2347
[1808.00023] The Measure and Mismeasure of Fairness: A Critical Review of Fair Machine Learning
The nascent field of fair machine learning aims to ensure that decisions guided by algorithms are equitable. Over the last several years, three formal definitions of fairness have gained prominence: (1) anti-classification, meaning that protected attributes---like race, gender, and their proxies---are not explicitly used to make decisions; (2) classification parity, meaning that common measures of predictive performance (e.g., false positive and false negative rates) are equal across groups defined by the protected attributes; and (3) calibration, meaning that conditional on risk estimates, outcomes are independent of protected attributes. Here we show that all three of these fairness definitions suffer from significant statistical limitations. Requiring anti-classification or classification parity can, perversely, harm the very groups they were designed to protect; and calibration, though generally desirable, provides little guarantee that decisions are equitable. In contrast to these formal fairness criteria, we argue that it is often preferable to treat similarly risky people similarly, based on the most statistically accurate estimates of risk that one can produce. Such a strategy, while not universally applicable, often aligns well with policy objectives; notably, this strategy will typically violate both anti-classification and classification parity. In practice, it requires significant effort to construct suitable risk estimates. One must carefully define and measure the targets of prediction to avoid retrenching biases in the data. But, importantly, one cannot generally address these difficulties by requiring that algorithms satisfy popular mathematical formalizations of fairness. By highlighting these challenges in the foundation of fair machine learning, we hope to help researchers and practitioners productively advance the area.
machine_learning  algorithms  bias  ethics  privacy  review  for_friends
2 days ago
Algorithm for generating a Brownian motion on a sphere - IOPscience
We present a new algorithm for generation of a random walk on a two-dimensional sphere. The algorithm is obtained by viewing the 2-sphere as the equator in the 3-sphere surrounded by an infinitesimally thin band with boundary which reflects Brownian particles and then applying known effective methods for generating Brownian motion on the 3-sphere. To test the method, the diffusion coefficient was calculated in computer simulations using the new algorithm and, for comparison, also using a commonly used method in which the particle takes a Brownian step in the tangent plane to the 2-sphere and is then projected back to the spherical surface. The two methods are in good agreement for short time steps, while the method presented in this paper continues to give good results also for larger time steps, when the alternative method becomes unstable.

https://pdfs.semanticscholar.org/6901/148c0e70f94c4f50a691185f02af92d2489c.pdf
differential_geometry  brownian  stochastic_processes  simulation
4 days ago
Open Science by Design: Realizing a Vision for 21st Century Research | The National Academies Press
Openness and sharing of information are fundamental to the progress of science and to the effective functioning of the research enterprise. The advent of scientific journals in the 17th century helped power the Scientific Revolution by allowing researchers to communicate across time and space, using the technologies of that era to generate reliable knowledge more quickly and efficiently. Harnessing today’s stunning, ongoing advances in information technologies, the global research enterprise and its stakeholders are moving toward a new open science ecosystem. Open science aims to ensure the free availability and usability of scholarly publications, the data that result from scholarly research, and the methodologies, including code or algorithms, that were used to generate those data.

Open Science by Design is aimed at overcoming barriers and moving toward open science as the default approach across the research enterprise. This report explores specific examples of open science and discusses a range of challenges, focusing on stakeholder perspectives. It is meant to provide guidance to the research enterprise and its stakeholders as they build strategies for achieving open science and take the next steps.
nap  report  science_as_a_social_process  public_policy  institutions  sociology_of_science
6 days ago
Universal Method to Sort Complex Information Found | Quanta Magazine
-- ignore the typically hyperbolic title! ... contains information on soon to be published results relating graph embedding in normed spaces to nearest neighbor queries in databases.
algorithms  graph_theory  metric_spaces  computational_complexity  quanta_mag
6 days ago
[1705.02801] Graph Embedding Techniques, Applications, and Performance: A Survey
Graphs, such as social networks, word co-occurrence networks, and communication networks, occur naturally in various real-world applications. Analyzing them yields insight into the structure of society, language, and different patterns of communication. Many approaches have been proposed to perform the analysis. Recently, methods which use the representation of graph nodes in vector space have gained traction from the research community. In this survey, we provide a comprehensive and structured analysis of various graph embedding techniques proposed in the literature. We first introduce the embedding task and its challenges such as scalability, choice of dimensionality, and features to be preserved, and their possible solutions. We then present three categories of approaches based on factorization methods, random walks, and deep learning, with examples of representative algorithms in each category and analysis of their performance on various tasks. We evaluate these state-of-the-art methods on a few common datasets and compare their performance against one another. Our analysis concludes by suggesting some potential applications and future directions. We finally present the open-source Python library we developed, named GEM (Graph Embedding Methods, available at this https URL), which provides all presented algorithms within a unified interface to foster and facilitate research on the topic.

--Ok as a survey. Works from the following definition of graph embedding.

(Graph embedding) Given a graph G=(V,E),a graph embedding is a mapping f:vi→yi∈Rd∀∈[n]such that d |V |and the function f preserves some proximity measure defined on graph G
graph_theory  survey  geometry  topological_data_analysis  networks  ?
6 days ago
Shtetl-Optimized » Blog Archive » Summer recapitulates life
-- contains links to lectures. One of the lectures uses classical and quantum information theory to derive positive mass conditions in general relativity... also contains Aaronson's lectures on quantum computation, AdS-CFT and all that.
blog  scott.aaronson  information_theory  quantum_computing  general_relativity  computational_complexity
10 days ago
Stochastic Turing patterns in a synthetic bacterial population | PNAS
The origin of biological morphology and form is one of the deepest problems in science, underlying our understanding of development and the functioning of living systems. In 1952, Alan Turing showed that chemical morphogenesis could arise from a linear instability of a spatially uniform state, giving rise to periodic pattern formation in reaction–diffusion systems but only those with a rapidly diffusing inhibitor and a slowly diffusing activator. These conditions are disappointingly hard to achieve in nature, and the role of Turing instabilities in biological pattern formation has been called into question. Recently, the theory was extended to include noisy activator–inhibitor birth and death processes. Surprisingly, this stochastic Turing theory predicts the existence of patterns over a wide range of parameters, in particular with no severe requirement on the ratio of activator–inhibitor diffusion coefficients. To explore whether this mechanism is viable in practice, we have genetically engineered a synthetic bacterial population in which the signaling molecules form a stochastic activator–inhibitor system. The synthetic pattern-forming gene circuit destabilizes an initially homogenous lawn of genetically engineered bacteria, producing disordered patterns with tunable features on a spatial scale much larger than that of a single cell. Spatial correlations of the experimental patterns agree quantitatively with the signature predicted by theory. These results show that Turing-type pattern-forming mechanisms, if driven by stochasticity, can potentially underlie a broad range of biological patterns. These findings provide the groundwork for a unified picture of biological morphogenesis, arising from a combination of stochastic gene expression and dynamical instabilities.
pattern_formation  pde  dynamical_system  teaching  ecology  spatio-temporal_statistics
19 days ago
‘Infotaxis’ as a strategy for searching without gradients | Nature
Chemotactic bacteria rely on local concentration gradients to guide them towards the source of a nutrient1. Such local cues pointing towards the location of the source are not always available at macroscopic scales because mixing in a flowing medium breaks up regions of high concentration into random and disconnected patches. Thus, animals sensing odours in air or water detect them only intermittently as patches sweep by on the wind or currents2,3,4,5,6. A macroscopic searcher must devise a strategy of movement based on sporadic cues and partial information. Here we propose a search algorithm, which we call ‘infotaxis’, designed to work under such conditions. Any search process can be thought of as acquisition of information on source location; for infotaxis, information plays a role similar to concentration in chemotaxis. The infotaxis strategy locally maximizes the expected rate of information gain. We demonstrate its efficiency using a computational model of odour plume propagation and experimental data on mixing flows7. Infotactic trajectories feature ‘zigzagging’ and ‘casting’ paths similar to those observed in the flight of moths8. The proposed search algorithm is relevant to the design of olfactory robots9,10,11, but the general idea of infotaxis can be applied more broadly in the context of searching with sparse information.

