Path Length Distribution Research Articles

Brain networks modeling for studying the mechanism underlying the development of Alzheimer's disease.

Alzheimer's disease is a primary age-related neurodegenerative disorder that can result in impaired cognitive and memory functions. Although connections between changes in brain networks of Alzheimer's disease patients have been established, the mechanisms that drive these alterations remain incompletely understood. This study, which was conducted in 2018 at Northeastern University in China, included data from 97 participants of the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset covering genetics, imaging, and clinical data. All participants were divided into two groups: normal control (n = 52; 20 males and 32 females; mean age 73.90 ± 4.72 years) and Alzheimer's disease (n = 45, 23 males and 22 females; mean age 74.85 ± 5.66). To uncover the wiring mechanisms that shaped changes in the topology of human brain networks of Alzheimer's disease patients, we proposed a local naïve Bayes brain network model based on graph theory. Our results showed that the proposed model provided an excellent fit to observe networks in all properties examined, including clustering coefficient, modularity, characteristic path length, network efficiency, betweenness, and degree distribution compared with empirical methods. This proposed model simulated the wiring changes in human brain networks between controls and Alzheimer's disease patients. Our results demonstrate its utility in understanding relationships between brain tissue structure and cognitive or behavioral functions. The ADNI was performed in accordance with the Good Clinical Practice guidelines, US 21CFR Part 50-Protection of Human Subjects, and Part 56-Institutional Review Boards (IRBs)/Research Good Clinical Practice guidelines Institutional Review Boards (IRBs)/Research Ethics Boards (REBs).

On the free path length distribution for linear motion in an n-dimensional box

We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as the free path length coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we give an explicit formula (piecewise real analytic) for the probability density function in dimension two and three.In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as , and give an explicit (again piecewise real analytic) formula for its probability density function.Further, in both models we can recover the side lengths of the box from the location of the discontinuities of the probability density functions.

Estimation of forest leaf area index using terrestrial laser scanning data and path length distribution model in open-canopy forests

Terrestrial Laser Scanning (TLS) is an active technology that can acquire the finest characteristics of canopy structure and plays an increasing role in estimating Leaf Area Index (LAI) in forest canopies. However, 3D information is not directly used in conventional TLS-based methods using the gap fraction theory. In addition, quantifying clumping effect within canopies is still a difficult task. In this paper, we presented a method to reduce clumping effect and estimate LAI using TLS data. Our recently proposed path length distribution model was applied to TLS data. Instead of converting 3D points to 2D image, the path length distribution can be extracted using the TLS-recorded 3D data and the crown models built with the alpha shapes algorithm. Two simulated scenes and one actual forest plot were utilized for validation. The results of the proposed method agree well with both the true LAI (in the simulated scenes) and the extracted PAI by the digital hemispherical photography (in the actual plot). This LAI estimation method using TLS and the path length distribution model provides a novel way for ground-based LAI measurements and shows its great potential.

Gas Flow at the Ultra-nanoscale: Universal Predictive Model and Validation in Nanochannels of Ångstrom-Level Resolution.

Gas transport across nanoscale pores is determinant in molecular exchange in living organisms as well as in a broad spectrum of technologies. Here, we report an unprecedented theoretical and experimental analysis of gas transport in a consistent set of confining nanochannels ranging in size from the ultra-nanoscale to the sub-microscale. A generally applicable theoretical approach quantitatively predicting confined gas flow in the Knudsen and transition regime was developed. Unlike current theories, specifically designed for very simple channel geometries, our approach can be applied to virtually all geometries, for which the probability distribution of path lengths for particle-interface collisions can be computed, either analytically or by numerical simulations. To generate a much needed benchmark experimental model, we manufactured extremely reproducible membranes with two-dimensional nanochannels. Channel sizes ranged from 2.5 to 250 nm, and angstrom level of size control and interface tolerances were achieved using leading-edge nanofabrication techniques. We then measured gas flow in the Knudsen number range from 0.2 to 20. Excellent agreement between theoretical predictions and experimental data was found, demonstrating the validity and potential of our approach.

