Analysis of contagion maps on a class of networks that are spatially embedded in a torus

A spreading process on a network is influenced by the network's underlying spatial structure, and it is insightful to study the extent to which a spreading process follows such structure. We consider a threshold contagion on a network whose nodes are embedded in a manifold and where the network has both `geometric edges', which respect the geometry of the underlying manifold, and `non-geometric edges' that are not constrained by that geometry. Building on ideas from Taylor et al. \cite{Taylor2015}, we examine when a contagion propagates as a wave along a network whose nodes are embedded in a torus and when it jumps via long non-geometric edges to remote areas of the network. We build a `contagion map' for a contagion spreading on such a `noisy geometric network' to produce a point cloud; and we study the dimensionality, geometry, and topology of this point cloud to examine qualitative properties of this spreading process. We identify a region in parameter space in which the contagion propagates predominantly via wavefront propagation. We consider different probability distributions for constructing non-geometric edges --- reflecting different decay rates with respect to the distance between nodes in the underlying manifold --- and examine the effect of such choices on the qualitative properties of the spreading dynamics. Our work generalizes the analysis in Taylor et al. and consolidates contagion maps both as a tool for investigating spreading behavior on spatial networks and as a technique for manifold learning

Recent (2008–10) Water Quality in the Barton Springs Segment of the Edwards Aquifer and Its Contributing Zone, Central Texas, with Emphasis on Factors Affecting Nutrients and Bacteria

The Barton Springs zone, which comprises the Barton Springs segment of the Edwards aquifer and the watersheds to the west that contribute to its recharge, is in south-central Texas, an area with rapid growth in population and increasing amounts of land area affected by development. During November 2008-March 2010, an investigation of factors affecting the fate and transport of nutrients and bacteria in the Barton Springs zone was conducted by the U.S. Geological Survey (USGS), in cooperation with the Texas Commission on Environmental Quality. The primary objectives of the study were to characterize occurrence of nutrients and bacteria in the Barton Springs zone under a range of flow conditions; to improve understanding of the interaction between surface-water quality and groundwater quality; and to evaluate how factors such as streamflow variability and dilution affect the fate and transport of nutrients and bacteria in the Barton Springs zone. The USGS collected and analyzed water samples from five streams (Barton, Williamson, Slaughter, Bear, and Onion Creeks), two groundwater wells (Marbridge and Buda), and the main orifice of Barton Springs in Austin, Texas. During the period of the study, during which the hydrologic conditions transitioned from exceptional drought to wetter than normal, water samples were collected routinely (every 3 to 4 weeks) from the streams, wells, and spring and, in response to storms, from the streams and spring. All samples were analyzed for major ions, nutrients, the bacterium Escherichia coli, and suspended sediment. During the dry period, the geochemistry of groundwater at the two wells and at Barton Springs was dominated by flow from the aquifer matrix and was relatively similar and unchanging at the three sites. At the onset of the wet period, when the streams began to flow, the geochemistry of groundwater samples from the Marbridge well and Barton Springs changed rapidly, and concentrations of most major ions and nutrients and densities of Escherichia coli became more similar

Prediction and control of nitrate concentrations in groundwater by implementing a model based on GIS and artificial neural networks (ANN)

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