Traditional studies of networks generally assume that nodes are connected to each other by a single type of static edge that encapsulates all connections between them.  This assumption is almost always a gross oversimplification, and it can lead to misleading results and even the sheer inability to address certain problems.  For example, ignoring time-dependence throws away the ordering of pairwise human contacts in transmission of diseases, and ignoring the presence of multiple types of edges (which is known as "multiplexity") makes it hard to take into account the simultaneous presence and relevance of multiple modes of transportation or communication.

Mathematical formulation of multilayer networks


Multiplex networks explicitly incorporate multiple channels of connectivity in a system, and they are particularly interesting because they provide a natural description for systems in which entities interact with a different neighborhood of entities (depending on e.g., task, activity, or category). A fundamental aspect of describing multiplex networks is defining and quantifying the interconnectivity between different categories of connections. This amounts to switching between layers in a multilayer system, and the associated inter-layer connections in a network are responsible for the emergence of new phenomena in multiplex networks. Inter-layer connections can generate new structural and dynamical correlations between components of a system, so it is important to develop a framework that takes them into account. Note that multiplex networks are not simply a special case of interdependent networks: in multiplex systems, many or even all of the nodes in each layer are the same, so one can associate a vector of states to each node. For example, a person might currently be logged into Facebook (and hence able to receive information there) but not logged into some other social networking site. The presence of nodes in multiple layers of a system also entails the possibility of self-interactions.
This feature has no counterpart in interdependent networks, which were conceived as interconnected communities within a single, larger network.

To study multiplex and/or temporal networks systematically, it is necessary to develop a precise mathematical representation for them as well as appropriate tools to go with such a representation. We have developed a mathematical framework for multilayer networks using tensor algebra. Our framework can be used to study all types of multilayer networks (including multiplex networks, temporal networks, cognitive social structures, multivariate networks, interdependent networks, etc).



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Centrality in multilayer networks


We have generalized the concept of classical node's centrality to multilayer networks and we have applied it to real biological, social, socio-technical, technological and transportation systems. We have demonstrated that the combination of existing techniques or the aggregation of the underlying information might lead to dramatically wrong results.





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Triadic closure and clustering in multilayer networks


Triadic closure is not trivial in the multilayer domain. We have extended this concept and proposed a mathematical approach to quantify it in real multiplex systems.






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Community detection in multilayer networks


We have proposed a method for identifying communities in a multilayer system and to quantify the underlying mesoscale organization. This information-theoretic method is based on the compression of information flowing through the network. We have shown that multilayer communities, naturally interpreted as overlapping ones, provide much more insights about the system with respect to aggregated approaches.






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Random walks and diffusion in multilayer networks


We have defined the concept of random walks on the top of multilayer networks, finding new physical phenomena related to exploration and navigability of such systems. Our mathematical formulation in terms of tensors allowed us to write the rich and complex dynamics of diffusive processes in a compact way.










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Structural and dynamical resilience of multilayer networks


We have exploited the concept of random walk to define a measure of dynamical resilience for multilayer networks to random and systemic failures, with application to empirical urban transportation systems.

More recently, we have proposed the interconnectedness among layers as an indicator of structural resilience, and, as a practical application, we have analyzed some social-ecological systems from the Arctic area, demonstrating how societal changes affect the robustness of the system more than resource depletion.



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Structural reducibility of multilayer networks


To cope with the additional complexity of multilayer systems, we have proposed an algorithm, grounded on information theory and inspired by quantum computing, to reduce the structure of such networks by aggregating appropriately its layers.




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Computational tools


We have developed a free and open source software platform for multilayer analysis and visualization of complex networks, named muxViz (visit the official Web site).




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WIKIPEDIA might provide further information about multilayer networks.

HERE you can find an updated list of my pubblications about multiplex and multilayer networks.