Generalized Shannon entropies on graphs with applications

Networks are ubiquitous models of complex systems, from the Internet to protein interactions, to neuronal connectivity in the brain, and in countless other instances. In many contexts, it is useful to assess which node is more important in a network, a seemingly trivial task that, however, depends crucially on which definition of "important" we are using. This notion not only depends on the specific application but also on the metric used to rank the nodes in the network. There are many metrics used in the literature, ranging from classical measures like degree, betweenness, and closeness, to eigenvector centrality, and so on. In this thesis, we focus on how effective a node is at transmitting information through a network. To do so, we define and study two new centrality measures: Generalized Shannon entropy (S) and Generalized Shannon information (I). First, we compare these new metrics to classical centrality measures and to two other measures recently proposed in the literature, namely Accessibility and Generalized Accessibility, and quantify the correlation between our proposals and the classical ones. We then focus on applications, particularly on attacks on networks, where we investigate whether the new centralities predict the most critical node in the case of node deletion. Finally, we study SIR models to determine whether our centralities can predict or prevent the outcome of an epidemic spreading across the network.