Underlying Diffusion Process Research Articles

Identification of causal diffusion effects using placebo outcomes under structural stationarity

Abstract Social and biomedical scientists have long been interested in the process through which ideas and behaviours diffuse. In this article, we study an urgent social problem, the spatial diffusion of hate crimes against refugees in Germany, which has admitted more than 1 million asylum seekers since the 2015 refugee crisis. Despite its importance, identification of causal diffusion effects, also known as peer and contagion effects, remains challenging because the commonly used assumption of no omitted confounders is often untenable due to contextual confounding and homophily bias. To address this long-standing problem, we examine causal identification using placebo outcomes under a new assumption of structural stationarity, which formalizes the underlying diffusion process with a class of nonparametric structural equation models with recursive structure. We show under structural stationarity that a lagged dependent variable is a general, valid placebo outcome for detecting a wide range of biases, including the 2 types mentioned above. We then propose a difference-in-differences style estimator that can directly correct biases under an additional causal assumption. Analysing fine-grained geo-coded hate crime data from Germany, we show when and how the proposed methods can detect and correct unmeasured confounding in spatial causal diffusion analysis.

Narrow Pore Crossing of Active Particles under Stochastic Resetting

We propose a two-dimensional model of biochemical activation process, whereby self-propelling particles of finite correlation times are injected at the center of a circular cavity with constant rate equal to the inverse of their lifetime; activation is triggered when one such particle hits a receptor on the cavity boundary, modeled as a narrow pore. We numerically investigated this process by computing the particle mean-first exit times through the cavity pore as a function of the correlation and injection time constants. Due to the breach of the circular symmetry associated with the positioning of the receptor, the exit times may depend on the orientation of the self-propelling velocity at injection. Stochastic resetting appears to favor activation for large particle correlation times, where most of the underlying diffusion process occurs at the cavity boundary.

Evolution of Microstructure and Hardness of the Nitrided Zone during Plasma Nitriding of High-Alloy Tool Steel

Plasma nitriding is widely used in various industrial applications to improve surface hardness and wear properties. Especially for tool steels, it is also used to improve the support and adhesion of diamond-like carbon (DLC) coatings. The properties of the nitrided zone produced by plasma nitriding are influenced by the applied process parameter, in particular temperature and time. However, for high-alloy tool steels, a deeper understanding of the underlying diffusion processes of the nitrogen and the interaction with the existing microstructure, as well as the effects on the case depth is still lacking. Therefore, in this study, specimens of high-alloy tool steel X153CrMoV12 were plasma nitrided at varying temperatures (480 °C, 520 °C, 560 °C) and treatment times (2 h, 4 h, 16 h). The resulting nitrided zones were investigated by optical and scanning electron microscopy (OM and SEM), depth-dependent glow discharge optical emission spectroscopy (GDOES), X-ray diffraction (XRD), and hardness measurements to characterize their microstructure, chemical composition, and hardness depending on the process parameters. The distribution of carbides (M7C3), e.g., chromium carbides, affects the diffusion of the nitrogen and the layer growth. An increase of temperature and duration leads to an increased layer thickness. The composition of the compound layer is, e.g., influenced by the process parameters: ε nitrides (Fe2–3N) occurred preferentially at lower temperatures, while γ′ nitrides (Fe4N) appeared mostly at higher temperatures. In order to investigate the influence of the carbides of the high-alloy tool steel on the nitriding process, a new methodology was developed by means of finite element analysis (FE), which makes it possible to analyze this influence on the development of the nitrogen concentration profile. This methodology makes it possible for the first time to map the heterogeneous nitrogen evolution and distribution.

Fundamental solution to 1D degenerate diffusion equation with locally bounded coefficients

In this work we study the degenerate diffusion equation ∂t=xαa(x)∂x2+b(x)∂x for (x,t)∈(0,∞)2, equipped with a Cauchy initial data and the Dirichlet boundary condition at 0. We assume that the order of degeneracy at 0 of the diffusion operator is α∈(0,2), and the coefficient functions (and their derivatives) are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution p(x,y,t) and prove several properties for p(x,y,t); by conducting a localization procedure, we obtain an approximation for p(x,y,t) for x,y in a neighborhood of 0 and t sufficiently small, where the error estimates only rely on the local bounds of a(x) and b(x) (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of α=1. Our work extends part of the existing results to the cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probabilistic view (e.g., wellposedness of stochastic differential equations).