Path length distribution in two-dimensional causal sets

We study the distribution of maximal-chain lengths between two elements of a causal set and its relationship with the embeddability of the causal set in a region of flat spacetime. We start with causal sets obtained from uniformly distributed points in Minkowski space. After some general considerations we focus on the 2-dimensional case and derive a recursion relation for the expected number of maximal chains n_k as a function of their length k and the total number of points N between the maximal and minimal elements. By studying these theoretical distributions as well as ones generated from simulated sprinklings in Minkowski space we identify two features, the most probable path length or peak of the distribution k_0 and its width varDelta , which can be used both to provide a measure of the embeddability of the causal set as a uniform distribution of points in Minkowski space and to determine its dimensionality, if the causal set is manifoldlike in that sense. We end with a few simple examples of n_k distributions for non-manifoldlike causal sets.

Estimating the leaf area of an individual tree in urban areas using terrestrial laser scanner and path length distribution model

Urban leaf area measurement is crucial to properly determining the effect of urban trees on micro-climate regulation, heat island effect, building cooling, air quality improvement, and ozone formation. Previous works on the leaf area measurement have mainly focused on the stand level, although the presence of individual trees is more common than forests in urban areas. The only feasible ways for an operational non-destructive leaf area measurement, namely, optical indirect methods, are mostly limited in urban areas because light path is constantly intercepted by surrounding buildings or other objects. A terrestrial laser scanner (TLS), which can extract an individual tree by using its unique distance information, provides a possibility for indirectly measuring the leaf area index (LAI) in urban areas. However, indirect LAI measurement theory, which uses the cosine of an observation zenith angle for path-length correction, is incompatible for an individual tree because the representative projected area of LAI changes while the observation zenith angle changes, thus making the results incomparable and ambiguous. Therefore, we modified a path length distribution model for the leaf area measurement of an individual tree by replacing the traditional cosine path length correction for a continuous canopy with real path length distribution. We reconstructed the tree crown envelope from a TLS point cloud and calculated a real path length distribution through laser pulse-envelope intersections. Consequently, leaf area density was separated from the path length distribution model for leaf area calculation. Comparisons with reference measurement for an individual tree showed that the TLS-derived leaf area using the path length distribution is insensitive to the scanning resolution and agrees well with an allometric measurement with an overestimation from 5 m2 to 18 m2 (3–10%, respectively). Results from different stations are globally consistent, and using a weighted mean for different stations by sample numbers further improves the universality and efficiency of the proposed method. Further automation of the proposed method can facilitate a rapid and operational leaf area extraction of an individual tree for urban climate modeling.

Common network characteristics of four bus transport networks in Northeast China based on a perfect space P

The space P and space L representation models are often adopted together in the empirical analysis of bus transport networks (BTNs). In this paper, we develop a new representation model for BTNs using double edge weights and node weights, namely perfect space P. The model incorporates the number and lengths of directed routes between the two stations and the number of routes that connect to each station. Based on the model, we develop an empirical study on four large BTNs in Northeast China. Some common network characteristics of the BTNs are revealed and discussed in detail. The results show that several empirical distributions follow an exponential law. Moreover, the unweighted path length distributions can be fitted by a Gaussian function, while the weighted path length distributions can be fitted by a composite Gaussian function. Specifically, we introduce the weighted-degree distribution with different lengths and the special edge weight distribution, given the new evidence for the random behavior during the expansion of BTNs, including new stations and routes.

Distribution of shortest path lengths in subcritical Erdős-Rényi networks.

