Preferential attachment with choice

We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when $r=2$ and shows condensation-like behaviour when $r>2$.Comment: 17 pages, 1 figure. Accepted for publication in Random Structures and Algorithm

Impact of Resonance on Thermal Targets for Invisible Dark Photon Searches

Dark photons in the MeV to GeV mass range are important targets for experimental searches. We consider the case where dark photons $A'$ decay invisibly to hidden dark matter $X$ through $A' \to XX$. For generic masses, proposed accelerator searches are projected to probe the thermal target region of parameter space, where the $X$ particles annihilate through $XX \to A' \to \text{SM}$ in the early universe and freeze out with the correct relic density. However, if $m_{A'} \approx 2m_X$, dark matter annihilation is resonantly enhanced, shifting the thermal target region to weaker couplings. For $\sim 10\%$ degeneracies, we find that the annihilation cross section is generically enhanced by four (two) orders of magnitude for scalar (pseudo-Dirac) dark matter. For such moderate degeneracies, the thermal target region drops to weak couplings beyond the reach of all proposed accelerator experiments in the scalar case and becomes extremely challenging in the pseudo-Dirac case. Proposed direct detection experiments can probe moderate degeneracies in the scalar case. For greater degeneracies, the effect of the resonance can be even more significant, and both scalar and pseudo-Dirac cases are beyond the reach of all proposed accelerator and direct detection experiments. For scalar dark matter, we find an absolute minimum that sets the ultimate experimental sensitivity required to probe the entire thermal target parameter space, but for pseudo-Dirac fermions, we find no such thermal target floor.Comment: 17 pages, 2 figures; v2: improved agreement with existing non-resonant results, added extensive discussion of implications for direct detection experiment

The spectra of the laplacians of fractal graphs not satisfying spectral decimation

We consider the spectra of the Laplacians of two sequences of fractal graphs in the context of the general theory introduced by Sabot in 2003. For the sequence of graphs associated with the pentagasket, we give a description of the eigenvalues in terms of the iteration of a map from (C-2)(3) to itself. For the sequence of graphs introduced in a previous paper by the author, we show that the results found therein can be related to Sabot's theory

Undermining and Strengthening Social Networks through Network Modification

Social networks have well documented effects at the individual and aggregate level. Consequently it is often useful to understand how an attempt to influence a network will change its structure and consequently achieve other goals. We develop a framework for network modification that allows for arbitrary objective functions, types of modification (e.g. edge weight addition, edge weight removal, node removal, and covariate value change), and recovery mechanisms (i.e. how a network responds to interventions). The framework outlined in this paper helps both to situate the existing work on network interventions but also opens up many new possibilities for intervening in networks. In particular use two case studies to highlight the potential impact of empirically calibrating the objective function and network recovery mechanisms as well as showing how interventions beyond node removal can be optimised. First, we simulate an optimal removal of nodes from the Noordin terrorist network in order to reduce the expected number of attacks (based on empirically predicting the terrorist collaboration network from multiple types of network ties). Second, we simulate optimally strengthening ties within entrepreneurial ecosystems in six developing countries. In both cases we estimate ERGM models to simulate how a network will endogenously evolve after intervention

Dark Photons from the Center of the Earth: Smoking-Gun Signals of Dark Matter

Dark matter may be charged under dark electromagnetism with a dark photon that kinetically mixes with the Standard Model photon. In this framework, dark matter will collect at the center of the Earth and annihilate into dark photons, which may reach the surface of the Earth and decay into observable particles. We determine the resulting signal rates, including Sommerfeld enhancements, which play an important role in bringing the Earth's dark matter population to their maximal, equilibrium value. For dark matter masses $m_X \sim$ 100 GeV - 10 TeV, dark photon masses $m_{A'} \sim$ MeV - GeV, and kinetic mixing parameters $\varepsilon \sim 10^{-10} - 10^{-8}$, the resulting electrons, muons, photons, and hadrons that point back to the center of the Earth are a smoking-gun signal of dark matter that may be detected by a variety of experiments, including neutrino telescopes, such as IceCube, and space-based cosmic ray detectors, such as Fermi-LAT and AMS. We determine the signal rates and characteristics, and show that large and striking signals---such as parallel muon tracks---are possible in regions of the $(m_{A'}, \varepsilon)$ plane that are not probed by direct detection, accelerator experiments, or astrophysical observations.Comment: 26 pages, 10 figures. v2: minor revisions to match published version; v3: updated direct detection and CMB constraints and corrected decay length in code, moving the region of experimental sensitivity to values of epsilon that are lower by an order of magnitud

Solar Hydrogen Production by Solar Thermal Decoupled Electrolysis: Analysis of FE3O4 in Solution

The Valparaiso University Solar Research Project utilizes a solar thermal decoupled electrolysis process for the production of H2 from water. This study focuses on the electrochemical conversion of magnetite to hematite during the H2 production process. Laboratory experiments show that the expected amount of H2 gas forms at the cathode of the electrolytic cell, but we have encountered difficulties recovering the expected amount of solid hematite at the anode. An intensive study of the complex solution chemistry using cyclic voltammetry, Mossbauer spectroscopy, and solubility modeling showed that several dissolved iron species potentially exist in solution. Further work must be done to clarify which species are key participants in the overall electrochemical reaction. This information will help us to determine and implement reaction conditions that are favorable for precipitation of hematite or other oxidized iron species