.2 Preferential Attachment Models . 10 1.2.3 The Configuration Model . 11 1.3 Dynamic Processes on Graphs . 13 2 Random Growth Models 15 2.1.1 First Passage Percolation . 16 d 2.1.2 The Frog Model on Z ....................... 18 3 Overview of Papers 21 3.1 Paper I . 21 3.2 Paper II . 22 3.3 Paper III .