Extreme Events on Higher Order Networks

Abstract

Random walks on complex networks have been studied as physical models for various processes such as diffusion, and also due to their increasing number of applications ranging from web search problems to recommender systems and classification problems in data sciences. They have been helpful in studying systems where Markovian property is assumed, such as the spread of diseases and information diffusion. However, this assumption is insufficient to accurately represent processes in many systems, such as natural language processing and web users' clickstream behaviours, which call for a finer representation of networks. As a result, there has been considerable interest in higher-order models of networks that capture far more information about diffusion processes on networks in recent times. Higher-order networks capture more information than the pair-wise interactions captured by the standard graph paradigms. This thesis aims to study what can be learned about extreme events on networks from the perspective of higher-order networks. We find some analytical results for extreme events on higher-order networks and compare them with numerical results. We also look for relations between properties of extreme events on higher-order representations of undirected networks with respect to standard complex undirected networks: do extreme events on higher-order nodes imply the extreme events on standard complex nodes, and vice-versa? We also study extreme events on directed networks. Finally, we look for extreme events in real taxi data from the city of Porto.