Dynkin Games with Incomplete and Asymmetric Information

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian setup, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result that allows us to find the value of the game and players’ optimal strategies by solving suitable quasi-variational inequalities with some nonstandard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.

On the generalizability of diffusion MRI signal representations across acquisition parameters, sequences and tissue types: Chronicles of the MEMENTO challenge

Diffusion MRI (dMRI) has become an invaluable tool to assess the microstructural organization of brain tissue. Depending on the specific acquisition settings, the dMRI signal encodes specific properties of the underlying diffusion process. In the last two decades, several signal representations have been proposed to fit the dMRI signal and decode such properties. Most methods, however, are tested and developed on a limited amount of data, and their applicability to other acquisition schemes remains unknown. With this work, we aimed to shed light on the generalizability of existing dMRI signal representations to different diffusion encoding parameters and brain tissue types. To this end, we organized a community challenge - named MEMENTO, making available the same datasets for fair comparisons across algorithms and techniques. We considered two state-of-the-art diffusion datasets, including single-diffusion-encoding (SDE) spin-echo data from a human brain with over 3820 unique diffusion weightings (the MASSIVE dataset), and double (oscillating) diffusion encoding data (DDE/DODE) of a mouse brain including over 2520 unique data points. A subset of the data sampled in 5 different voxels was openly distributed, and the challenge participants were asked to predict the remaining part of the data. After one year, eight participant teams submitted a total of 80 signal fits. For each submission, we evaluated the mean squared error, the variance of the prediction error and the Bayesian information criteria. The received submissions predicted either multi-shell SDE data (37%) or DODE data (22%), followed by cartesian SDE data (19%) and DDE (18%). Most submissions predicted the signals measured with SDE remarkably well, with the exception of low and very strong diffusion weightings. The prediction of DDE and DODE data seemed more challenging, likely because none of the submissions explicitly accounted for diffusion time and frequency. Next to the choice of the model, decisions on fit procedure and hyperparameters play a major role in the prediction performance, highlighting the importance of optimizing and reporting such choices. This work is a community effort to highlight strength and limitations of the field at representing dMRI acquired with trending encoding schemes, gaining insights into how different models generalize to different tissue types and fiber configurations over a large range of diffusion encodings.

Diffusion State Distances: Multitemporal Analysis, Fast Algorithms, and Applications to Biological Networks

Data-dependent metrics are powerful tools for learning the underlying structure of high-dimensional data. This article further develops and analyzes a data-dependent metric known as diffusion state distance (DSD), which compares points using a data-driven diffusion process. Unlike related diffusion methods, DSDs incorporate information across time scales, which allows for the intrinsic data structure to be inferred in a parameter-free manner. This article develops a theory for DSD based on the multitemporal emergence of mesoscopic equilibria in the underlying diffusion process. New algorithms for denoising and dimension reduction with DSD are also proposed and analyzed. These approaches are based on a weighted spectral decomposition of the underlying diffusion process, and experiments on synthetic datasets and real biological networks illustrate the efficacy of the proposed algorithms in terms of both speed and accuracy. Throughout, comparisons with related methods are made in order to illustrate the distinct advantages of DSD for datasets exhibiting multiscale structure.

A Convergent Discretization Method for Transition Path Theory for Diffusion Processes

Transition path theory (TPT) for diffusion processes is a framework for analysing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretisation of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretisation. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.

Growth kinetics of σ-phase precipitates and underlying diffusion processes in CrMnFeCoNi high-entropy alloys

