Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Drawing Big Graphs using Spectral Sparsification

Spectral sparsification is a general technique developed by Spielman et al. to reduce the number of edges in a graph while retaining its structural properties. We investigate the use of spectral sparsification to produce good visual representations of big graphs. We evaluate spectral sparsification approaches on real-world and synthetic graphs. We show that spectral sparsifiers are more effective than random edge sampling. Our results lead to guidelines for using spectral sparsification in big graph visualization.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes

When network and graph theory are used in the study of complex systems, a typically finite set of nodes of the network under consideration is frequently either explicitly or implicitly considered representative of a much larger finite or infinite region or set of objects of interest. The selection procedure, e.g., formation of a subset or some kind of discretization or aggregation, typically results in individual nodes of the studied network representing quite differently sized parts of the domain of interest. This heterogeneity may induce substantial bias and artifacts in derived network statistics. To avoid this bias, we propose an axiomatic scheme based on the idea of node splitting invariance to derive consistently weighted variants of various commonly used statistical network measures. The practical relevance and applicability of our approach is demonstrated for a number of example networks from different fields of research, and is shown to be of fundamental importance in particular in the study of spatially embedded functional networks derived from time series as studied in, e.g., neuroscience and climatology.Comment: 21 pages, 13 figure

Microjets in the penumbra of a sunspot

Penumbral Microjets (PMJs) are short-lived jets found in the penumbra of sunspots, first observed in wide-band Ca H-line observations as localized brightenings, and are thought to be caused by magnetic reconnection. Earlier work on PMJs has been focused on smaller samples of by-eye selected events and case studies. It is our goal to present an automated study of a large sample of PMJs to place the basic statistics of PMJs on a sure footing and to study the PMJ Ca II 8542 Angstrom spectral profile in detail. High spatial resolution and spectrally well-sampled observations in the Ca II 8542 Angstrom line obtained from the Swedish 1-m Solar Telescope (SST) are reduced by a Principle Component Analysis and subsequently used in the automated detection of PMJs using the simple learning algorithm k-Nearest Neighbour. PMJ detections were verified with co-temporal Ca H-line observations. A total of 453 tracked PMJ events were found, or 4253 PMJs detections tallied over all timeframes and a detection rate of 21 events per timestep. From these, an average length, width and lifetime of 640 km, 210 km and 90 s were obtained. The average PMJ Ca II 8542 Angstrom line profile is characterized by enhanced inner wings, often in the form of one or two distinct peaks, and a brighter line core as compared to the quiet Sun average. Average blue and red peak positions were determined at -10.4 km/s and +10.2 km/s offsets from the Ca II 8542 Angstrom line core. We found several clusters of PMJ hotspots within the sunspot penumbra, where PMJ events occur in the same general area repeatedly over time. Our results indicate smaller average PMJs sizes and longer lifetimes compared to previously published values, but with statistics still in the same orders of magnitude. The investigation and analysis of the PMJ line profiles strengthen the proposed heating of PMJs to transition region temperatures.Comment: Figures 1, 2, 3, 4 and 11 exhibited artifacts in some pdf-readers, and have been replotted with new graphical settings to remedy this. Apart from slight changes in sizing and fonts, the figures are the same. The arXiv abstract has had tex-syntax removed for better readabilit

Optimal Data Collection For Informative Rankings Expose Well-Connected Graphs

Given a graph where vertices represent alternatives and arcs represent pairwise comparison data, the statistical ranking problem is to find a potential function, defined on the vertices, such that the gradient of the potential function agrees with the pairwise comparisons. Our goal in this paper is to develop a method for collecting data for which the least squares estimator for the ranking problem has maximal Fisher information. Our approach, based on experimental design, is to view data collection as a bi-level optimization problem where the inner problem is the ranking problem and the outer problem is to identify data which maximizes the informativeness of the ranking. Under certain assumptions, the data collection problem decouples, reducing to a problem of finding multigraphs with large algebraic connectivity. This reduction of the data collection problem to graph-theoretic questions is one of the primary contributions of this work. As an application, we study the Yahoo! Movie user rating dataset and demonstrate that the addition of a small number of well-chosen pairwise comparisons can significantly increase the Fisher informativeness of the ranking. As another application, we study the 2011-12 NCAA football schedule and propose schedules with the same number of games which are significantly more informative. Using spectral clustering methods to identify highly-connected communities within the division, we argue that the NCAA could improve its notoriously poor rankings by simply scheduling more out-of-conference games.Comment: 31 pages, 10 figures, 3 table