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Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such graphs, can we rank how similar they are in such a way that we capture their geometric “shape” in the plane? We explore a method to compare two such embedded graphs, via a simplified combinatorial representation called a tail-less merge tree which encodes the structure based on a fixed direction. First, we examine the properties of a distance designed to compare merge trees called the branching distance, and show that the distance as defined in previous work fails to satisfy some of the requirements of a metric. We incorporate this into a new distance function called average branching distance to compare graphs by looking at the branching distance for merge trees defined over many directions. Despite the theoretical issues, we show that the definition is still quite useful in practice by using our open-source code to cluster data sets of embedded graphs. 

Abstract We report sequential triggered slip at 271–384 km distances on the San Andreas, Superstition Hills, and Imperial faults with an apparent travel-time speed of 2.2 ± 0.1  km/s, following the passage of surface waves from the 4 July 2019 (17:33:49 UTC) Mw 6.4 and 6 July 2019 (03:19:53 UTC) Mw 7.1 Ridgecrest earthquakes. Slip on remote faults was not triggered instantaneously but developed over several minutes, increasing in duration with distance. Maximum slip amplitudes varied from 10  μm to 5 mm within minutes of slip nucleation, but on the southernmost San Andreas fault slip continued for two months and was followed on 16 September 2019 by a swarm of microearthquakes (Mw≤3.8) near Bombay Beach. These observations add to a growing body of evidence that fault creep may result in delayed triggered seismicity. Displacements across surface faults in the southern epicentral region and on the Garlock fault in the months following the Ridgecrest earthquakes were negligible (<1.1  mm), and they are interpreted to characterize surface strain adjustments in the epicentral region, rather than to result from discrete slip on surface faults. 

Abstract Significant sediment flux and deposition in a sedimentary system are influenced by climate changes, tectonics, lithology, and the sedimentary system's internal dynamics. Identifying the timing of depositional periods from stratigraphic records is a first step to critically evaluate the controls of sediment flux and deposition. Here, we show that ages of single‐grain K‐feldspar luminescence subpopulations may provide information on the timing of previous major depositional periods. We analyzed 754 K‐feldspar single‐grains from 17 samples from the surface to ∼9 m‐depth in a trench located downstream of the Mission Creek catchment. Single‐grain luminescence subpopulation ages significantly overlap at least eight times since ∼12.0 ka indicating a common depositional history. These depositional periods correspond reasonably well with the Holocene intervals of wetter than average climate conditions based on hydroclimatic proxies from nearby locations. Our findings imply a first‐order climatic control on sediment depositional history in southern California on a millennial timescale.