  A network representation is useful for describing the structure of a large
variety of complex systems. However, most real and engineered systems have
multiple subsystems and layers of connectivity, and the data produced by such
systems is very rich. Achieving a deep understanding of such systems
necessitates generalizing "traditional" network theory, and the newfound deluge
of data now makes it possible to test increasingly general frameworks for the
study of networks. In particular, although adjacency matrices are useful to
describe traditional single-layer networks, such a representation is
insufficient for the analysis and description of multiplex and time-dependent
networks. One must therefore develop a more general mathematical framework to
cope with the challenges posed by multi-layer complex systems. In this paper,
we introduce a tensorial framework to study multi-layer networks, and we
discuss the generalization of several important network descriptors and
dynamical processes --including degree centrality, clustering coefficients,
eigenvector centrality, modularity, Von Neumann entropy, and diffusion-- for
this framework. We examine the impact of different choices in constructing
these generalizations, and we illustrate how to obtain known results for the
special cases of single-layer and multiplex networks. Our tensorial approach
will be helpful for tackling pressing problems in multi-layer complex systems,
such as inferring who is influencing whom (and by which media) in multichannel
social networks and developing routing techniques for multimodal transportation
systems.
