Tensor-based mathematical framework and new centralities for temporal multilayer networks.

• A novel mathematical model, referred to as a temporal multilayer network, is proposed for exploring complex networked systems with time and space. • Using the mathematical formulation of the fifth-order tensor to represent the temporal multilayer networks. • Based on tensor framework, four important topological metrics are proposed to quantitatively evaluate the temporal multilayer networks. • Two novel iterative refinement centralities are proposed to quantify importance of nodes in temporal multilayer networks. • Using the theory of multilinear algebra and matrix analysis to prove the convergence of two iterative refinement algorithm. Although networks provide a powerful methodology to study a variety of real-world and engineered systems, their current formulation does not typically account for multiple interaction types that vary in space and time, or address interconnected systems such as networks of networks. Unavoidably, ignoring time-dependence and multiple interactions of topological structures might lead to important temporal and structural information loss and may obscure the actual organization. To achieve a deep understanding of the time-varying and interconnected complex networked systems, in this paper, we develop a more general mathematical model, referred to as a temporal multilayer network, which explicitly incorporates time-dependence and multiple relationships of topological structures into a system, and provides a natural and reasonable description for real-world complex systems. Furthermore, using a fifth-order tensorial framework to represent temporal multilayer networks, we generalize four important topological metrics, including overlapping degree, entropy, degree correlation and link overlap, to quantitatively evaluate the temporal multilayer networks. In particular, based on the tensorial framework, we propose two iterative refinement centralities for temporal multilayer networks, referred to as TM-eigenvector and TM-PageRank centralities, which are used to quantitatively evaluate the importance of nodes in real-world complex systems. Moreover, we use the theory of multilinear algebra and matrix analysis to strictly prove the convergence of these iterative algorithms. The proposed mathematical frameworks are validated using two real-world temporal multilayer networked systems related to complex diseases, i.e., influenza and heart diseases. [ABSTRACT FROM AUTHOR]