--I'll surprised if this is not presented in one of Herb Simon's papers as a side note. But still, interesting.....
heuristics  animal_behavior  ethology  information_theory
20 days ago
[1305.2167] The nonlinear heat equation on W-random graphs
For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial value problems (IVPs) for the discrete model converge to the solution of the IVP for its continuum limit. These results combined with the analysis of nonlocally coupled deterministic networks in [9] justify the continuum (thermodynamic) limit for a large class of coupled dynamical systems on convergent families of graphs.
graph_limit  dynamical_system  networks
20 days ago
[1302.5804] The nonlinear heat equation on dense graphs and graph limits
We use the combination of ideas and results from the theory of graph limits and nonlinear evolution equations to provide a rigorous mathematical justification for taking continuum limit for certain nonlocally coupled networks and to extend this method to cover many complex networks, for which it has not been applied before. Specifically, for dynamical networks on convergent sequences of simple and weighted graphs, we prove convergence of solutions of the initial-value problems for discrete models to those of the limiting continuous equations. In addition, for sequences of simple graphs converging to {0, 1}-valued graphons, it is shown that the convergence rate depends on the fractal dimension of the boundary of the support of the graph limit. These results are then used to study the regions of continuity of chimera states and the attractors of the nonlocal Kuramoto equation on certain multipartite graphs. Furthermore, the analytical tools developed in this work are used in the rigorous justification of the continuum limit for networks on random graphs that we undertake in a companion paper (Medvedev, 2013).
As a by-product of the analysis of the continuum limit on deterministic and random graphs, we identify the link between this problem and the convergence analysis of several classical numerical schemes: the collocation, Galerkin, and Monte-Carlo methods. Therefore, our results can be used to characterize convergence of these approximate methods of solving initial-value problems for nonlinear evolution equations with nonlocal interactions.
graph_limit  dynamical_system  networks
20 days ago
[1710.01696] Maximum likelihood estimation of the Latent Class Model through model boundary decomposition
The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.
latent_variables  graphical_models  algebraic_statistics  statistics  machine_learning
22 days ago
[1605.02114] The semilinear heat equation on sparse random graphs
Using the theory of Lp-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we derive the continuum limit for coupled dynamical systems on power law graphs. The latter is the main motivation for this work.

