Characterizing Distances of Networks on the Tensor Manifold

At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a family of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a point on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of geodesics amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems.Comment: This paper is accepted at 8th International Conference on Complex Networks 201

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Attention and Anticipation in Fast Visual-Inertial Navigation

We study a Visual-Inertial Navigation (VIN) problem in which a robot needs to estimate its state using an on-board camera and an inertial sensor, without any prior knowledge of the external environment. We consider the case in which the robot can allocate limited resources to VIN, due to tight computational constraints. Therefore, we answer the following question: under limited resources, what are the most relevant visual cues to maximize the performance of visual-inertial navigation? Our approach has four key ingredients. First, it is task-driven, in that the selection of the visual cues is guided by a metric quantifying the VIN performance. Second, it exploits the notion of anticipation, since it uses a simplified model for forward-simulation of robot dynamics, predicting the utility of a set of visual cues over a future time horizon. Third, it is efficient and easy to implement, since it leads to a greedy algorithm for the selection of the most relevant visual cues. Fourth, it provides formal performance guarantees: we leverage submodularity to prove that the greedy selection cannot be far from the optimal (combinatorial) selection. Simulations and real experiments on agile drones show that our approach ensures state-of-the-art VIN performance while maintaining a lean processing time. In the easy scenarios, our approach outperforms appearance-based feature selection in terms of localization errors. In the most challenging scenarios, it enables accurate visual-inertial navigation while appearance-based feature selection fails to track robot's motion during aggressive maneuvers.Comment: 20 pages, 7 figures, 2 table

Collective chaos in pulse-coupled neural networks

We study the dynamics of two symmetrically coupled populations of identical leaky integrate-and-fire neurons characterized by an excitatory coupling. Upon varying the coupling strength, we find symmetry-breaking transitions that lead to the onset of various chimera states as well as to a new regime, where the two populations are characterized by a different degree of synchronization. Symmetric collective states of increasing dynamical complexity are also observed. The computation of the the finite-amplitude Lyapunov exponent allows us to establish the chaoticity of the (collective) dynamics in a finite region of the phase plane. The further numerical study of the standard Lyapunov spectrum reveals the presence of several positive exponents, indicating that the microscopic dynamics is high-dimensional.Comment: 6 pages, 5 eps figures, to appear on Europhysics Letters in 201

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

Lurching Toward Chernobyl: Dysfunctions of Real-Time Computation

Cognitive biological structures, social organizations, and computing machines operating in real time are subject to Rate Distortion Theorem constraints driven by the homology between information source uncertainty and free energy density. This exposes the unitary structure/environment system to a relentless entropic torrent compounded by sudden large deviations causing increased distortion between intent and impact, particularly as demands escalate. The phase transitions characteristic of information phenomena suggest that, rather than graceful decay under increasing load, these structures will undergo punctuated degradation akin to spontaneous symmetry breaking in physical systems. Rate distortion problems, that also affect internal structural dynamics, can become synergistic with limitations equivalent to the inattentional blindness of natural cognitive process. These mechanisms, and their interactions, are unlikely to scale well, so that, depending on architecture, enlarging the structure or its duties may lead to a crossover point at which added resources must be almost entirely devoted to ensuring system stability -- a form of allometric scaling familiar from biological examples. This suggests a critical need to tune architecture to problem type and system demand. A real-time computational structure and its environment are a unitary phenomenon, and environments are usually idiosyncratic. Thus the resulting path dependence in the development of pathology could often require an individualized approach to remediation more akin to an arduous psychiatric intervention than to the traditional engineering or medical quick fix. Failure to recognize the depth of these problems seems likely to produce a relentless chain of the Chernobyl-like failures that are necessary, bot often insufficient, for remediation under our system