"Networks are fundamental tools for the analysis of complex systems. In fact, we often state questions and develop analysis in terms of nodes and edges. For example, we can describe a social system by modeling individuals as nodes and social interactions as edges between pair of nodes. Similarly, we can model a biochemical reaction by assigning a node to each protein and edges between them to model physical contacts of high specificity, so-called protein-protein interactions. Due to the broad scope of this modeling paradigm,
the analysis of systems or datasets as networks has enjoyed tremendous success over the last decade.
This project has focused on the task known as network centrality where one wants to quantify the importance of the nodes of a network by exploiting only its topological structure, i.e.\ the nodes and the edges between them, without assuming any other external information about the data.
Being able to identify important nodes is a very relevant issue which is useful in numerous contexts. For example, it can be used to identify the most important individuals in a communication system and thus maximize the effect of commercial campaigns, to locate infected nodes in a population with diseases and thus to predict the spreading of diseases, to improve the ordering of the results of search engines, to identify common patterns in digital datasets to improve text-to-speech computer systems, or to discover and predict anomalous financial quantity behaviors.
From a mathematical point of view, network centrality requires to address two main issues: to design the model that defines the importance of the nodes and to develop algorithms that allow to efficiently compute such importances on large network datasets.
One of the oldest and most successful centrality models, known as ""eigenvector centrality"", is based on the following LINEAR mutual reinforcing argument: the importance of a node is proportional to the sum of the importances of the nodes it is connected to. For example, this is the model used by the Google search engine to decide the ordering of the pages result of a search query. Together with its well understood mathematical model, the success of eigenvector centrality is due to a simple algorithm that can be used for its computation: the so-called ""power method"".
However, because of advances in technology and the increased digitisation of
human behaviour, network data is growing rapidly in terms of size and variety and the linearity of the eigenvector centrality model is a big limitation that prevents to fully exploit the complexity of modern network data.
This project had two major overall objectives:
1) Introduce new nonlinear eigenvector centrality models able to capture network structure details and complex higher-order node interactions that are overlooked by simpler linear models.
2) Design bespoke nonlinear versions of the power method in order to efficiently compute the newly introduced nonlinear centrality scores for large real-world complex networks."