--See all the recent work of
http://www.math.drexel.edu/~medvedev/
graph_limit  dynamical_system  networks
22 days ago
[1807.03573] Power Network Dynamics on Graphons
Power grids are undergoing major changes from a few large producers to smart grids build upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture, when dynamic nodes can achieve synchronization to a common grid frequency on complex network topologies. In this paper we study a second-order rotator model in the large network limit. We merge the recent theory of random graph limits for complex small-world networks with approaches to first-order systems on graphons. We prove that there exists a well-posed continuum limit integral equation approximating the large finite-dimensional case power grid network dynamics. Then we analyse the linear stability of synchronized solutions and prove linear stability. However, on small-world networks we demonstrate that there are topological parameters moving the spectrum arbitrarily close to the imaginary axis leading to potential instability on finite time scales.
graph_limit  dynamical_system  networks
22 days ago
Religious change preceded economic change in the 20th century | Science Advances
The decline in the everyday importance of religion with economic development is a well-known correlation, but which phenomenon comes first? Using unsupervised factor analysis and a birth cohort approach to create a retrospective time series, we present 100-year time series of secularization in different nations, derived from recent global values surveys, which we compare by decade to historical gross domestic product figures in those nations. We find evidence that a rise in secularization generally has preceded economic growth over the past century. Our multilevel, time-lagged regressions also indicate that tolerance for individual rights predicted 20th century economic growth even better than secularization. These findings hold when we control for education and shared cultural heritage.
time_series  secularism  economic_history  the_civilizing_process  economic_sociology  via:pinker
25 days ago
Stochastic Processes: From Applications to Theory - CRC Press Book
Unlike traditional books presenting stochastic processes in an academic way, this book includes concrete applications that students will find interesting such as gambling, finance, physics, signal processing, statistics, fractals, and biology. Written with an important illustrated guide in the beginning, it contains many illustrations, photos and pictures, along with several website links. Computational tools such as simulation and Monte Carlo methods are included as well as complete toolboxes for both traditional and new computational techniques.
book  stochastic_processes  probability
27 days ago
Crane : The Ubiquitous Ewens Sampling Formula
Ewens’s sampling formula exemplifies the harmony of mathematical theory, statistical application, and scientific discovery. The formula not only contributes to the foundations of evolutionary molecular genetics, the neutral theory of biodiversity, Bayesian nonparametrics, combinatorial stochastic processes, and inductive inference but also emerges from fundamental concepts in probability theory, algebra, and number theory. With an emphasis on its far-reaching influence throughout statistics and probability, we highlight these and many other consequences of Ewens’s seminal discovery.
probability  statistics  review  combinatorics
28 days ago
Statistical and Computational Guarantees for the Baum-Welch Algorithm
The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known to be susceptible to local optima. In this paper, we first give a general characterization of the basin of attraction associated with any global optimum of the population likelihood. By exploiting this characterization, we provide non-asymptotic finite sample guarantees on the Baum-Welch updates and show geometric convergence to a small ball of radius on the order of the minimax rate around a global optimum. As a concrete example, we prove a linear rate of convergence for a hidden Markov mixture of two isotropic Gaussians given a suitable mean separation and an initialization within a ball of large radius around (one of) the true parameters. To our knowledge, these are the first rigorous local convergence guarantees to global optima for the Baum-Welch algorithm in a setting where the likelihood function is nonconvex. We complement our theoretical results with thorough numerical simulations studying the convergence of the Baum-Welch algorithm and illustrating the accuracy of our predictions.
algorithms  statistics  machine_learning  martin.wainwright
29 days ago
Density Estimation in Infinite Dimensional Exponential Families
In this paper, we consider an infinite dimensional exponential family  of probability densities, which are parametrized by functions in a reproducing kernel Hilbert space , and show it to be quite rich in the sense that a broad class of densities on ℝd can be approximated arbitrarily well in Kullback-Leibler (KL) divergence by elements in . Motivated by this approximation property, the paper addresses the question of estimating an unknown density p0 through an element in . Standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves), which are based on minimizing the KL divergence between p0 and , do not yield practically useful estimators because of their inability to efficiently handle the log-partition function. We propose an estimator p̂ n based on minimizing the Fisher divergence, J(p0‖p) between p0 and p∈, which involves solving a simple finite-dimensional linear system. When p0∈, we show that the proposed estimator is consistent, and provide a convergence rate of n−min{23,2β+12β+2} in Fisher divergence under the smoothness assumption that logp0∈(Cβ) for some β≥0, where C is a certain Hilbert-Schmidt operator on  and (Cβ) denotes the image of Cβ. We also investigate the misspecified case of p0∉ and show that J(p0‖p̂ n)→infp∈J(p0‖p) as n→∞, and provide a rate for this convergence under a similar smoothness condition as above. Through numerical simulations we demonstrate that the proposed estimator outperforms the non- parametric kernel density estimator, and that the advantage of the proposed estimator grows as d increases.
kernel  machine_learning  exponential_family
29 days ago
Faithfulness of Probability Distributions and Graphs
A main question in graphical models and causal inference is whether, given a probability distribution P (which is usually an underlying distribution of data), there is a graph (or graphs) to which P is faithful. The main goal of this paper is to provide a theoretical answer to this problem. We work with general independence models, which contain probabilistic independence models as a special case. We exploit a generalization of ordering, called preordering, of the nodes of (mixed) graphs. This allows us to provide sufficient conditions for a given independence model to be Markov to a graph with the minimum possible number of edges, and more importantly, necessary and sufficient conditions for a given probability distribution to be faithful to a graph. We present our results for the general case of mixed graphs, but specialize the definitions and results to the better-known subclasses of undirected (concentration) and bidirected (covariance) graphs as well as directed acyclic graphs.
causal_inference  graphical_models  machine_learning  statistics
29 days ago
Ding , Li : Causal Inference: A Missing Data Perspective
Inferring causal effects of treatments is a central goal in many disciplines. The potential outcomes framework is a main statistical approach to causal inference, in which a causal effect is defined as a comparison of the potential outcomes of the same units under different treatment conditions. Because for each unit at most one of the potential outcomes is observed and the rest are missing, causal inference is inherently a missing data problem. Indeed, there is a close analogy in the terminology and the inferential framework between causal inference and missing data. Despite the intrinsic connection between the two subjects, statistical analyses of causal inference and missing data also have marked differences in aims, settings and methods. This article provides a systematic review of causal inference from the missing data perspective. Focusing on ignorable treatment assignment mechanisms, we discuss a wide range of causal inference methods that have analogues in missing data analysis, such as imputation, inverse probability weighting and doubly robust methods. Under each of the three modes of inference—Frequentist, Bayesian and Fisherian randomization—we present the general structure of inference for both finite-sample and super-population estimands, and illustrate via specific examples. We identify open questions to motivate more research to bridge the two fields.
causal_inference  statistics  review
29 days ago
Aldous : Elo Ratings and the Sports Model: A Neglected Topic in Applied Probability?
In a simple model for sports, the probability A beats B is a specified function of their difference in strength. One might think this would be a staple topic in Applied Probability textbooks (like the Galton–Watson branching process model, for instance) but it is curiously absent. Our first purpose is to point out that the model suggests a wide range of questions, suitable for “undergraduate research” via simulation but also challenging as professional research. Our second, more specific, purpose concerns Elo-type rating algorithms for tracking changing strengths. There has been little foundational research on their accuracy, despite a much-copied “30 matches suffice” claim, which our simulation study casts doubt upon.
probability  statistics  review
29 days ago
[1611.00814] Information-theoretic thresholds from the cavity method
Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results [Contucci et al.: Communications in Mathematical Physics 2013]. Further, we prove the conjecture from [Krzakala et al.: PNAS 2007] about the condensation phase transition in the random graph coloring problem for any number q≥3 of colors. Moreover, we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model [Decelle et al.: Phys. Rev. E 2011]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes [Montanari: IEEE Transactions on Information Theory 2005].
review  spin_glass  information_theory  phase_transition
29 days ago
Intervention and Identifiability in Latent Variable Modelling | SpringerLink
"We consider the use of interventions for resolving a problem of unidentified statistical models. The leading examples are from latent variable modelling, an influential statistical tool in the social sciences. We first explain the problem of statistical identifiability and contrast it with the identifiability of causal models. We then draw a parallel between the latent variable models and Bayesian networks with hidden nodes. This allows us to clarify the use of interventions for dealing with unidentified statistical models. We end by discussing the philosophical and methodological import of our result."
causal_inference  graphical_models  latent_variables  identifiability  via:cshalizi
29 days ago
On the nature and use of models in network neuroscience | Nature Reviews Neuroscience
Network theory provides an intuitively appealing framework for studying relationships among interconnected brain mechanisms and their relevance to behaviour. As the space of its applications grows, so does the diversity of meanings of the term network model. This diversity can cause confusion, complicate efforts to assess model validity and efficacy, and hamper interdisciplinary collaboration. In this Review, we examine the field of network neuroscience, focusing on organizing principles that can help overcome these challenges. First, we describe the fundamental goals in constructing network models. Second, we review the most common forms of network models, which can be described parsimoniously along the following three primary dimensions: from data representations to first-principles theory; from biophysical realism to functional phenomenology; and from elementary descriptions to coarse-grained approximations. Third, we draw on biology, philosophy and other disciplines to establish validation principles for these models. We close with a discussion of opportunities to bridge model types and point to exciting frontiers for future pursuits.
review  networks  neuroscience
4 weeks ago
[1309.6928] Structure and dynamics of core-periphery networks
Recent studies uncovered important core/periphery network structures characterizing complex sets of cooperative and competitive interactions between network nodes, be they proteins, cells, species or humans. Better characterization of the structure, dynamics and function of core/periphery networks is a key step of our understanding cellular functions, species adaptation, social and market changes. Here we summarize the current knowledge of the structure and dynamics of "traditional" core/periphery networks, rich-clubs, nested, bow-tie and onion networks. Comparing core/periphery structures with network modules, we discriminate between global and local cores. The core/periphery network organization lies in the middle of several extreme properties, such as random/condensed structures, clique/star configurations, network symmetry/asymmetry, network assortativity/disassortativity, as well as network hierarchy/anti-hierarchy. These properties of high complexity together with the large degeneracy of core pathways ensuring cooperation and providing multiple options of network flow re-channelling greatly contribute to the high robustness of complex systems. Core processes enable a coordinated response to various stimuli, decrease noise, and evolve slowly. The integrative function of network cores is an important step in the development of a large variety of complex organisms and organizations. In addition to these important features and several decades of research interest, studies on core/periphery networks still have a number of unexplored areas.
review  networks  dynamics
4 weeks ago
MAKING REPLICATION MAINSTREAM | Behavioral and Brain Sciences | Cambridge Core
Many philosophers of science and methodologists have argued that the ability to repeat studies and obtain similar results is an essential component of science. A finding is elevated from single observation to scientific evidence when the procedures that were used to obtain it can be reproduced and the finding itself can be replicated. Recent replication attempts show that some high profile results---most notably in psychology, but in many other disciplines as well---cannot be replicated consistently. These replication attempts have generated a considerable amount of controversy and the issue of whether direct replications have value has, in particular, proven to be contentious. However, much of this discussion has occurred in published commentaries and social media outlets, resulting in a fragmented discourse. To address the need for an integrative summary, we review various types of replication studies and then discuss the most commonly voiced concerns about direct replication. We provide detailed responses to these concerns and consider different statistical ways to evaluate replications. We conclude there are no theoretical or statistical obstacles to making direct replication a routine aspect of psychological science.
replication_of_studies  meta-analysis  statistics  philosophy_of_science  via:gelman
4 weeks ago
[0902.3561] Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials
We investigate the construction of diffusions consisting of infinitely numerous Brownian particles moving in ℝd and interacting via logarithmic functions (two-dimensional Coulomb potentials). These potentials are very strong and act over a long range in nature. The associated equilibrium states are no longer Gibbs measures. We present general results for the construction of such diffusions and, as applications thereof, construct two typical interacting Brownian motions with logarithmic interaction potentials, namely the Dyson model in infinite dimensions and Ginibre interacting Brownian motions. The former is a particle system in ℝ, while the latter is in ℝ2. Both models are translation and rotation invariant in space, and as such, are prototypes of dimensions d=1,2, respectively. The equilibrium states of the former diffusion model are determinantal or Pfaffian random point fields with sine kernels. They appear in the thermodynamical limits of the spectrum of the ensembles of Gaussian random matrices such as GOE, GUE and GSE. The equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non-Hermitian Gaussian random matrices known as the Ginibre ensemble.
interating_particle_system  brownian  diffusion  random_matrix  thermodynamics
4 weeks ago
Large‐Deviation Principle for Interacting Brownian Motions - Seo - 2017 - Communications on Pure and Applied Mathematics - Wiley Online Library
We prove the large‐deviation principle for the empirical process in a system of locally interacting Brownian motions in the nonequilibrium. Such a phenomenon has been proven only for two lattice systems: the symmetric simple exclusion process and the zero‐range process. Therefore, we have achieved the third result in this context and moreover the first result for the diffusion‐type interacting particle system.© 2016 Wiley Periodicals, Inc.
large_deviation  interating_particle_system  brownian  diffusion
4 weeks ago
[1802.09679] A guide to Brownian motion and related stochastic processes
This is a guide to the mathematical theory of Brownian motion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical theory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley and Durrett , and who wants a broader perspective on the theory of Brownian motion and related stochastic processes than can be found in these texts.
tutorial  review  brownian  stochastic_processes
4 weeks ago
[1401.4770] Opinion Exchange Dynamics
We survey a range of models of opinion exchange. From the introduction: "The exchange of opinions between individuals is a fundamental social interaction... Moreover, many models in this field are an excellent playground for mathematicians, especially those working in probability, algorithms and combinatorics. The goal of this survey is to introduce such models to mathematicians, and especially to those working in discrete mathematics, information theory, optimization, probability and statistics."
opinion_dynamics  opinion_formation  interating_particle_system  game_theory  review
4 weeks ago
The Non-Existence of Representative Agents by Matthew O. Jackson, Leeat Yariv :: SSRN
We characterize environments in which there exists a representative agent: an agent who inherits the structure of preferences of the population that she represents. The existence of such a representative agent imposes strong restrictions on individual utility functions -- requiring them to be linear in the allocation and additively separable in any parameter that characterizes agents' preferences (e.g., a risk aversion parameter, a discount factor, etc.). Commonly used classes of utility functions (exponentially discounted utility functions, CRRA or CARA utility functions, logarithmic functions, etc.) do not admit a representative agent.