Networks that are fragmented into small disconnected components are prevalent in a large variety of systems. These include the secure communication networks of commercial enterprises, government agencies, and illicit organizations, as well as networks that suffered multiple failures, attacks, or epidemics. The structural and statistical properties of such networks resemble those of subcritical random networks, which consist of finite components, whose sizes are nonextensive. Surprisingly, such networks do not exhibit the small-world property that is typical in supercritical random networks, where the mean distance between pairs of nodes scales logarithmically with the network size. Unlike supercritical networks whose structure has been studied extensively, subcritical networks have attracted relatively little attention. A special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies. The overall statistics of the network can be obtained by a summation over all the components with suitable weights. We use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths (DSPL) on components of given sizes and topologies in subcritical Erdős-Rényi (ER) networks. From this expansion we obtain an exact analytical expression for the DSPL of the entire subcritical network, in the asymptotic limit. The DSPL, which accounts for all the pairs of nodes that reside on the same finite component (FC), is found to follow a geometric distribution of the form P_{FC}(L=ℓ|L<∞)=(1-c)c^{ℓ-1}, where c<1 is the mean degree. Using computer simulations we calculate the DSPL in subcritical ER networks of increasing sizes and confirm the convergence to this asymptotic result. We also obtain exact asymptotic results for the mean distance, 〈L〉_{FC}, and for the standard deviation of the DSPL, σ_{L,FC}, and show that the simulation results converge to these asymptotic results. Using the duality relations between subcritical and supercritical ER networks, we obtain the DSPL on the nongiant components of ER networks above the percolation transition.

Nonlocal mechanisms of attosecond interferometry and implications for condensed-phase experiments

Attosecond interferometry relies on the application of a train of attosecond extreme-ultraviolet (XUV) pulses and a synchronized low-intensity infrared (IR) field to measure the dynamics of photoemission via the interference signal in the spectrum of emitted electrons. So far, all condensed-phase experiments relying on this technique have investigated metallic, i.e. nontransparent, systems and have revealed the importance of transport and final-state effects. Here, we demonstrate the existence of a nonlocal mechanism of attosecond interferometry general to all condensed-matter systems and particularly important for those that are transparent to the IR. Key to the process is that, after XUV absorption, additional emission pathways result from the absorption or emission of an IR photon at remote positions due to the potential of neighboring atoms (or molecules). By solving the time-dependent Schr\odinger equation (TDSE) in one dimension, we show that interference of the resulting local and remote pathways leads to a mapping of the atomic or molecular environment onto the attosecond interference signal. We derive an analytical theory for the local and remote pathways as well as the resulting response and find excellent agreement with the TDSE. Our analytical theory shows that the nonlocal mechanism generally encodes both mean-free paths and scattering delays into the experimental observables. We generalize the analytical theory to the case of multiple collisions and study the effect of path-length distributions typical of electron scattering in condensed matter.

Using Airborne Laser Scanner and Path Length Distribution Model to Quantify Clumping Effect and Estimate Leaf Area Index

The airborne laser scanner (ALS) provides great potential for mapping the leaf area index (LAI) at the landscape scale using grid cell statistics, while its application is restricted by the lack of clumping information, which has been an unsolved issue highlighted for a long time. ALS generally provides an effective LAI because its footprint is too large to capture small gaps to apply traditional ground-based clumping correction methods. Here, we present a grid cell method based on path length distribution model to calculate the clumping-corrected LAI using ALS data without the requirement of additional field measurements. We separated the within- and between-crown areas to consider between-crown clumping, and used the path length distribution as estimated by local canopy height distribution to consider 3-D foliage profile and within-crown clumping. The path length distribution model takes advantage of the 3-D information rather than the gap size distribution, thus avoiding the limitation of large ALS footprint. With the 0.4-m-footprint ALS data, the results are generally promising and a multilevel clumping analysis is consistent with landscape flown. The ALS LAIs of different resolutions are consistent, with a difference of less than 5% from 5- to 250-m resolutions. Due to its consistency and simple configuration, the method provides an opportunity to map the clumping-corrected LAI operationally and strengthens the ability of airborne lidar to monitor vegetation change and validate the satellite product. This grid cell method based on path length distribution is worth further testing and application using more recent laser technology.

Revealing the microstructure of the giant component in random graph ensembles.

The microstructure of the giant component of the Erdős-Rényi network and other configuration model networks is analyzed using generating function methods. While configuration model networks are uncorrelated, the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree-degree correlations of all orders. We present exact analytical results for the degree distributions as well as higher-order degree-degree correlations on the giant components of configuration model networks. We show that the degree-degree correlations are essential for the integrity of the giant component, in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component. To demonstrate the importance and broad applicability of these results, we apply them to the study of the distribution of shortest path lengths on the giant component, percolation on the giant component, and spectra of sparse matrices defined on the giant component. We show that by using the degree distribution on the giant component one obtains high quality results for these properties, which can be further improved by taking the degree-degree correlations into account. This suggests that many existing methods, currently used for the analysis of the whole network, can be adapted in a straightforward fashion to yield results conditioned on the giant component.

