INFERENCE OF HIDDEN HIERARCHIES FROM OBSERVABLE NETWORKS

Abstract

In this dissertation I propose probabilistic models and computational estimation methods to infer the hierarchical organization of network systems from the observable patterns of interactions among their elements. My formulation is based on the general idea of statistical network reconstruction (Peixoto (2019a)). Current advancements in this area allowed us to infer some forms of system organization but not hierarchical. Hierarchical systems that manifest as networks are pervasive and statistical models dedicated to their characterization are scarce. This work fills the existing gap that prevents us from understanding several observable phenomena that might be naturally explained by the existence of a hierarchy among their components. Chapter 1 has two parts. First we go through the minimal algebraic and graph theory background on which the addressed problem builds up. We also discuss the general problem we are trying to solve, which is network reconstruction. The second part focuses on hierarchies, as we need clear understanding of what hierarchy means. We discuss why they are important within the network reconstruction framework. Chapter 2 goes through the specific characterizations for hierarchies and hierarchical systems I pick for this dissertation. We focus on those hierarchies, but will also have a high-level introduction to the observables those hierarchies generate. We discuss the general statistical framework that will be later used. Chapter 3 is totally inferential. We will learn how hierarchies can be treated as random objects to enable inference of them from network data. Computational estimation methods as well as hierarchy estimators are discussed. Chapter 4 focuses in one particular model for hierarchical observable networks that unipartite, which is dominance hierarchies. This is an intuitive problem that exemplifies how hierarchies can actually be reconstructed. We derive the theoretical model and fit it to data. Chapter 5 focuses in another particular model for hierarchical observable networks that are bipartite. This problem is more complex and requires more modeling tools. We derive the theoretical 1 model and fit it to data. Chapter 6 summarizes the statistical thinking that drove this dissertation. The main structural features of this family of models are discussed, The focus is on how the hierarchy drives the information passed to the likelihood on these models. This is key to understand all the possibilities this framework offer to explain a variety of real hierarchical systems. We will conclude discussing the contributions of this thesis and its future extensions.