--
non-equilibrium  statistical_mechanics  interating_particle_system  game_theory  economics  macro_from_micro  collective_dynamics  matthew.jackson
4 weeks ago
A Transition to Sharp Timing in Stochastic Leaky Integrate-and-Fire Neurons Driven by Frozen Noisy Input | Neural Computation | MIT Press Journals
The firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current (Mainen & Sejnowski, 1995). This suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit. Modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking (Cecchi et al., 2000; Tiesinga, 2002; Fellous, Rudolph, Destexhe, & Sejnowski, 2003). In order to investigate this interplay, we focus on the stochastic leaky integrate-and-fire neuron and identify the Hölder exponent H of the integrated input as the key mathematical property dictating the regime of firing of a single-unit neuron. We have recently provided numerical evidence (Taillefumier & Magnasco, 2013) for the existence of a phase transition when becomes less than the statistical Hölder exponent associated with internal gaussian white noise (H=1/2). Here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent H>0. In doing so, we account for the existence of a transition between two regimes of firing when H=1/2, and we show that spiking times have a continuous density when the Hölder exponent satisfies H>1/2. The transition at H=1/2 formally separates rate codes, for which the neural firing probability varies smoothly, from temporal codes, for which the neuron fires at sharply defined times regardless of the intensity of internal noise.
first-passage-time  neural_coding_and_decoding  neural_representation
5 weeks ago
[1408.6596] Emergence of Clustering in an Acquaintance Model without Homophily
We introduce an agent-based acquaintance model in which social links are created by processes in which there is no explicit homophily. In spite of the homogeneous nature of the social interactions, highly-clustered social networks can arise. The crucial feature of our model is that of variable transitive interactions. Namely, when an agent introduces two unconnected friends, the rate at which a connection actually occurs between them depends on the number of their mutual acquaintances. As this transitive interaction rate is varied, the social network undergoes a dramatic clustering transition. Close to the transition, the network consists of a collection of well-defined communities. As a function of time, the network can also undergo an \emph{incomplete} gelation transition, in which the gel, or giant cluster, does not constitute the entire network, even at infinite time. Some of the clustering properties of our model also arise, but in a more gradual manner, in Facebook networks. Finally, we discuss a more realistic variant of our original model in which there is a soft cutoff in the rate of transitive interactions. With this variant, one can construct network realizations that quantitatively match Facebook networks.
networks  teaching  sidney.redner
5 weeks ago
Economic Consequences of Network Structure
We survey the literature on the economic consequences of the structure of social networks. We develop a taxonomy of "macro" and "micro" characteristics of social-interaction networks and discuss both the theoretical and empirical findings concerning the role of those characteristics in determining learning, diffusion, decisions, and resulting behaviors. We also discuss the challenges of accounting for the endogeneity of networks in assessing the relationship between the patterns of interactions and behaviors.
economics  social_networks  networks  review  matthew.jackson  teaching
5 weeks ago
The Benefits and Pitfalls of Google Scholar | PS: Political Science & Politics | Cambridge Core
Google Scholar (GS) is an important tool that faculty, administrators, and external reviewers use to evaluate the scholarly impact of candidates for jobs, tenure, and promotion. This article highlights both the benefits of GS—including the reliability and consistency of its citation counts and its platform for disseminating scholarship and facilitating networking—and its pitfalls. GS has biases because citation is a social and political process that disadvantages certain groups, including women, younger scholars, scholars in smaller research communities, and scholars opting for risky and innovative work. GS counts also reflect practices of strategic citation that exacerbate existing hierarchies and inequalities. As a result, it is imperative that political scientists incorporate other data sources, especially independent scholarly judgment, when making decisions that are crucial for careers. External reviewers have a unique obligation to offer a reasoned, rigorous, and qualitative assessment of a scholar’s contributions and therefore should not use GS.
sociology_of_science  bibliometry  scientometry  via:randw
5 weeks ago
Predictive modeling of U.S. health care spending in late life | Science
That one-quarter of Medicare spending in the United States occurs in the last year of life is commonly interpreted as waste. But this interpretation presumes knowledge of who will die and when. Here we analyze how spending is distributed by predicted mortality, based on a machine-learning model of annual mortality risk built using Medicare claims. Death is highly unpredictable. Less than 5% of spending is accounted for by individuals with predicted mortality above 50%. The simple fact that we spend more on the sick—both on those who recover and those who die—accounts for 30 to 50% of the concentration of spending on the dead. Our results suggest that spending on the ex post dead does not necessarily mean that we spend on the ex ante “hopeless.”
health  statistics  prediction  mortality_risk  econometrics  sendhil.mullainathan  for_friends
5 weeks ago
[1807.01771] Direct Uncertainty Prediction with Applications to Healthcare
Large labeled datasets for supervised learning are frequently constructed by assigning each instance to multiple human evaluators, and this leads to disagreement in the labels associated with a single instance. Here we consider the question of predicting the level of disagreement for a given instance, and we find an interesting phenomenon: direct prediction of uncertainty performs better than the two-step process of training a classifier and then using the classifier outputs to derive an uncertainty. We show stronger performance for predicting disagreement via this direct method both in a synthetic setting whose parameters we can fully control, and in a paradigmatic healthcare application involving multiple labels assigned by medical domain experts. We further show implications for allocating additional labeling effort toward instances with the greatest levels of predicted disagreement.
sendhil.mullainathan  machine_learning  prediction  health  for_friends
5 weeks ago
[1607.02441] Generalized Hypergeometric Ensembles: Statistical Hypothesis Testing in Complex Networks
Statistical ensembles of networks, i.e., probability spaces of all networks that are consistent with given aggregate statistics, have become instrumental in the analysis of complex networks. Their numerical and analytical study provides the foundation for the inference of topological patterns, the definition of network-analytic measures, as well as for model selection and statistical hypothesis testing. Contributing to the foundation of these data analysis techniques, in this Letter we introduce generalized hypergeometric ensembles, a broad class of analytically tractable statistical ensembles of finite, directed and weighted networks. This framework can be interpreted as a generalization of the classical configuration model, which is commonly used to randomly generate networks with a given degree sequence or distribution. Our generalization rests on the introduction of dyadic link propensities, which capture the degree-corrected tendencies of pairs of nodes to form edges between each other. Studying empirical and synthetic data, we show that our approach provides broad perspectives for model selection and statistical hypothesis testing in data on complex networks.
networks  statistics  network_data_analysis
6 weeks ago
Statistical Causality from a Decision-Theoretic Perspective | Annual Review of Statistics and Its Application
We present an overview of the decision-theoretic framework of statistical causality, which is well suited for formulating and solving problems of determining the effects of applied causes. The approach is described in detail, and it is related to and contrasted with other current formulations, such as structural equation models and potential responses. Topics and applications covered include confounding, the effect of treatment on the treated, instrumental variables, and dynamic treatment strategies.
causal_inference  decison_theory  statistics  review  philip.dawid
6 weeks ago
Brothers of the Gun: A Memoir of the Syrian War
A bracingly immediate memoir of the Syrian war from its inception to the present by a young man coming of age and finding his voice as a journalist, whose friends traveled divergent paths through the carnage. An intimate lens into the century’s bloodiest conflict, and a profound meditation on kinship, home, and freedom. Illustrated with over 80 ink drawings by Molly Crabapple.

