Graph guided predictions

Abstract

Graphs provide a natural abstraction to model relational and strategic data in domains as diverse as biology (e.g., molecules), multiagent settings (e.g., online vendors on ecommerce platforms), and distributed systems (e.g., Internet). Graphs also find much use as theoretical objects (e.g., probabilistic graphical models), and several important algorithms (e.g., max-flow algorithm for image segmentation) can be used when tasks are formulated in terms of graphs. In this thesis, we focus on three important issues that arise when using graph structured data for prediction: (a) compressing graphs to facilitate predictions, (b) understanding the capability of state-of-the-art algorithms (graph neural networks) operating on graphs, and (c) inferring interaction graphs so as to predict strategic outcomes. Our approach to graph compression builds on the idea of optimal transport specifying the cost of mapping a large graph to a smaller one. The cost decomposes as a flow on the edges, and the selection of the subgraph to retain can be optimized via convex Boolean relaxations. Graph neural networks (GNNs) are naturally suited for making predictions based on graphs but they remain poorly understood in terms of what they can and cannot do. We analyze whether GNNs can distinguish graphs that differ in properties such as cycles, but have similar local structure. We also investigate data dependent generalization bounds for GNNs. In many cases the graph structure is not given but must be inferred. This is the case, for example, in trying to understand how a set of players arrive at their decisions. We study the role of structure -- interaction graph -- in the context of predicting outcomes of such games. We analyze conditions under which players converge to an equilibrium, when the samples from equilibrium are themselves at (near) equilibrium, and when the unknown interaction graph can be identified from a data set of sampled context-dependent outcomes.