Key mechanisms and elementary diffusion processes that control the growth kinetics of σ precipitates in high-entropy alloys were investigated in the present study. For this purpose, an off-equiatomic Cr26Mn20Fe20Co20Ni14 alloy with an initially single-phase FCC structure was subjected to isothermal heat treatments, which are known to promote the formation of σ phase, i.e., aging between 600 °C and 1000 °C for times ranging from 0.1 h to 1000 h. The growth kinetics of σ precipitates at grain boundaries of the FCC matrix and those located within the interior of the grains were analyzed separately. The latter precipitates are found to grow through direct substitutional diffusion of Cr-solutes towards and Mn, Fe, Co, and Ni away from them and the growth rate of the allotriomorphs can be rationalized by the collector plate mechanism of interfacial diffusion-aided growth. From the growth-kinetics data obtained in the present study, lattice interdiffusion coefficients as well as diffusivities along crystalline defects were obtained. Above 800 °C, the growth kinetics are dominated by lattice interdiffusion of Cr in the FCC matrix described by DL = 9.8 × 10-4 exp[(-300 kJ/mol)/(RT)] m2/s. At lower temperatures, the growth kinetics are enhanced by fast interdiffusion along dislocation pipes, which temperature dependence is given by DD = 5.0 × 10-3 exp[(-205 kJ/mol)/(RT)] m2/s. The Cr-diffusivity along σ/FCC interphase boundaries deduced from the thickening kinetics of grain boundary precipitates can be represented by the Arrhenius relationship DI = 0.5 × 10-4 exp[(-145 kJ/mol)/(RT)] m2/s, which is similar to that found for grain boundary interdiffusion in metals and alloys.

Spatial variation in the frequency of knockdown resistance genotypes in Florida Aedes aegypti populations

BackgroundThe development of insecticide resistance in disease-vectoring mosquito species can lead to vector control failure and disease resurgence. However, insecticide applications remain an essential public health intervention. In Florida, insecticide resistance in Aedes aegypti, an anthropophilic mosquito species capable of transmitting dengue, chikungunya, and Zika virus, is a major concern. Understanding the location, scale, and driving factors of insecticide resistance can enhance the ability of vector control organizations to target populations effectively.MethodsWe used previously collected data on frequencies of mutations that confer resistance to commonly used pyrethroid insecticides in Ae. aegypti specimens from 62 sites distributed across 18 counties in Florida. To determine the scale of clustering for the most resistant variant, we used a Ripley’s K function. We also used a spatial scanning statistic technique to identify locations of clusters where higher than expected frequencies of susceptible or resistant mosquitoes occurred. We then tested for associations between landscape, demographic, and insecticide-use factors using a beta regression modelling approach and evaluated the effect of spatial lag and spatial error terms on overall explanatory power of these models.ResultsThe scale at which maximum clustering of the most resistant variant occurs is approximately 20 kilometers. We identified statistically significant clusters of genotypes associated with resistance in several coastal cities, although some of these clusters were near significant clusters of susceptible mosquitoes, indicating selection pressures vary at the local scale. Vegetation density, distance from roads, and pyrethroid-use by vector control districts were consistently significant predictors of knockdown resistance genotype frequency in the top-performing beta regression models, although pyrethroid use surprisingly had a negatively associated with resistance. The incorporation of spatial lags resulted in improvements to the fit and explanatory power of the models, indicating an underlying diffusion process likely explains some of the spatial patterns observed.ConclusionsThe genetic mutations that confer resistance to pyrethroids in Ae. aegypti mosquitoes in Florida exhibit spatial autocorrelation and patterns that can be partially explained by landscape and insecticide-use factors. Further work at local scales should be able to identify the mechanisms by which these variables influence selection for alleles associated with resistance.

Spatially regularized active diffusion learning for high-dimensional images

An active learning method for the classification of high-dimensional images is proposed in which spatially-regularized nonlinear diffusion geometry is used to characterize cluster cores. The proposed method samples from estimated cluster cores in order to generate a small but potent set of training labels which propagate to the remainder of the dataset via the underlying diffusion process. By spatially regularizing the rich, high-dimensional spectral information of the image to efficiently estimate the most significant and influential points in the data, our approach avoids redundancy in the training dataset. This allows it to produce high-accuracy labelings with a very small number of training labels. The proposed algorithm admits an efficient numerical implementation that scales essentially linearly in the number of data points under a suitable data model and enjoys state-of-the-art performance on real hyperspectral images.

Joint Learning of Embedding-Based Parent Components and Information Diffusion for Social Networks