In 2011, Marwan Hisham and his two friends–fellow working-class college students–Nael and Tareq, joined the first protests of the Arab Spring in Syria, in response to a recent massacre. Arm-in-arm they marched, poured Coke into each other’s eyes to blunt the effects of tear gas, ran from the security forces, and cursed the country’s president, Bashar al-Assad. It was ecstasy. A long-bottled revolution was finally erupting, and freedom from a brutal dictator seemed, at last, imminent. Five years later, the three young friends were scattered: one now an Islamist revolutionary; another dead at the hands of government soldiers; and the last, Marwan, now a journalist in Turkish exile, trying to find a way back to a homeland reduced to rubble.

Brothers of the Gun is the story of a young man coming of age during the Syrian war from its inception to the present. Marwan watched from the rooftops as regime warplanes bombed rebels; as revolutionary activist groups, for a few dreamy days, spray-painted hope on Raqqa; as his friends died or threw in their lot with Islamist fighters. He became a journalist by courageously tweeting out news from a city under siege by ISIS, the Russians, and the Americans, all at once. He watched the country that ran through his veins–the country that held his hopes, dreams, and fears–be destroyed in front of him, and eventually joined the relentless stream of refugees risking their lives to escape.

With vivid illustrations that bring to life the beauty and chaos, Brothers of the Gun offers a ground-level reflection on the Syrian revolution–and how it bled into international catastrophe and global war. This is a story of pragmatism and idealism, impossible violence and repression, and, even in the midst of war, profound acts of courage, creativity, and hope
book  syria
6 weeks ago
Handbook of Graphical Models - CRC Press Book
A graphical model is a statistical model that is represented by a graph. The factorization properties underlying graphical models facilitate tractable computation with multivariate distributions, making the models a valuable tool with a plethora of applications. Furthermore, directed graphical models allow intuitive causal interpretations and have become a cornerstone for causal inference.

While there exist a number of excellent books on graphical models, the field has grown so much that individual authors can hardly cover its entire scope. Moreover, the field is interdisciplinary by nature. Through chapters by leading researchers from different areas, this handbook provides a broad and accessible overview of the state of the art.

The handbook is targeted at a wide audience, including graduate students, applied researchers, and experts in graphical models.

