Tensor decomposition techniques for analysing time-varying networks

The aim of this Ph.D thesis is the study of time-varying networks via theoretical and data-driven approaches. Networks are natural objects to represent a vast variety of systems in nature, e.g., communication networks (phone calls and e-mails), online social networks (Facebook, Twitter), infrastructural networks, etc. Considering the temporal dimension of networks helps to better understand and predict complex phenomena, by taking into account both the fact that links in the network are not continuously active over time and the potential relation between multiple dimensions, such as space and time. A fundamental challenge in this area is the definition of mathematical models and tools able to capture topological and dynamical aspects and to reproduce properties observed on the real dynamics of networks. Thus, the purpose of this thesis is threefold: 1) we will focus on the analysis of the complex mesoscale patterns, as community like structures and their evolution in time, that characterize time-varying networks; 2) we will study how these patterns impact dynamical processes that occur over the network; 3) we will sketch a generative model to study the interplay between topological and temporal patterns of time-varying networks and dynamical processes occurring over the network, e.g., disease spreading. To tackle these problems, we adopt and extend an approach at the intersection between multi-linear algebra and machine learning: the decomposition of time-varying networks represented as tensors (multi-dimensional arrays). In particular, we focus on the study of Non-negative Tensor Factorization (NTF) techniques to detect complex topological and temporal patterns in the network. We first extend the NTF framework to tackle the problem of detecting anomalies in time-varying networks. Then, we propose a technique to approximate and reconstruct time-varying networks affected by missing information, to both recover the missing values and to reproduce dynamical processes on top of the network. Finally, we focus on the analysis of the interplay between the discovered patterns and dynamical processes. To this aim, we use the NTF as an hint to devise a generative model of time-varying networks, in which we can control both the topological and temporal patterns, to identify which of them has a major impact on the dynamics.