Diffusion on social networks refers to the phenomena that opinions are spread via connected nodes. Given a set of observed cascades, the underlying diffusion process can be inferred for social network analysis. Earlier studies for modeling the diffusion process often assume that the activation of a node depends independently on the activations of its neighbors (or called parent nodes). Nevertheless, the activation of a node also depends on the connectivity of its neighbors. For instance, the opinions from the neighbors of the same closely connected social group are often similar, and thus those neighbors exhibit similar influence. Some recent studies incorporate the structural dependency of neighbors as connected components, which allow more accurate diffusion models to be inferred. However, the effectiveness of such component-based models often depends on how the components are identified. Existing methods are not designed to directly preserve the local connectivity of neighbors. In this paper, we propose to incorporate network embedding to enhance the performance of component-based diffusion models in social networks. In particular, we embed nodes in a social network in a latent vector space with local connectivity of the nodes preserved. Parent component identification then becomes a clustering task in the embedding space. A united probabilistic framework is proposed so that the parent components and the component-based diffusion models can be inferred simultaneously using a two-level EM algorithm based on observed information cascades. For performance evaluation, we apply the proposed model to both synthetic and real world data sets with promising results obtained. The empirical results also show how the use of the embedding-based framework can enhance both the component identification and the diffusion model.

Growth Kinetics of Σ-Phase Precipitates and Underlying Diffusion Processes in Crmnfeconi High-Entropy Alloys

Key mechanisms and elementary diffusion processes that control the growth kinetics of σ precipitates in high-entropy alloys were investigated in the present study. For this purpose, an off-equiatomic Cr26Mn20Fe20Co20Ni14 alloy with an initially single-phase FCC structure was subjected to isothermal heat treatments, which are known to promote the formation of σ phase, i.e. aging between 600 °C and 1000 °C for times ranging from 0.1 h to 1000 h. The growth kinetics of σ precipitates at grain boundaries of the FCC matrix and those located within the interior of the grains were analyzed separately. The latter precipitates are found to grow through direct substitutional diffusion of Cr-solutes towards and Mn, Fe, Co, and Ni away from them and the growth rate of the allotriomorphs can be rationalized by the collector plate mechanism of interfacial diffusion-aided growth. From the growth-kinetics data obtained in the present study, lattice interdiffusion coefficients as well as diffusivities along crystalline defects were obtained. Above 800 °C, the growth kinetics are dominated by lattice interdiffusion of Cr in the FCC matrix described by DL = 9.8 × 10-4 exp[( 300 kJ/mol)/(RT)] m2/s. At lower temperatures, the growth kinetics are enhanced by fast interdiffusion along dislocation pipes, which temperature dependence is given by DD = 2.5 × 10-3 exp[( 210 kJ/mol)/(RT)] m2/s. The Cr-diffusivity along σ/FCC interphase boundaries deduced from the thickening kinetics of grain boundary precipitates can be represented by the Arrhenius relationship DI = 0.5 × 10-4 exp[( 145 kJ/mol)/(RT)] m2/s, which is similar to that found for grain boundary interdiffusion in metals and alloys.

LEARNING ABOUT EDGE EFFECTS AND ULTRAMICROELECTRODES IN ELECTROCHEMISTRY: SYNERGY BETWEEN EXPERIMENTS AND SIMULATIONS

A short experimental activity aimed at upper level undergraduate students or beginner researchers is presented to introduce the concepts of edge effects and hemispherical diffusion at ultramicroelectrodes. Students will perform simple cyclic voltammetry experiments varying the scan rate and using disk electrodes with different radii. They are then asked to interpret the results based on their previous knowledge. Simulations are also performed, which provide a clear visual interpretation of the underlying diffusion process and show the transition between linear and hemispherical diffusion regimes. The simulations are also presented as a tool to extend the study beyond the range of experimentally accessible conditions. The interplay between voltammetry experiments and simulations allows a natural comprehension of these concepts without the need of a previous rigorous theoretical introduction.

A 2D and 3D nanostructural study of naturally deformed pyrite: assessing the links between trace element mobility and defect structures

The links between deformation-induced micro- and nanostructures and trace element mobility in sulphide minerals have recently become a popular subject of research in the Earth sciences due to its connections with metallic ore paragenesis. It has been shown that plastic deformation in pyrite creates diffusion pathways in the form of low-angle grain boundaries that act as traps for base- and precious-metals. However, the plastic behavior of pyrite and the physiochemical processes that concentrate these trace elements in deformation-induced micro- and nanostructures remain poorly understood. In this study, we develop strategies for 2D and 3D analysis of naturally deformed sulphides by combining electron backscatter diffraction, electron channeling contrast imaging and atom probe tomography on pyrite in an attempt to better understand the underlying diffusion processes that mobilize trace elements. The combined results reveal structures associated with crystal-plastic deformation in the form of dislocations, stacking faults, and low-angle grain boundaries that are decorated by As and Co. Although our data support a dislocation-impurity pair diffusion model, we have evidence that multiple diffusion mechanisms may have acted simultaneously. In this study, we applied new data processing techniques that allow for orientation measurement of nanostructural crystal defects from atom probe tomography data. Dislocations within our studied sample occur along the (110) planes suggesting glide on {110}.