https://stat.ethz.ch/~maathuis/papers/Handbook.pdf
graphical_models  book  statistics  algorithms
6 weeks ago
[1801.08364] Model selection and local geometry
We consider problems in model selection caused by the geometry of models close to their points of intersection. In some cases, including common classes of causal or graphical models as well as time series models, distinct models may nevertheless have identical tangent spaces. This has two immediate consequences: first, in order to obtain constant power to reject one model in favour of another we need local alternative hypotheses that decrease to the null at a slower rate than the usual parametric n−1/2 (typically we will require n−1/4 or slower); in other words, to distinguish between the models we need large effect sizes or very large sample sizes. Second, we show that under even weaker conditions on their tangent cones, models in these classes cannot be made simultaneously convex by a reparameterization.
This shows that Bayesian network models, amongst others, cannot be learned directly with a convex method similar to the graphical lasso. However, we are able to use our results to suggest methods for model selection that learn the tangent space directly, rather than the model itself. In particular, we give a generic algorithm for learning discrete ancestral graph models, which includes Bayesian network models as a special case.
graphical_models  algebraic_statistics  differential_geometry
6 weeks ago
Nicolas Rashevsky's Mathematical Biophysics | SpringerLink
This paper explores the work of Nicolas Rashevsky, a Russian émigré theoretical physicist who developed a program in “mathematical biophysics” at the University of Chicago during the 1930s. Stressing the complexity of many biological phenomena, Rashevsky argued that the methods of theoretical physics – namely mathematics – were needed to “simplify” complex biological processes such as cell division and nerve conduction. A maverick of sorts, Rashevsky was a conspicuous figure in the biological community during the 1930s and early 1940s: he participated in several Cold Spring Harbor symposia and received several years of funding from the Rockefeller Foundation. However, in contrast to many other physicists who moved into biology, Rashevsky's work was almost entirely theoretical, and he eventually faced resistance to his mathematical methods. Through an examination of the conceptual, institutional, and scientific context of Rashevsky's work, this paper seeks to understand some of the reasons behind this resistance.
history  bio-physics  social_science  animal_behavior  mathematics
6 weeks ago
Order statistics for first passage times in diffusion processes | SpringerLink
We consider the problem of the first passage times for absorption (trapping) of the firstj (j = 1,2, ....) ofk, j <k, identical and independent diffusing particles for the asymptotic case k≫>1. Our results are a special case of the theory of order statistics. We show that in one dimension the mean time to absorption at a boundary for the first ofk diffusing particles, μ1,k, goes as (lnk)−1 for the set of initial conditions in which none of thek particles is located at a boundary and goes ask−2 for the set of initial conditions in which some of thek particles may be located at the boundary. We demonstrate that in one dimension our asymptotic results (k21) are independent of the potential field in which the diffusion takes place for a wide class of potentials. We conjecture that our results are independent of dimension and produce some evidence supporting this conjecture. We conclude with a discussion of the possible import of these results on diffusion-controlled rate processes.
first-passage-time  brownian  diffusion
6 weeks ago
Determination of Firing Times for the Stochastic Fitzhugh-Nagumo Neuronal Model | Neural Computation | MIT Press Journals
We present for the first time an analytical approach for determining the time of firing of multicomponent nonlinear stochastic neuronal models. We apply the theory of first exit times for Markov processes to the Fitzhugh-Nagumo system with a constant mean gaussian white noise input, representing stochastic excitation and inhibition. Partial differential equations are obtained for the moments of the time to first spike. The observation that the recovery variable barely changes in the prespike trajectory leads to an accurate one-dimensional approximation. For the moments of the time to reach threshold, this leads to ordinary differential equations that may be easily solved. Several analytical approaches are explored that involve perturbation expansions for large and small values of the noise parameter. For ranges of the parameters appropriate for these asymptotic methods, the perturbation solutions are used to establish the validity of the one-dimensional approximation for both small and large values of the noise parameter. Additional verification is obtained with the excellent agreement between the mean and variance of the firing time found by numerical solution of the differential equations for the one-dimensional approximation and those obtained by simulation of the solutions of the model stochastic differential equations. Such agreement extends to intermediate values of the noise parameter. For the mean time to threshold, we find maxima at small noise values that constitute a form of stochastic resonance. We also investigate the dependence of the mean firing time on the initial values of the voltage and recovery variables when the input current has zero mean.
first-passage-time  neuroscience  stochastic_process  dynamical_system
6 weeks ago
[1806.07149] Stochastic FitzHugh-Nagumo neuron model in excitable regime embeds a leaky integrate-and-fire model
In this paper, we provide a complete mathematical construction for a stochastic leaky-integrate-and-fire model (LIF) mimicking the interspike interval (ISI) statistics of a stochastic FitzHugh-Nagumo neuron model (FHN) in the excitable regime, where the unique fixed point is stable. Under specific types of noises, we prove that there exists a global random attractor for the stochastic FHN system. The linearization method is then applied to estimate the firing time and to derive the associated radial equation representing a LIF equation. This result confirms the previous prediction in [Ditlevsen, S. and Greenwood, P. (2012). The Morris-Lecar neuron model embeds a leaky integrate-and-fire model. Journal of Mathematical Biology, 67(2):239-259] for the Morris-Lecar neuron model in the bistability regime consisting of a stable fixed point and a stable limit cycle.
first-passage-time  neuroscience  dynamical_system
6 weeks ago
[1801.09909] Properties of additive functionals of Brownian motion with resetting
We study the distribution of time-additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate according to a Poisson process and placed back to a given reset position. For three examples of functionals (occupation time, area, absolute area), we investigate the effect of the resetting by computing various moments and distributions, using a recent result that links the generating function with resetting to the generating function without reset. We also derive a general variational formula for the large deviation rate function, which we use to analyze two different large deviation regimes appearing in the presence of resetting.
satya.majumdar  hugo.touchette  markov_models  brownian  stochastic_processes
6 weeks ago
[1806.10701] Empirical Risk Minimization and Stochastic Gradient Descent for Relational Data
Empirical risk minimization is the principal tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent advances in graph sampling theory. We (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this risk that are automatically unbiased. The key ingredient is to consider the method by which data is sampled from a graph as an explicit component of model design. Theoretical results establish that the choice of sampling scheme is critical. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we are able to solve the risk minimization in a plug-and-play fashion even on large datasets. We demonstrate empirically that relational ERM models achieve state-of-the-art results on semi-supervised node classification tasks. The experiments also confirm the importance of the choice of sampling scheme.
graph_limit  machine_learning  prediction  via:droy
6 weeks ago
[1703.10007] A Course in Interacting Particle Systems
These lecture notes give an introduction to the theory of interacting particle systems. The main subjects are the construction using generators and graphical representations, the mean field limit, stochastic order, duality, and the relation to oriented percolation. An attempt is made to give a large number of examples beyond the classical voter, contact and Ising processes and to illustrate these based on numerical simulations.
interating_particle_system  tutorial  review
6 weeks ago
Active Matter | The MIT Press
The past few decades brought a revolution in computer software and hardware; today we are on the cusp of a materials revolution. If yesterday we programmed computers and other machines, today we program matter itself. This has created new capabilities in design, computing, and fabrication, which allow us to program proteins and bacteria, to generate self-transforming wood products and architectural details, and to create clothing from “intelligent textiles” that grow themselves. This book offers essays and sample projects from the front lines of the emerging field of active matter.

