One of the most challenging problems in the engineering of complex networks (Small Cell Networks, Smart Grid Networks or Wireless Sensor Networks among others) is to manage complexity. The key is to develop :

- the right abstractions to reason about the spatial and temporal dynamics of complex systems.
- understand how information and energy can be processed, stored, and transferred in the system with bounded delay.

The theoretical foundations of complex networks pose very challenging theoretical and practical problems which are the focus of the European Research Council Starting Grant sponsored project **MORE (Advanced Mathematical Tools for Complex Network Engineering). **The theoretical research is highly interdisciplinary and is a blend of many tools such as Random Matrix Theory, Mean Field Games or Decentralized Stochastic Optimization just to name a few. The goal of the team of the project MORE led by the Principal Investigator Prof. Mérouane DEBBAH is to provide a unified framework to address specific problems of Large Dimensional Stochastic Networks.

- ”A Mathematical Theory of Communication”, Bell System Technical Journal, 1948, C. E. Shannon
- ”Cybernetics, or Control and Communication in the Animal and the Machine”, Herman et Cie/The Technology Press, 1948, N. Wiener

**Heterogeneity:**The various inputs and outputs of the systems can be of different nature**Limited information:**there may be limited or no communication between different systems and decisions have to be made based on such distributed information.**Temporal requirements:**systems change rapidly and the system needs to adapt fast.

**Random matrix theory:**Historically used to analyze the performance of large point-to-point communication systems, RMT has recently been identified as a new means to characterize the performance of communication networks and a new tool to address signal processing problems in large sensor arrays (array processing, failure detection in networks, etc.). The mathematical foundations of RMT will be further developed within the scope of MORE to tackle the question of performance, optimization, and algorithm development for large dimensional networks.**Decentralized stochastic optimization:**This tool, initiated in the area of decentralized computing, allows for the development and the analysis of decentralized algorithms and techniques in stochastic environments. In MORE, the decentralized stochastic optimization methods will be further developed to tackle the problem of delay-limited or communication-limited computation in large networks.**Game theory and mean field games:**Game theory techniques have recently developed to study the performance and equilibria of stochastic decentralized systems with individual objectives. In large network conditions, mean field games are prevalent as they turn large but finite dimensional games into infinite-size games easier to characterize. In MORE, the strong mathematical limitations associated with these techniques will be explored and moved towards more practical system configurations.**Stochastic geometry:**Stochastic geometry has developed in the scope of large communication networks to analyze the performance of individual nodes in homogeneous but random environments. Our objective in MORE is to extend and connect these methods with other tools to tackle the problem of inhomogeneous environments and cooperative network nodes.**Network information theory and coding:**Network information theory analyzes the performance of communication networks with more than two nodes. In MORE, these techniques will be explored focusing on the problem of stochastic fast-changing environments.**Statistical mechanics:**Statistical mechanics have historically known a large success in the physical study of large dimensional systems. These tools are often necessary to provide intuitive rather than accurate tools to tackle difficult large dimensional problems. In MORE, these approaches will be explored in the objective of connecting large dimensional systems to large physical systems well-known in classical physics.**Advanced signal processing methods:**Classical signal processing methods used in networks and usually assuming large quantities of observations will be explored in the scope of large, fast-evolving environments, with limited number of observations.

**Communication networks:**Communication networks become increasingly complex as the legacy point-to-point interference limited approach (single large cell viewpoint) is being sequentially replaced by multipoint cooperative schemes (multiple small cells network). The mathematical tools developed in MORE will allow for a better understanding of the performance of such involved networks and for the development of optimization methods for these schemes.**Energy distribution networks:**Power networks, formerly static and hierarchical, are also being increasingly replaced by decentralized autonomous nodes of a very large inter-dependent network. The understanding of the limitations and the development of new signal processing algorithms for these power networks (failure identification) is a goal of the project MORE.**Sensor arrays:**The performance of sensor arrays with limited individual node capabilities (both in terms of storage and processing) in stochastic environment can be analyzed through distributed optimization. This is one goal of the MORE project aiming at developing tools for their analysis.