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**Abstract**

Complex systems are made up by many interacting, non-identical components, whose individual dynamics are usually governed by simple rules that operate at multiple levels. The structure of interactions between the system’s components is defined through networks, the study of which represent one of the most fascinating topics in modern science. Network science has revolutionized our classical understanding of physical, biological, social and technological systems. Nevertheless, there are several challenges hindering significant advances in the theoretical and computational characterization of complex networks, the most important one being the lack of a mathematical formalism for coping with the multi-level (both in space and time) nature of many real

systems. The vision of PLEXMATH relies on formulating a brand new mathematical framework for the analysis of multi-level networks in terms of tensors, in particular rank-four tensors that represent with four indices the most general structure of possible connections. We therefore will accommodate current and future theoretical and algorithmic needs by adopting a radically new point of view. Capitalizing on tensorial algebra we will reformulate all network descriptors and will propose dynamical equations to represent diffusive processes on multiplex networks. In doing this, we will generate new mathematical models that will be validated on unparalleled amounts of

ICT data that describe relevant socioeconomic and techno-social systems. PLEXMATH constitutes a vital step towards a more general formalism for real-world networks, as the generated knowledge will substantially improve our understanding of complex systems, and will directly impact the way we deal with structural and dynamical patterns in many systems, including ICT.

**Aims and Objectives:**

The main objectives of this project are of scientific foundational character, fitting the nature of the specific call Dynamics of Multi-Level Complex Systems (DyM-CS)

http://cordis.europa.eu/fp7/ict/fet-proactive/dymcs_en.html.

The specific objectives of the project are:

- To create a comprehensive mathematical formalism for describing the general class of multiplex network structures, including time-varying networks. The class of multiplex networks will be sufficiently large to include all existing models in the literature, and will enable us to develop a unified theory for real-world networks.

- To extend the understanding of standard network structure diagnostics and descriptors, such as (to give just a few examples) strength distributions, correlations, clustering coefficients, betweenness etc., to the general multiplex framework.

- To develop a method for analyzing the mesoscopic structure of time varying multiplex networks, and so identify communities of closely linked nodes, and track these through time.

- To determine how particular dynamical processes (random walks and complex contagions) act on multiplex networks. We will develop analytical methods for these cases, and use these to highlight the novel effects of the multiplexity.

- To validate our theoretical approaches on real-world multiplex data sets, and to use the new analytical tools to understand the complexities of these networks and the dynamics that take place upon them.