Active matter and programmable materials are at the intersection of science, art, design, and engineering, with applications in fields from biology and computer science to architecture and fashion. These essays contextualize current work and explore recent research. Sample projects, generously illustrated in color, show the range of possibilities envisioned by their makers. Contributors explore the design of active material at scales from nano to micro, kilo, and even planetary. They investigate processes of self-assembly at a microscopic level; test new materials that can sense and actuate themselves; and examine the potential of active matter in the built environment and in living and artificial systems. Active Matter is an essential guide to a field that could shape the future of design.
book  active_matter  self_organization  collective_dynamics  robotics  artificial_life  artificial_intelligence
7 weeks ago
Network structure of the human musculoskeletal system shapes neural interactions on multiple time scales | Science Advances
Human motor control requires the coordination of muscle activity under the anatomical constraints imposed by the musculoskeletal system. Interactions within the central nervous system are fundamental to motor coordination, but the principles governing functional integration remain poorly understood. We used network analysis to investigate the relationship between anatomical and functional connectivity among 36 muscles. Anatomical networks were defined by the physical connections between muscles, and functional networks were based on intermuscular coherence assessed during postural tasks. We found a modular structure of functional networks that was strongly shaped by the anatomical constraints of the musculoskeletal system. Changes in postural tasks were associated with a frequency-dependent reconfiguration of the coupling between functional modules. These findings reveal distinct patterns of functional interactions between muscles involved in flexibly organizing muscle activity during postural control. Our network approach to the motor system offers a unique window into the neural circuitry driving the musculoskeletal system.
networks  dynamics  connectome
7 weeks ago
Phys. Rev. X 8, 021071 (2018) - Key Features of Turing Systems are Determined Purely by Network Topology
Turing’s theory of pattern formation is a universal model for self-organization, applicable to many systems in physics, chemistry, and biology. Essential properties of a Turing system, such as the conditions for the existence of patterns and the mechanisms of pattern selection, are well understood in small networks. However, a general set of rules explaining how network topology determines fundamental system properties and constraints has not been found. Here we provide a first general theory of Turing network topology, which proves why three key features of a Turing system are directly determined by the topology: the type of restrictions that apply to the diffusion rates, the robustness of the system, and the phase relations of the molecular species.

--seems similar to Golubitsky's work on synchronization but ....
pattern_formation  nonlinear_dynamics  networks
7 weeks ago
Persistence and first-passage properties in nonequilibrium systems: Advances in Physics: Vol 62, No 3
In this review, we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spin models undergoing phase-ordering dynamics, diffusion equation, fluctuating interfaces, etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalizations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.
first-passage-time  non-equilibrium  dynamics  review  satya.majumdar
7 weeks ago
[1805.06826] The Blessings of Multiple Causes
Causal inference from observational data often assumes "strong ignorability," that all confounders are observed. This assumption is standard yet untestable. However, many scientific studies involve multiple causes, different variables whose effects are simultaneously of interest. We propose the deconfounder, an algorithm that combines unsupervised machine learning and predictive model checking to perform causal inference in multiple-cause settings. The deconfounder infers a latent variable as a substitute for unobserved confounders and then uses that substitute to perform causal inference. We develop theory for when the deconfounder leads to unbiased causal estimates, and show that it requires weaker assumptions than classical causal inference. We analyze its performance in three types of studies: semi-simulated data around smoking and lung cancer, semi-simulated data around genomewide association studies, and a real dataset about actors and movie revenue. The deconfounder provides a checkable approach to estimating close-to-truth causal effects.
causal_discovery  statistics  machine_learning  david.blei
7 weeks ago
The Exit Problem for Randomly Perturbed Dynamical Systems | SIAM Journal on Applied Mathematics | Vol. 33, No. 2 | Society for Industrial and Applied Mathematics
The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system has an asymptotically stable equilibrium point, random effects can cause the trajectories of the system to leave any bounded domain with probability one. In this paper we consider the effect of small random perturbations of the type referred to as Gaussian white noise, on a (deterministic) dynamical system $\dot x = b(x)$. The vector $x(t)$ then becomes a stochastic process $x_\varepsilon (t)$ which satisfies the stochastic differential equation $dx_\varepsilon = b(x_\varepsilon )dt + \varepsilon \sigma (x_\varepsilon )dw$. Here $w(t)$ is the n dimensional Wiener process (Brownian motion), $b(x)$ is a vector field, $\sigma (x)$ is the diffusion matrix and $\varepsilon \ne 0$ is a small real parameter. We give the first complete formal solution of the following problem originally posed by Kolmogorov: Find the asymptotic expansion in $\varepsilon$ of (i) the probability distribution of the points on the boundary of a domain, where trajectories of the perturbed system first exit, and (ii) of the expected exit times. Our method is to relate the solutions of the above problems to the solutions of various singularly perturbed elliptic boundary value problems with turning points, whose solutions are then constructed asymptotically.
review  first-passage-time
7 weeks ago
The Exit Problem: A New Approach to Diffusion Across Potential Barriers | SIAM Journal on Applied Mathematics | Vol. 36, No. 3 | Society for Industrial and Applied Mathematics
We consider the problem of a Brownian particle confined in a potential well of forces, which escapes the potential barrrier as the result of white noise forces acting on it. The problem is characterized by a diffusion process in a force field and is described by Langevin’s stochastic differential equation. We consider potential wells with many transition states and compute the expected exit time of the particle from the well as well as the probability distribution of the exit points. Our method relates these quantities to the solutions of certain singularly perturbed elliptic boundary value problems which are solved asymptotically. Our results are then applied to the calculation of chemical reaction rates by considering the breaking of chemical bonds caused by random molecular collisions, and to the calculation of the diffusion matrix in crystals by considering random atomic migration in the periodic force field of the crystal lattice, caused by thermal vibrations of the lattice.
review  first-passage-time
7 weeks ago
A Direct Approach to the Exit Problem | SIAM Journal on Applied Mathematics | Vol. 50, No. 2 | Society for Industrial and Applied Mathematics
This paper considers the problem of exit for a dynamical system driven by small white noise, from the domain of attraction of a stable state. A direct singular perturbation analysis of the forward equation is presented, based on Kramers’ approach, in which the solution to the stationary Fokker–Planck equation is constructed, for a process with absorption at the boundary and a source at the attractor. In this formulation the boundary and matching conditions fully determine the uniform expansion of the solution, without resorting to “external” selection criteria for the expansion coefficients, such as variational principles or the Lagrange identity, as in our previous theory. The exit density and the mean first passage time to the boundary are calculated from the solution of the stationary Fokker–Planck equation as the probability current density and as the inverse of the total flux on the boundary, respectively. As an application, a uniform expansion is constructed for the escape rate in Kramers’ problem of activated escape from a potential well for the full range of the dissipation parameter.
review  first-passage-time
7 weeks ago
[1702.08446] Monte Carlo on manifolds: sampling densities and integrating functions
We describe and analyze some Monte Carlo methods for manifolds in Euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by un-normalized densities on such manifolds. The sampler uses a specific orthogonal projection to the surface that requires only information about the tangent space to the manifold, obtainable from first derivatives of the constraint functions, hence avoiding the need for curvature information or second derivatives. Second, we use the sampler to develop a multi-stage algorithm to compute integrals over such manifolds. We provide single-run error estimates that avoid the need for multiple independent runs. Computational experiments on various test problems show that the algorithms and error estimates work in practice. The method is applied to compute the entropies of different sticky hard sphere systems. These predict the temperature or interaction energy at which loops of hard sticky spheres become preferable to chains.
numerical  markov_process  simulation
7 weeks ago
Non-assortative community structure in resting and task-evoked functional brain networks | bioRxiv
Brain networks exhibit community structure that reconfigures during cognitively demanding tasks. Extant work has emphasized a single class of communities: those that are assortative, or internally dense and externally sparse. Other classes that may play key functional roles in brain function have largely been ignored, leading to an impoverished view in the best case and a mischaracterization in the worst case. Here, we leverage weighted stochastic blockmodeling, a community detection method capable of detecting diverse classes of communities, to study the community structure of functional brain networks while subjects either rest or perform cognitively demanding tasks. We find evidence that the resting brain is largely assortative, although higher order association areas exhibit non-assortative organization, forming cores and peripheries. Surprisingly, this assortative structure breaks down during tasks and is supplanted by core, periphery, and disassortative communities. Using measures derived from the community structure, we show that it is possible to classify an individual's task state with an accuracy that is well above average. Finally, we show that inter-individual differences in the composition of assortative and non-assortative communities is correlated with subject performance on in-scanner cognitive tasks. These findings offer a new perspective on the community organization of functional brain networks and its relation to cognition.