Distance Entropy Cartography Characterises Centrality in Complex Networks.

We introduce distance entropy as a measure of homogeneity in the distribution of path lengths between a given node and its neighbours in a complex network. Distance entropy defines a new centrality measure whose properties are investigated for a variety of synthetic network models. By coupling distance entropy information with closeness centrality, we introduce a network cartography which allows one to reduce the degeneracy of ranking based on closeness alone. We apply this methodology to the empirical multiplex lexical network encoding the linguistic relationships known to English speaking toddlers. We show that the distance entropy cartography better predicts how children learn words compared to closeness centrality. Our results highlight the importance of distance entropy for gaining insights from distance patterns in complex networks.

Transmission versus reflectance spectroscopy for quantitation.

The objective of this work was to compare the accuracy of analyte concentration estimation when using transmission versus diffuse reflectance spectroscopy of a scattering medium. Monte Carlo ray tracing of light through the medium was used in conjunction with pure component absorption spectra and Beer-Lambert absorption along each ray's pathlength to generate matched sets of pseudoabsorbance spectra, containing water and six analytes present in skin. PLS regression models revealed an improvement in accuracy when using transmission compared to reflectance for a range of medium thicknesses and instrument noise levels. An analytical expression revealed the source of the accuracy degradation with reflectance was due both to the reduced collection efficiency for a fixed instrument etendue and to the broad pathlength distribution that detected light travels in the medium before exiting from the incident side.

Rational graph: a model for complex networks

Real life networks like social, biological and technical networks have amazing characteristics. These networks grow rapidly by the continuous addition of new nodes without following any organisation. Despite their huge sizes, these networks have a short path length, high clustering and power law degree distribution. Here, we investigate the topological characteristics of these types of networks. We also explore the main graph models that have been proposed to represent the dynamics and growth of these networks. This work provides an original contribution to build a complex network model. Motivated by the structural characteristics of complex networks and the deficiencies in the existing graph models, we are proposing the rational model, which is a model that imitates the characteristics of complex networks. The model was implemented and examined against other complex network models. The results showed that the rational model imitates the growth behaviour of the network, while maintaining the expected topological properties of the network.

Fluorescein Transport Assay to Assess Bulk Flow of Molecules Through the Hypocotyl in Arabidopsis thaliana.

The bulk transport of molecules through plant tissues underpins growth and development. The stem acts as a conduit between the upper and low domains of the plant, facilitating transport of solutes and water from the roots to the shoot system, and sugar plus other elaborated metabolites towards the non-photosynthetic organs. In order to perform this function efficiently, the stem needs to be optimized for transport. This is achieved through the formation of vasculature that connects the whole plant but also through connectivity signatures that reduce path length distributions outside the vascular system. This protocol was devised to characterize how cell connectivity affects the bulk flow of molecules traversing the stem. This is achieved by exposing young seedlings to fluorescein, for which no specific transporter is assumed to be present in A. thaliana, and assessing the relative concentration of this fluorescent compound in individual cells of the embryonic stem (hypocotyl) using confocal microscopy and quantitative 3D image analysis after a given exposure time.

Fractional diffusion limits of non-classical transport equations

We establish asymptotic diffusion limits of the non-classical transport equation derived in [E. W. Larsen, A generalized Boltzmann equation for non-classical particle transport, Joint international topical meeting on mathematics & computation and supercomputing in nuclear applications, 2007]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis uses the Fourier transform combined with a moment method. We conclude with remarks on the diffusion limit of the periodic Lorentz gas equation.

Diffusing-wave spectroscopy in a standard dynamic light scattering setup.

