Evaluating links through spectral decomposition

Spectral decomposition has been rarely used to investigate complex networks. In this work we apply this concept in order to define two types of link-directed attacks while quantifying their respective effects on the topology. Several other types of more traditional attacks are also adopted and compared. These attacks had substantially diverse effects, depending on each specific network (models and real-world structures). It is also showed that the spectral-based attacks have special effect in affecting the transitivity of the networks

Integrating cross-frequency and within band functional networks in resting-state MEG: A multi-layer network approach

Neuronal oscillations exist across a broad frequency spectrum, and are thought to provide a mechanism of interaction between spatially separated brain regions. Since ongoing mental activity necessitates the simultaneous formation of multiple networks, it seems likely that the brain employs interactions within multiple frequency bands, as well as cross-frequency coupling, to support such networks. Here, we propose a multi-layer network framework that elucidates this pan-spectral picture of network interactions. Our network consists of multiple layers (frequency-band specific networks) that influence each other via inter-layer (cross-frequency) coupling. Applying this model to MEG resting-state data and using envelope correlations as connectivity metric, we demonstrate strong dependency between within layer structure and inter-layer coupling, indicating that networks obtained in different frequency bands do not act as independent entities. More specifically, our results suggest that frequency band specific networks are characterised by a common structure seen across all layers, superimposed by layer specific connectivity, and inter-layer coupling is most strongly associated with this common mode. Finally, using a biophysical model, we demonstrate that there are two regimes of multi-layer network behaviour; one in which different layers are independent and a second in which they operate highly dependent. Results suggest that the healthy human brain operates at the transition point between these regimes, allowing for integration and segregation between layers. Overall, our observations show that a complete picture of global brain network connectivity requires integration of connectivity patterns across the full frequency spectrum

Fundamental limits of symmetric low-rank matrix estimation

We consider the high-dimensional inference problem where the signal is a low-rank symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic model for the low-rank matrix, we compute the limit in the large dimension setting for the mutual information between the signal and the observations, as well as the matrix minimum mean square error, while the rank of the signal remains constant. We also show that our model extends beyond the particular case of additive Gaussian noise and we prove an universality result connecting the community detection problem to our Gaussian framework. We unify and generalize a number of recent works on PCA, sparse PCA, submatrix localization or community detection by computing the information-theoretic limits for these problems in the high noise regime. In addition, we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal (i.e. scalar in the case of rank one). Finally, we connect our work with the hard but detectable conjecture in statistical physics

Holographic tensor network models and quantum error correction: A topical review

Recent progress in studies of holographic dualities, originally motivated by insights from string theory, has led to a confluence with concepts and techniques from quantum information theory. A particularly successful approach has involved capturing holographic properties by means of tensor networks which not only give rise to physically meaningful correlations of holographic boundary states, but also reproduce and refine features of quantum error correction in holography. This topical review provides an overview over recent successful realizations of such models. It does so by building on an introduction of the theoretical foundations of AdS/CFT and necessary quantum information concepts, many of which have themselves developed into independent, rapidly evolving research fields.Comment: 43 pages, 12 figure

Observability and observer design for switched linear systems

Hybrid vehicles, HVAC systems in new/old buildings, power networks, and the like require safe, robust control that includes switching the mode of operation to meet environmental and performance objectives. Such switched systems consist of a set of continuous-time dynamical behaviors whose sequence of operational modes is driven by an underlying decision process. This thesis investigates feasibility conditions and a methodology for state and mode reconstruction given input-output measurements (not including mode sequence). An application herein considers insulation failures in permanent magnet synchronous machines (PMSMs) used in heavy hybrid vehicles. Leveraging the feasibility literature for switched linear time-invariant systems, this thesis introduces two additional feasibility results: 1) detecting switches from safe modes into failure modes and 2) state and mode estimation for switched linear time-varying systems. This thesis also addresses the robust observability problem of computing the smallest structured perturbations to system matrices that causes observer infeasibility (with respect to the Frobenius norm). This robustness framework is sufficiently general to solve related robustness problems including controllability, stabilizability, and detectability. Having established feasibility, real-time observer reconstruction of the state and mode sequence becomes possible. We propose the embedded moving horizon observer (EMHO), which re-poses the reconstruction as an optimization using an embedded state model which relaxes the range of the mode sequence estimates into a continuous space. Optimal state and mode estimates minimize an L2-norm between the measured output and estimated output of the associated embedded state model. Necessary conditions for observer convergence are developed. The EMHO is adapted to solve the surface PMSM fault detection problem