-- for class discussions and/or student projects. Not sure about their implications... need to read more carefully.
networks  connectome  community_detection  via:clauset  teaching
7 weeks ago
[1711.04024] How fragile are information cascades?
It is well known that sequential decision making may lead to information cascades. That is, when agents make decisions based on their private information, as well as observing the actions of those before them, then it might be rational to ignore their private signal and imitate the action of previous individuals. If the individuals are choosing between a right and a wrong state, and the initial actions are wrong, then the whole cascade will be wrong. This issue is due to the fact that cascades can be based on very little information.
We show that if agents occasionally disregard the actions of others and base their action only on their private information, then wrong cascades can be avoided. Moreover, we study the optimal asymptotic rate at which the error probability at time t can go to zero. The optimal policy is for the player at time t to follow their private information with probability pt=c/t, leading to a learning rate of c′/t, where the constants c and c′ are explicit.
self_organization  information_diffusion  contagion  decision_making  models_of_behavior  networks  ?
7 weeks ago
Continuum limits for classical sequential growth models - Brightwell - 2010 - Random Structures &amp; Algorithms - Wiley Online Library
A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,…,n − 1} and putting i below j if there is a path i = i1…ik = j in the graph with i1 < … < ik.

Rideout and Sorkin 14 provide computational evidence that suitably normalized sequences of random graph orders have a “continuum limit.” We confirm that this is the case and show that the continuum limit is always a semiorder.

Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large‐scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,…,n ‐ 1} such that the random partial order on {0,…,n ‐ 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur.

Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension ≥ 2.
poset  combinatorics
7 weeks ago
[1708.03633] Properties of the Promotion Markov Chain on Linear Extensions
The Tsetlin library is a very well studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. This result has been generalized in different ways by various people. In this work we investigate one of the generalizations given by the extended promotion Markov Chain on linear extensions of a poset P introduced by Ayyer, Klee, and Schilling in 2014. They showed that if the poset P is a rooted forest, the transition matrix of this Markov chain has eigenvalues that are linear in the transition probabilities and described their multiplicities. We show that the same property holds for a larger class of posets for which we also derive convergence to stationarity results.
poset  markov_process  combinatorics
7 weeks ago
[1512.03563] Markov chains on graded posets: Compatibility of up-directed and down-directed transition probabilities
We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an \emph{up chain} or \emph{down chain}). The second type toggles between two adjacent rank levels (an \emph{up-and-down chain}).
We introduce two compatibility concepts between the up-directed transition probabilities (an \emph{up rule}) and the down-directed (a \emph{down rule}), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice.
Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.
poset  markov_process  combinatorics
7 weeks ago
The lumpability property for a family of Markov chains on poset block structures - ScienceDirect
We construct different classes of lumpings for a family of Markov chain products which reflect the structure of a given finite poset. We essentially use combinatorial methods. We prove that, for such a product, every lumping can be obtained from the action of a suitable subgroup of the generalized wreath product of symmetric groups, acting on the underlying poset block structure, if and only if the poset defining the Markov process is totally ordered, and one takes the uniform Markov operator in each factor state space. Finally we show that, when the state space is a homogeneous space associated with a Gelfand pair, the spectral analysis of the corresponding lumped Markov chain is completely determined by the decomposition of the group action into irreducible submodules
poset  markov_process  combinatorics
7 weeks ago
Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics | SIAM Review | Vol. 22, No. 2 | Society for Industrial and Applied Mathematics
Stochastic differential equations are used as models for various physical phenomena, such as chemical reactions, atomic migration in crystals, thermal fluctuations in electrical networks, noisy signals in radio transmission, etc. First passage times of solutions of such equations from certain domains and the distribution of the exit points are computed from the solutions of singularly perturbed elliptic boundary value problems. Physical interpretation of these quantities is given. Applications in communication theory and in reliability of structures are shown.
review  first-passage-time  stochastic_processes  dynamical_system
8 weeks ago
Asymptotics of Elliptic and Parabolic PDEs - and their Applications in Statistical Physics, Computational Neuroscience, and Biophysics | David Holcman | Springer
This is a monograph on the emerging branch of mathematical biophysics combining asymptotic analysis with numerical and stochastic methods to analyze partial differential equations arising in biological and physical sciences.

In more detail, the book presents the analytic methods and tools for approximating solutions of mixed boundary value problems, with particular emphasis on the narrow escape problem. Informed throughout by real-world applications, the book includes topics such as the Fokker-Planck equation, boundary layer analysis, WKB approximation, applications of spectral theory, as well as recent results in narrow escape theory. Numerical and stochastic aspects, including mean first passage time and extreme statistics, are discussed in detail and relevant applications are presented in parallel with the theory.
Including background on the classical asymptotic theory of differential equations, this book is written for scientists of various backgrounds interested in deriving solutions to real-world problems from first principles.
markov_process  pde  book
8 weeks ago
On the stochastic properties of bursts of single ion channel openings and of clusters of bursts | Philosophical Transactions of the Royal Society B: Biological Sciences
Characteristics of observed bursts of single channel openings were derived recently for two particular ion channel mechanisms. In this paper these methods are generalized so that the observable characteristics of bursts can be calculated directly for any mechanism that has transition probabilities that are independent of time as long as the process is at equilibrium or is maintained in a steady state by an energy supply. General expressions are given for the distributions of the open time, the number of openings per burst, the total open time per burst, the gaps within and between bursts, and so on. With the aid of these general results a single computer program can be written that will provide numerical values for such distributions for postulated mechanism, given only the transition rates between the various states. The results are illustrated by a numerical example of a mechanism in which two agonist molecules can bind sequentially, and either singly or doubly occupied receptor ion channels may open. The analogous theory is also given for the case where bursts of channel openings are grouped into clusters; many of the results bear a close analogy with those found for simple bursts.
bio-physics  bio-chemistry  markov_models  automata  ion_channels  stochastic_processes
8 weeks ago
Stochastic Actor-Oriented Models for Network Dynamics | Annual Review of Statistics and Its Application
This article discusses the stochastic actor-oriented model for analyzing panel data of networks. The model is defined as a continuous-time Markov chain, observed at two or more discrete time moments. It can be regarded as a generalized linear model with a large amount of missing data. Several estimation methods are discussed. After presenting the model for evolution of networks, attention is given to coevolution models. These use the same approach of a continuous-time Markov chain observed at a small number of time points, but now with an extended state space. The state space can be, for example, the combination of a network and nodal variables, or a combination of several networks. This leads to models for the dynamics of multivariate networks. The article emphasizes the approach to modeling and algorithmic issues for estimation; some attention is given to comparison with other models.
review  networks  dynamics
8 weeks ago

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