Nonequilibrium phase transition in the coevolution of networks and opinions

Models of the convergence of opinion in social systems have been the subject of a considerable amount of recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of personal acquaintances, and those in which, conversely, network connections form between individuals of similar beliefs. While both of these processes can give rise to realistic levels of agreement between acquaintances, practical experience suggests that opinion formation in the real world is not a result of one process or the other, but a combination of the two. Here we present a simple model of this combination, with a single parameter controlling the balance of the two processes. We find that the model undergoes a continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one in which most individuals hold the same opinion. We characterize the static and dynamical properties of this transition

Relativistic Images in Randall-Sundrum II Braneworld Lensing

In this paper, we explore the properties of gravitational lensing by black holes in the Randall-Sundrum II braneworld. We use numerical techniques to calculate lensing observables using the Tidal Reissner-Nordstrom (TRN) and Garriga-Tanaka metrics to examine supermassive black holes and primordial black holes. We introduce a new way tp parameterize tidal charge in the TRN metric which results in a large increase in image magnifications for braneworld primordial black holes compared to their 4 dimensional analogues. Finally, we offer a mathematical analysis that allows us to analyze the validity of the logarithmic approximation of the bending angle for any static, spherically symmetric metric. We apply this to the TRN metric and show that it is valid for any amount of tidal charge.Comment: 13 pages, 3 figures; Accepted for Publication in Physical Review

Neutral theory of chemical reaction networks

To what extent do the characteristic features of a chemical reaction network reflect its purpose and function? In general, one argues that correlations between specific features and specific functions are key to understanding a complex structure. However, specific features may sometimes be neutral and uncorrelated with any system-specific purpose, function or causal chain. Such neutral features are caused by chance and randomness. Here we compare two classes of chemical networks: one that has been subjected to biological evolution (the chemical reaction network of metabolism in living cells) and one that has not (the atmospheric planetary chemical reaction networks). Their degree distributions are shown to share the very same neutral system-independent features. The shape of the broad distributions is to a large extent controlled by a single parameter, the network size. From this perspective, there is little difference between atmospheric and metabolic networks; they are just different sizes of the same random assembling network. In other words, the shape of the degree distribution is a neutral characteristic feature and has no functional or evolutionary implications in itself; it is not a matter of life and death.Comment: 13 pages, 8 figure

A generalization of the Heine--Stieltjes theorem

We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question on how to generalize their results to higher degrees. Many of the results are new even for the classical case.Comment: 12 pages, typos corrected and refined the interlacing theorem