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MaNet - Mathematical engineering tools for Networks: Data driven mining, identification, control and optimization

From 01-10-2009 to 30-09-2015

Description

The central objective of this proposal is the analysis, design and implementation of data-driven mathematical engineering methods and numerical algorithms for the analysis, modelling and optimization of networks. While inspired by cases from the aforementioned application areas, the real focus will be on tackling ‘generic’ network problems starting from quantitative measured data collected from network nodes and links.

In this proposal, acquired expertise on electrical networks, utility nets, information networks, biological networks and acoustic networks will be synergetically combined with the availability of large numerical databases in all of these fields (‘high-throughput’) and the current scientific research developments in data-driven, large scale, numerical algorithms.

The proposal contains four work packages (WP).

WP I will investigate the applicability to network problems, of concepts and insights deriving from linear and multilinear algebra for matrices, tensors and kernels.

Networks and graphs can be represented as matrices describing either the network topology, quantifying certain attributes of nodes or exhibiting the correlation and distances between certain node features. This matrix representation implies two things: Many network problems can be formulated as numerical data matrix problems, which can then be tackled with the insights and tools from numerical (multi-)linear algebra and optimization. The network problems relevant for our application domains, include methods for unraveling network topologies, graph realization problems, dimensionality reduction and clustering, graph multi-partitioning and community detection problems, effective ranking and prioritization problems. All of these methods to applications in which multiple and complementarity data sources are available, leading to heterogeneous data integration approaches will be extended. We will also further develop the machinery to numerically solve these problems, which will include spectral methods, simultaneous matrix decompositions, tensor decomposition algorithms, nonnegative matrix factorization, classification, clustering and pattern recognition algorithms The work on kernel approaches will allow to ‘kernelize’ many of the methods solving, using convex optimization approaches, nonlinear classification and clustering, data fusion, prioritization and graph partitioning problems on networks. Building on these results will lead to identifying network topologies via Bayesian network learning for a case in genetic network identification. In all of these problems, we are confronted with specific large-scale ‘high-throughput’ challenges and issues of scarcity and matrix structure.

WP II will investigate networks as dynamical systems. Networks provide a powerful metaphor for describing the dynamic behavior of systems in – among others - biology, computer science, information technology and engineering. This implies that all important system theoretic notions, such as stability and dissipativity, adaptivity, synchronization, self-organization and emerging properties, robustness and fragility, causality issues, etc., are highly relevant for networks as well. WP II will be dealing with some of these more fundamental issues. One of them is the description and modeling of networks as interconnected dynamical systems, and behavioral framework for dynamical systems. Another problem will consider derives from some of our applications (e.g. genetic networks, brain connectivity networks) and evolves around causality detection of information flows between nodes. A third issue concerns the dynamic evolution of the network topology itself, in trying to ‘understand’ mechanisms such as ‘preferential attachment’ and other dynamics of the topology. A fourth research issue will be to elaborate on mechanisms of emerging behavior in certain networks, such as synchronization phenomena. Finally, we will investigate how perturbation analysis methods from electrical circuit theory, can be useful in analyzing the sensitivity of general networks and how approaches from robust control theory can be transferred to robustness issues over networks.

WP III will investigate network identification, estimation and filtering problems. While in WP II, we concentrate on the conceptual dynamic properties of networks, in WP III we will tackle dynamic, ‘longitudinal data’-driven problems on networks, that can be solved with system identification approaches. Typically the challenges here are in clustering, forecasting, or ‘normalization’. Fundamental problems here include: how to model time series obtained from nodes for linear systems and structured total least squares problems can be solved as an eigen value problem, which would shed a new light on the nonlinear optimization issues that are a bottleneck in PEM and STLS), the estimation of time-dependent origin-destination matrices, how to cluster time series over nodes (how to generalize distance measures such as the cepstral distance to vector time series, is it best to first model and then cluster or to first cluster and then model,…), how to design static, but even better, dynamic experiments that provide data that lead to correct network topology determination ‘understanding’ dimensionality reduction approaches such as proper orthogonal decomposition, unkown state and input estimation in the context of data assimilation (sensor grid networks) via Kalman filtering and moving horizon estimation, develop distributed estimation algorithms for a number of non-standard least-squares problems in sensor networks with various network topologies, and analyse their performance and optimality, and numerical algorithms to identify network nodes and links. For this last problem, we will develop further our subspace identification inspired Hidden Markov Model (HMM) realization algorithms, for those applications (e.g. genetic networks), where the data are sequences of symbols (e.g. DNA). These algorithms could be useful to identify fundamental nodes in unraveling genetic networks (e.g. by detecting regulatory elements). HMM as such are closely related to random walk interpretations of network matrix representations via the Laplacian, on which we will also elaborate.

WP IV will investigate network optimization and distributed control problems. Many network problems can be formulated as numerical optimization problems. When (parts of) a network model (are) is parametrized, several important properties can be optimized over the parameters of the model. Also typical in these distributed modeling and optimization problems, is the fact that there are issues of network coupling constraints and hierarchical levels, which lead to sparse and structured matrix representations that could be exploited in numerical algorithms. We will develop optimization based algorithms for hierarchical, distributed model-predictive control problems over networks, and for its dual problem, distributed moving horizon estimation of unknown states and inputs.

Throughout the work packages, we will gain inspiration from conceptual and mathematical thinking as well as from the deployment of our algorithms in a few application domains. Discussed cases will derive from: bioinformatics and systems biology , information networks neuronal brain connectivity networks, modeling and control of utility networks and signal processing in acoustic sensor networks.