Diffusing-wave spectroscopy (DWS) extends dynamic light scattering measurements to samples with strong multiple scattering. DWS treats the transport of photons through turbid samples as a diffusion process, thereby making it possible to extract the dynamics of scatterers from measured correlation functions. The analysis of DWS data requires knowledge of the path length distribution of photons traveling through the sample. While for flat sample cells this path length distribution can be readily calculated and expressed in analytical form; no such expression is available for cylindrical sample cells. DWS measurements have therefore typically relied on dedicated setups that use flat sample cells. Here we show how DWS measurements, in particular DWS-based microrheology measurements, can be performed in standard dynamic light scattering setups that use cylindrical sample cells. To do so we perform simple random-walk simulations that yield numerical predictions of the path length distribution as a function of both the transport mean free path and the detection angle. This information is used in experiments to extract the mean-square displacement of tracer particles in the material, as well as the corresponding frequency-dependent viscoelastic response. An important advantage of our approach is that by performing measurements at different detection angles, the average path length through the sample can be varied. For measurements performed on a single sample cell, this gives access to a wider range of length and time scales than obtained in a conventional DWS setup. Such angle-dependent measurements also offer an important consistency check, as for all detection angles the DWS analysis should yield the same tracer dynamics, even though the respective path length distributions are very different. We validate our approach by performing measurements both on aqueous suspensions of tracer particles and on solidlike gelatin samples, for which we find our DWS-based microrheology data to be in good agreement with rheological measurements performed on the same samples.

Distribution of shortest cycle lengths in random networks.

We present analytical results for the distribution of shortest cycle lengths (DSCL) in random networks. The approach is based on the relation between the DSCL and the distribution of shortest path lengths (DSPL). We apply this approach to configuration model networks, for which analytical results for the DSPL were obtained before. We first calculate the fraction of nodes in the network which reside on at least one cycle. Conditioning on being on a cycle, we provide the DSCL over ensembles of configuration model networks with degree distributions which follow a Poisson distribution (Erdős-Rényi network), degenerate distribution (random regular graph), and a power-law distribution (scale-free network). The mean and variance of the DSCL are calculated. The analytical results are found to be in very good agreement with the results of computer simulations.

Distribution of shortest path lengths in a class of node duplication network models.

We present analytical results for the distribution of shortest path lengths (DSPL) in a network growth model which evolves by node duplication (ND). The model captures essential properties of the structure and growth dynamics of social networks, acquaintance networks, and scientific citation networks, where duplication mechanisms play a major role. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network, forming a link to the mother node, and with probability p to each one of its neighbors. The degree distribution of the resulting network turns out to follow a power-law distribution, thus the ND network is a scale-free network. To calculate the DSPL we derive a master equation for the time evolution of the probability P_{t}(L=ℓ), ℓ=1,2,⋯, where L is the distance between a pair of nodes and t is the time. Finding an exact analytical solution of the master equation, we obtain a closed form expression for P_{t}(L=ℓ). The mean distance 〈L〉_{t} and the diameter Δ_{t} are found to scale like lnt, namely, the ND network is a small-world network. The variance of the DSPL is also found to scale like lnt. Interestingly, the mean distance and the diameter exhibit properties of a small-world network, rather than the ultrasmall-world network behavior observed in other scale-free networks, in which 〈L〉_{t}∼lnlnt.

Modified gap fraction model of individual trees for estimating leaf area using terrestrial laser scanner

Terrestrial laser scanners (TLS) have demonstrated great potential in estimating structural attributes of forest canopy, such as leaf area index (LAI). However, the inversion accuracy of LAI is highly dependent on the measurement configuration of TLS and spatial characteristics of the scanned tree. Therefore, a modified gap fraction model integrating the path length distribution is developed to improve the accuracy of retrieved single-tree leaf area (LA) by considering the shape of a single-tree crown. The sensitivity of TLS measurement configurations on the accuracy of the retrieved LA is also discussed by using the modified gap fraction model based on several groups of simulated and field-measured point clouds. We conclude that (1) the modified gap fraction model has the potential to retrieve LA of an individual tree and (2) scanning distance has the enhanced impact on the accuracy of the retrieved LA than scanning step. A small scanning step for broadleaf trees reduces the scanning time, the storage volume, and postprocessing work in the condition of ensuring the accuracy of the retrieved LA. This work can benefit the design of an optimal survey configuration for the field campaign.