Convergence Rates in Dynamic Network Models

Abstract

The stochastic network model by Britton and Lindholm [BL10] describes a class of reasonably realistic dynamics for a complex system with an underlying network structure. In a closed social network, which is modeled by a dynamic random graph, the number of individuals evolves according to a linear birth and death process with per-capita birth rate λ and per-capita death rate µ < λ. A random social index is assigned to each individual at birth, which controls the rate at which connections to other individuals are created. Britton and Lindholm give a somewhat rough proof for the convergence of the degree distribution in this model towards a mixed Poisson distribution. We derive a rate for this convergence giving precise arguments. In order to do so, we deduce the degree distribution at finite time and derive an approximation result for mixed Poisson distributions to compute an upper bound for the total variation distance to the asymptotic degree distribution. We treat the pure birth case and the general case separately and obtain that the degree distribution converges exponentially fast in time in terms of the total variation distance. We reveal that the degree distribution converges at least of order √ te − 1 2 λt in the pure birth case and t 2 e − 1 6 (λ−µ)t in the general case. We compare the model to several other network models and find further interesting results for the model. In particular, we show that the asymptotic degree distribution can exhibit power law tails, which makes it an interesting alternative to the famous preferential attachment models.We finally add a spatial component to the model and find convergence rates for this extended model as well.We prove several general results about linear birth and death processes along the way. Most notably, we derive the age distribution of an individual picked uniformly at random at some finite time by exploiting a bijection between the birth and death tree and a contour process. I am very grateful to my friends who supported me greatly in various ways in the last years, especially to those who helped me with their useful remarks concerning this thesis. I would like to express my heartfelt thanks to my family, in particular my parents, for their constant unconditional support through the years. Finally, I would like to thank my girlfriend, Lara, for her patience and for being extraordinarily supportive.