Extreme Events on Complex Networks

Abstract

The aim of the project was to study extreme events on complex networks. The study of complex networks, especially that of scale free and small world networks, has been an important topic of research in recent times. In this project, extreme events on complex networks were studied by using random walks to model flow on the network. By considering a set of independent random walkers to be moving on the network simultaneously, an extreme event on the network was defined as an event when the number of walkers exceeds a certain threshold. The existence of a stationary probability distribution, that is, the probability that a walker will be found on a node at a given time, allows for this definition of an extreme event. There is abundant literature on stationary occupancy probability distributions of random walks on networks. Review of literature on the dynamics and stationary probability distributions in the case of discrete and continuous time random walks constituted the first part of the project. In the next part of the project, some statistics of extreme events were studied. The probability of extreme events was obtained through simulations on a scale free network both in the discrete and continuous random walk cases. Correlation between magnitude differences of two consecutive extreme events and the time interval between the occurrence of the two was computed analytically and through simulations on scale-free, small-world and random networks. As a separate part of the project, the spectral properties of the adjacency matrices of scale-free networks were studied.