Exponential-family Random Network Models

Random graphs, where the connections between nodes are considered random variables, have wide applicability in the social sciences. Exponential-family Random Graph Models (ERGM) have shown themselves to be a useful class of models for representing com- plex social phenomena. We generalize ERGM by also modeling nodal attributes as random variates, thus creating a random model of the full network, which we call Exponential-family Random Network Models (ERNM). We demonstrate how this framework allows a new formu- lation for logistic regression in network data. We develop likelihood-based inference for the model and an MCMC algorithm to implement it. This new model formulation is used to analyze a peer social network from the National Lon- gitudinal Study of Adolescent Health. We model the relationship between substance use and friendship relations, and show how the results differ from the standard use of logistic regression on network data

Analysis of Partially Observed Networks via Exponential-family Random Network Models

Exponential-family random network (ERN) models specify a joint representation of both the dyads of a network and nodal characteristics. This class of models allow the nodal characteristics to be modelled as stochastic processes, expanding the range and realism of exponential-family approaches to network modelling. In this paper we develop a theory of inference for ERN models when only part of the network is observed, as well as specific methodology for missing data, including non-ignorable mechanisms for network-based sampling designs and for latent class models. In particular, we consider data collected via contact tracing, of considerable importance to infectious disease epidemiology and public health

A New Two-Parameter Family of Potentials with a Tunable Ground State

In a previous paper we solved a countably infinite family of one-dimensional Schr\"odinger equations by showing that they were supersymmetric partner potentials of the standard quantum harmonic oscillator. In this work we extend these results to find the complete set of real partner potentials of the harmonic oscillator, showing that these depend upon two continuous parameters. Their spectra are identical to that of the harmonic oscillator, except that the ground state energy becomes a tunable parameter. We finally use these potentials to analyse the physical problem of Bose-Einstein condensation in an atomic gas trapped in a dimple potential.Comment: 15 pages, 5 figure

Unbinding of giant vortices in states of competing order

Funding: EPSRC (UK) via Grants No. EP/I031014/1 and No. EP/H049584/1.We consider a two-dimensional system with two order parameters, one with O(2) symmetry and one with O(M), near a point in parameter space where they couple to become a single O(2+M) order. While the O(2) sector supports vortex excitations, these vortices must somehow disappear as the high symmetry point is approached. We develop a variational argument which shows that the size of the vortex cores diverges as 1/root Delta and the Berezinskii-Kosterlitz-Thouless transition temperature of the O(2) order vanishes as 1/1n(1/Delta), where Delta denotes the distance from the high-symmetry point. Our physical picture is confirmed by a renormalization group analysis which gives further logarithmic corrections, and demonstrates full symmetry restoration within the cores.Publisher PDFPeer reviewe

Non-equilibrium Berezinskii-Kosterlitz-Thouless Transition in a Driven Open Quantum System

The Berezinskii-Kosterlitz-Thouless mechanism, in which a phase transition is mediated by the proliferation of topological defects, governs the critical behaviour of a wide range of equilibrium two-dimensional systems with a continuous symmetry, ranging from superconducting thin films to two-dimensional Bose fluids, such as liquid helium and ultracold atoms. We show here that this phenomenon is not restricted to thermal equilibrium, rather it survives more generally in a dissipative highly non-equilibrium system driven into a steady-state. By considering a light-matter superfluid of polaritons, in the so-called optical parametric oscillator regime, we demonstrate that it indeed undergoes a vortex binding-unbinding phase transition. Yet, the exponent of the power-law decay of the first order correlation function in the (algebraically) ordered phase can exceed the equilibrium upper limit -- a surprising occurrence, which has also been observed in a recent experiment. Thus we demonstrate that the ordered phase is somehow more robust against the quantum fluctuations of driven systems than thermal ones in equilibrium.Comment: 11 pages, 9 figure

Federal Court Special Masters: A Vital Resource in the Era of Complex Litigation

This article is dedicated to all those who have served as special masters in federal court. After serving as a judicial master, it is easy to believe in the importance of the role in our grand system of justice. After reading this article, we hope it will be clear how vital masters are to everyone receiving fair, just, and expedient civil justice

Federal Court Special Masters: A Vital Resource in the Era of Complex Litigation

This article is dedicated to all those who have served as special masters in federal court. After serving as a judicial master, it is easy to believe in the importance of the role in our grand system of justice. After reading this article, we hope it will be clear how vital masters are to everyone receiving fair, just, and expedient civil justice

Forty Years of Codification of Estates and Trusts Law: Lessons for the Next Generation

In this paper we develop two theses. First, we argue that uniform law proposals that ask courts and practitioners to abandon revered legal traditions and ways of thinking about estates and trusts, even when they are intent-furthering proposals, face resistance until in time the glories of the past and the risks of a new legal regime fade in importance in legal thought. Second, we argue that, especially within an environment in which states seek to gain competitive advantage over their counterparts in other states, the glories of the past and the risks of a new legal regime fade fastest when a uniform law proposal limits the effect of intent-defeating rules. Uniform laws tend to fall into three categories: (1) statutes that usurp older statutory-based laws; (2) statutes, typically remedial in nature, that reverse the common law; and (3) statutes that predominantly codify the common law. We look at examples of each to show how the interplay between revered legal traditions and donative freedom affects the reception of uniform law proposals. We also pay particular attention to intent-defeating common law doctrines and the risks that uniform law drafters face when they attempt to codify them in an environment where there is stiff jurisdictional competition for estate planning business