Sampling and inference of networked dynamics using Log-Koopman nonlinear graph fourier transform

Monitoring the networked dynamics via the subset of nodes is essential for a variety of scientific and operational purposes. When there is a lack of an explicit model and networked signal space, traditional observability analysis and non-convex methods are insufficient. Current data-driven Koopman linearization, although derives a linear evolution model for selected vector-valued observable of original state-space, may result in a large sampling set due to: (i) the large size of polynomial based observables (O(N2) , N number of nodes in network), and (ii) not factoring in the nonlinear dependency betweenobservables. In this work, to achieve linear scaling (O(N) ) and a small set of sampling nodes, wepropose to combine a novel Log-Koopman operator and nonlinear Graph Fourier Transform (NL-GFT) scheme. First, the Log-Koopman operator is able to reduce the size of observables by transforming multiplicative poly-observable to logarithm summation. Second, anonlinear GFT concept and sampling theory are provided to exploit the nonlinear dependence of observables for observability analysis using Koopman evolution model. The results demonstrate that the proposed Log-Koopman NL-GFT scheme can (i) linearize unknownnonlinear dynamics using O(N) observables, and (ii) achieve lower number of sampling nodes, compared with the state-of-the art polynomial Koopman based observability analysis

Sampling of time-varying network signals from equation-driven to data-driven techniques

Sampling and recovering the time-varying network signals via the subset of network vertices is essential for a wide range of scientific and engineering purposes. Current studies on sampling a single (continuous) time-series or a static network data, are not suitable for time-varying network signals. This will be even more challenging when there is a lack of explicit dynamic models and signal-space that indicate the time-evolution and vertex dependency. The work begins by bridging the time-domain sampling frequency and the network-domain sampling vertices, via the eigenvalues of the graph Fourier transform (GFT) operator composed by the combined dynamic equations and network topology. Then, for signals with hidden governing mechanisms, we propose a data-driven GFT sampling method using a prior signal-space. We characterize the signal dependency (among vertices) into the graph bandlimited frequency domain, and map such bandlimitedness into optimal sampling vertices. Furthermore, to achieve dynamic model and signal-space independent sensor placement, a Koopman based nonlinear GFT sampling is proposed. A novel data-driven Log-Koopman operator is designed to extract a linearized evolution model using small (M = O(N)) and decoupled observables defined on N original vertices. Then, nonlinear GFT is proposed to derive sampling vertices, by exploiting the inherent nonlinear dependence between M observables (defined on N < M vertices), and the time-evolved information presented by Log-Koopman evolution model. The work also informs the planned future work to formulate an easy-to-use and explainable neural network (NN) based sampling framework, for real-world industrial engineering and applications

A complex systems approach to education in Switzerland

The insights gained from the study of complex systems in biological, social, and engineered systems enables us not only to observe and understand, but also to actively design systems which will be capable of successfully coping with complex and dynamically changing situations. The methods and mindset required for this approach have been applied to educational systems with their diverse levels of scale and complexity. Based on the general case made by Yaneer Bar-Yam, this paper applies the complex systems approach to the educational system in Switzerland. It confirms that the complex systems approach is valid. Indeed, many recommendations made for the general case have already been implemented in the Swiss education system. To address existing problems and difficulties, further steps are recommended. This paper contributes to the further establishment complex systems approach by shedding light on an area which concerns us all, which is a frequent topic of discussion and dispute among politicians and the public, where billions of dollars have been spent without achieving the desired results, and where it is difficult to directly derive consequences from actions taken. The analysis of the education system's different levels, their complexity and scale will clarify how such a dynamic system should be approached, and how it can be guided towards the desired performance