A FORWARD EQUATION FOR COMPUTING DERIVATIVES EXPOSURE

We derive a forward equation for computing the expected exposure of financial derivatives. Under general assumptions about the underlying diffusion process, we give an explicit decomposition of the exposure into an intrinsic value which can be directly deduced from the term structure of the forward mark-to-market, and a time value which expresses the variability of the future mark-to-market. Our approach is inspired by Dupire’s equation for local volatility and leads to an ordinary differential equation qualifying the evolution of the expected exposure with respect to the observation dates. We show how this approach can be linked with local times theory in dimension one and to the co-area formula in a higher dimension. As for numerical considerations, we show how this approach leads to an efficient numerical method in the case of one or two risk factors. The accuracy and time-efficiency of this forward representation in small dimension are of special interest in benchmarking XVA valuation adjustments at trade level.

Dynamics and control for multi-agent networked systems: A finite-difference approach

We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time horizon and the size of the controls.

Real-world diffusion dynamics based on point process approaches: a review

Bursts in human and natural activities are highly clustered in time, suggesting that these activities are influenced by previous events within the social or natural system. Bursty behavior in the real world conveys information of underlying diffusion processes, which have been the focus of diverse scientific communities from online social media to criminology and epidemiology. However, universal components of real-world diffusion dynamics that cut across disciplines remain unexplored. Here, we introduce a wide range of diffusion processes across disciplines and propose universal components of diffusion frameworks. We apply these components to diffusion-based studies of human disease spread, through a case study of the vector-borne disease dengue. The proposed universality of diffusion can motivate transdisciplinary research and provide a fundamental framework for diffusion models.

Bootstrapped Graph Diffusions

Graph-based semi-supervised learning (SSL) algorithms predict labels for all nodes based on provided labels of a small set of seed nodes. Classic methods capture the graph structure through some underlying diffusion process that propagates through the graph edges. Spectral diffusion, which includes personalized page rank and label propagation, propagates through random walks. Social diffusion propagates through shortest paths. These diffusions are linear in the sense of not distinguishing between contributions of few "strong" relations or many "weak'' relations. Recent methods such as node embeddings and graph convolutional networks (GCN) attained significant gains in quality for SSL tasks. These methods vary on how the graph structure, seed label information, and other features are used, but do share a common thread of nonlinearity that suppresses weak relations and re-enforces stronger ones. Aiming for quality gain with more scalable methods, we revisit classic linear diffusion methods and place them in a self-training framework. The resulting bootstrapped diffusions are nonlinear in that they re-enforce stronger relations, as with the more complex methods. Surprisingly, we observe that SSL with bootstrapped diffusions not only significantly improves over the respective non-bootstrapped baselines but also outperform state-of-the-art SSL methods. Moreover, since the self-training wrapper retains the scalability of the base method, we obtain both higher quality and better scalability.

Bootstrapped Graph Diffusions

Graph-based semi-supervised learning (SSL) algorithms predict labels for all nodes based on provided labels of a small set of seed nodes. Classic methods capture the graph structure through some underlying diffusion process that propagates through the graph edges. Spectral diffusion, which includes personalized page rank and label propagation, propagates through random walks. Social diffusion propagates through shortest paths. These diffusions are linear in the sense of not distinguishing between contributions of few "strong" relations or many "weak" relations. Recent methods such as node embeddings and graph convolutional networks (GCN) attained significant gains in quality for SSL tasks. These methods vary on how the graph structure, seed label information, and other features are used, but do share a common thread of nonlinearity that suppresses weak relations and reenforces stronger ones. Aiming for quality gain with more scalable methods, we revisit classic linear diffusion methods and place them in a self-training framework. The resulting bootstrapped diffusions are nonlinear in that they re-enforce stronger relations, as with the more complex methods. Surprisingly, we observe that SSL with bootstrapped diffusions not only significantly improves over the respective non-bootstrapped baselines but also outperform state-of-the-art SSL methods. Moreover, since the self-training wrapper retains the scalability of the base method, we obtain both higher quality and better scalability.