Given enough choice, simple local rules percolate discontinuously

There is still much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set of $k$ randomly chosen candidate edges at each timestep until a giant component emerges. Several Achlioptas processes seem to yield a discontinuous percolation transition, but it was proven by Riordan and Warnke that the transition must be continuous in the thermodynamic limit. However, they also proved that if the number $k(n)$ of candidate edges increases with the number of nodes, then the percolation transition may be discontinuous. Here we attempt to find the simplest such process which yields a discontinuous transition in the thermodynamic limit. We introduce a process which considers only the degree of candidate edges and not component size. We calculate the critical point $t_{c}=(1-\theta(\frac{1}{k}))n$ and rigorously show that the critical window is of size $O(\frac{n}{k(n)})$. If $k(n)$ grows very slowly, for example $k(n)=\log n$, the critical window is barely sublinear and hence the phase transition is discontinuous but appears continuous in finite systems. We also present arguments that Achlioptas processes with bounded size rules will always have continuous percolation transitions even with infinite choice.Comment: Accepted to European Physical Journal

Improved Linear Algebra Methods for Redshift Computation from Limited Spectrum Data - II

Given photometric broadband measurements of a galaxy, Gaussian processes may be used with a training set to solve the regression problem of approximating the redshift of this galaxy. However, in practice solving the traditional Gaussian processes equation is too slow and requires too much memory. We employed several methods to avoid this difficulty using algebraic manipulation and low-rank approximation, and were able to quickly approximate the redshifts in our testing data within 17 percent of the known true values using limited computational resources. The accuracy of one method, the V Formulation, is comparable to the accuracy of the best methods currently used for this problem

The First Post-Kepler Brightness Dips of KIC 8462852

We present a photometric detection of the first brightness dips of the unique variable star KIC 8462852 since the end of the Kepler space mission in 2013 May. Our regular photometric surveillance started in October 2015, and a sequence of dipping began in 2017 May continuing on through the end of 2017, when the star was no longer visible from Earth. We distinguish four main 1-2.5% dips, named "Elsie," "Celeste," "Skara Brae," and "Angkor", which persist on timescales from several days to weeks. Our main results so far are: (i) there are no apparent changes of the stellar spectrum or polarization during the dips; (ii) the multiband photometry of the dips shows differential reddening favoring non-grey extinction. Therefore, our data are inconsistent with dip models that invoke optically thick material, but rather they are in-line with predictions for an occulter consisting primarily of ordinary dust, where much of the material must be optically thin with a size scale <<1um, and may also be consistent with models invoking variations intrinsic to the stellar photosphere. Notably, our data do not place constraints on the color of the longer-term "secular" dimming, which may be caused by independent processes, or probe different regimes of a single process

Phase Transitions on Static and Evolving Networks: Effects of Competition, Zealotry, and Growth

This thesis consists of studies of network processes with an emphasis on phase transitions. Various network models are studied and phase transitions are categorized as discontinuous, smooth, or continuous but non-differentiable. In one case we define a simple degree-based network model which exhibits a truly discontinuous phase transition to large-scale continuity, and prove this result rigorously. We also show that an altered version of the naming game exhibits critical points which appear to follow an exact log-normal curve. Additionally, we define a model which introduced edge competition to percolation on a growing network and a model which replicates features observed in weekly activity graphs from Facebook gifting applications. The goal is to determine which mechanisms lead to different categories of phase transitions and delay or enhance the onset of the phase transition. A second goal is to introduce growth and evolution into network models, in order to define models which better replicate features of real-world networks

Improved linear algebra methods for redshift computation from limited spectrum data

1 Redshift and photometric observations