The statistical mechanics of networks

We study the family of network models derived by requiring the expected properties of a graph ensemble to match a given set of measurements of a real-world network, while maximizing the entropy of the ensemble. Models of this type play the same role in the study of networks as is played by the Boltzmann distribution in classical statistical mechanics; they offer the best prediction of network properties subject to the constraints imposed by a given set of observations. We give exact solutions of models within this class that incorporate arbitrary degree distributions and arbitrary but independent edge probabilities. We also discuss some more complex examples with correlated edges that can be solved approximately or exactly by adapting various familiar methods, including mean-field theory, perturbation theory, and saddle-point expansions.Comment: 15 pages, 4 figure

Solution of the 2-star model of a network

The p-star model or exponential random graph is among the oldest and best-known of network models. Here we give an analytic solution for the particular case of the 2-star model, which is one of the most fundamental of exponential random graphs. We derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition.Comment: 5 pages, 3 figure

Creation and characterization of vortex clusters in atomic Bose-Einstein condensates

We show that a moving obstacle, in the form of an elongated paddle, can create vortices that are dispersed, or induce clusters of like-signed vortices in 2D Bose-Einstein condensates. We propose new statistical measures of clustering based on Ripley's K-function which are suitable to the small size and small number of vortices in atomic condensates, which lack the huge number of length scales excited in larger classical and quantum turbulent fluid systems. The evolution and decay of clustering is analyzed using these measures. Experimentally it should prove possible to create such an obstacle by a laser beam and a moving optical mask. The theoretical techniques we present are accessible to experimentalists and extend the current methods available to induce 2D quantum turbulence in Bose-Einstein condensates.Comment: 9 pages, 9 figure

Continuous inference for aggregated point process data

The paper introduces new methods for inference with count data registered on a set of aggregation units. Such data are omnipresent in epidemiology because of confidentiality issues: it is much more common to know the county in which an individual resides, say, than to know their exact location in space. Inference for aggregated data has traditionally made use of models for discrete spatial variation, e.g. conditional auto-regressive models. We argue that such discrete models can be improved from both a scientific and an inferential perspective by using spatiotemporally continuous models to model the aggregated counts directly. We introduce methods for delivering (limiting) continuous inference with spatiotemporal aggregated count data in which the aggregation units might change over time and are subject to uncertainty. We illustrate our methods by using two examples: from epidemiology, spatial prediction of malaria incidence in Namibia, and, from politics, forecasting voting under the proposed changes to parliamentary boundaries in the UK. © 2018 Royal Statistical Societ

Bayes-optimal inverse halftoning and statistical mechanics of the Q-Ising model

On the basis of statistical mechanics of the Q-Ising model, we formulate the Bayesian inference to the problem of inverse halftoning, which is the inverse process of representing gray-scales in images by means of black and white dots. Using Monte Carlo simulations, we investigate statistical properties of the inverse process, especially, we reveal the condition of the Bayes-optimal solution for which the mean-square error takes its minimum. The numerical result is qualitatively confirmed by analysis of the infinite-range model. As demonstrations of our approach, we apply the method to retrieve a grayscale image, such as standard image `Lenna', from the halftoned version. We find that the Bayes-optimal solution gives a fine restored grayscale image which is very close to the original.Comment: 13pages, 12figures, using elsart.cl

Spatially explicit species distribution models: A missed opportunity in conservation planning?

Aim: Systematic conservation planning is vital for allocating protected areas given the spatial distribution of conservation features, such as species. Due to incomplete species inventories, species distribution models (SDMs) are often used for predicting species habitat suitability and species probability of occurrence. Currently, SDMs mostly ignore spatial dependencies in species and predictor data. Here, we provide a comparative evaluation of how accounting for spatial dependencies, that is, autocorrelation, affects the delineation of optimized protected areas. Location: Southeast Australia, Southeast U.S. Continental Shelf, Danube River Basin. Methods: We employ Bayesian spatially explicit and non-spatial SDMs for terrestrial, marine and freshwater species, using realm-specific planning unit shapes (grid, hexagon and subcatchment, respectively). We then apply the software gurobi to optimize conservation plans based on species targets derived from spatial and non-spatial SDMs (10% 50% each to analyse sensitivity), and compare the delineation of the plans. Results: Across realms and irrespective of the planning unit shape, spatially explicit SDMs (a) produce on average more accurate predictions in terms of AUC, TSS, sensitivity and specificity, along with a higher species detection probability. All spatial optimizations meet the species conservation targets. Spatial conservation plans that use predictions from spatially explicit SDMs (b) are spatially substantially different compared to those that use non-spatial SDM predictions, but (c) encompass a similar amount of planning units. The overlap in the selection of planning units is smallest for conservation plans based on the lowest targets and vice versa. Main conclusions: Species distribution models are core tools in conservation planning. Not surprisingly, accounting for the spatial characteristics in SDMs has drastic impacts on the delineation of optimized conservation plans. We therefore encourage practitioners to consider spatial dependencies in conservation features to improve the spatial representation of future protected areas. © 2019 The Authors. Diversity and Distributions Published by John Wiley and Sons LtdThis study was funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 642317. SDL has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska‐Curie grant agreement No. 748625, and SCJ from the German Federal Ministry of Education and Research (BMBF) for the “GLANCE” project (Global Change Effects in River Ecosystems; 01 LN1320A). We wish to thank Gwen Iacona and two anonymous referees for their constructive comments on an earlier version of the manuscript

Bayesian Centroid Estimation for Motif Discovery

Biological sequences may contain patterns that are signal important biomolecular functions; a classical example is regulation of gene expression by transcription factors that bind to specific patterns in genomic promoter regions. In motif discovery we are given a set of sequences that share a common motif and aim to identify not only the motif composition, but also the binding sites in each sequence of the set. We present a Bayesian model that is an extended version of the model adopted by the Gibbs motif sampler, and propose a new centroid estimator that arises from a refined and meaningful loss function for binding site inference. We discuss the main advantages of centroid estimation for motif discovery, including computational convenience, and how its principled derivation offers further insights about the posterior distribution of binding site configurations. We also illustrate, using simulated and real datasets, that the centroid estimator can differ from the maximum a posteriori estimator.Comment: 24 pages, 9 figure

The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields

We consider an "elastic" version of the statistical mechanical monomer-dimer problem on the n-dimensional integer lattice. Our setting includes the classical "rigid" formulation as a special case and extends it by allowing each dimer to consist of particles at arbitrarily distant sites of the lattice, with the energy of interaction between the particles in a dimer depending on their relative position. We reduce the free energy of the elastic dimer-monomer (EDM) system per lattice site in the thermodynamic limit to the moment Lyapunov exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value and covariance function are the Boltzmann factors associated with the monomer energy and dimer potential. In particular, the classical monomer-dimer problem becomes related to the MLE of a moving average GRF. We outline an approach to recursive computation of the partition function for "Manhattan" EDM systems where the dimer potential is a weighted l1-distance and the auxiliary GRF is a Markov random field of Pickard type which behaves in space like autoregressive processes do in time. For one-dimensional Manhattan EDM systems, we compute the MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a compact transfer operator on a Hilbert space which is related to the annihilation and creation operators of the quantum harmonic oscillator and also recast it as the eigenvalue problem for a pantograph functional-differential equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue of DCDS-