Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction

Active Region Moss: Doppler Shifts from Hinode/EIS Observations

Studying the Doppler shifts and the temperature dependence of Doppler shifts in moss regions can help us understand the heating processes in the core of the active regions. In this paper we have used an active region observation recorded by the Extreme-ultraviolet Imaging Spectrometer (EIS) onboard Hinode on 12-Dec-2007 to measure the Doppler shifts in the moss regions. We have distinguished the moss regions from the rest of the active region by defining a low density cut-off as derived by Tripathi et al. (2010). We have carried out a very careful analysis of the EIS wavelength calibration based on the method described in Young et al. (2012). For spectral lines having maximum sensitivity between log T = 5.85 and log T = 6.25 K, we find that the velocity distribution peaks at around 0 km/s with an estimated error of 4-5 km/s. The width of the distribution decreases with temperature. The mean of the distribution shows a blue shift which increases with increasing temperature and the distribution also shows asymmetries towards blue-shift. Comparing these results with observables predicted from different coronal heating models, we find that these results are consistent with both steady and impulsive heating scenarios. However, the fact that there are a significant number of pixels showing velocity amplitudes that exceed the uncertainty of 5 km s$^{-1}$ is suggestive of impulsive heating. Clearly, further observational constraints are needed to distinguish between these two heating scenarios.Comment: 21 pages (single column), 7 figures, Accepted for Publication in The Astrophysical Journa

Alien Registration- Mason, James A. (Greenville, Piscataquis County)

https://digitalmaine.com/alien_docs/11406/thumbnail.jp

A Simple Generative Model of Collective Online Behaviour

Human activities increasingly take place in online environments, providing novel opportunities for relating individual behaviours to population-level outcomes. In this paper, we introduce a simple generative model for the collective behaviour of millions of social networking site users who are deciding between different software applications. Our model incorporates two distinct components: one is associated with recent decisions of users, and the other reflects the cumulative popularity of each application. Importantly, although various combinations of the two mechanisms yield long-time behaviour that is consistent with data, the only models that reproduce the observed temporal dynamics are those that strongly emphasize the recent popularity of applications over their cumulative popularity. This demonstrates---even when using purely observational data without experimental design---that temporal data-driven modelling can effectively distinguish between competing microscopic mechanisms, allowing us to uncover new aspects of collective online behaviour.Comment: Updated, with new figures and Supplementary Informatio

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

On Estimating the High-Energy Cutoff in the X-ray Spectra of Black Holes via Reflection Spectroscopy

The fundamental parameters describing the coronal spectrum of an accreting black hole are the slope $\Gamma$ of the power-law continuum and the energy $E_{cut}$ at which it rolls over. Remarkably, this parameter can be accurately measured for values as high as 1 MeV by modeling the spectrum of X-rays reflected from a black hole accretion disk at energies below 100 keV. This is possible because the details in the reflection spectrum, rich in fluorescent lines and other atomic features, are very sensitive to the spectral shape of the hardest coronal radiation illuminating the disk. We show that fitting simultaneous NuSTAR (3-79 keV) and low-energy (e.g., Suzaku) data with the most recent version of our reflection model RELXILL, one can obtain reasonable constraints on $E_{cut}$ at energies from tens of keV up to 1 MeV, for a source as faint as 1 mCrab in a 100 ks observation.Comment: Accepted for publication in ApJL, 6 pages, 5 figure

A Simple, Consistent Estimate for Disturbance Components in Financial Models

Many recent papers have estimated components of the disturbance term in the "market model" of equity returns. In particular, several studies of regulatory changes and other policy events have decomposed the event effects in order to allow for heterogeneity across firms. In this paper we demonstrate that the econometric method applied in some papers yields biased and inconsistent estimates of the model parameters. We demonstrate the consistency of a simple and easily-implemented alternative method.Research Seminar in International Economics, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100855/1/ECON309.pd