Alzheimer random walk model: Two previously overlooked diffusion regimes

A non-Markovian one-dimensional random walk model is studied with emphasis on the phase-diagram, showing all the diffusion regimes, along with the exactly determined critical lines. The model, known as the Alzheimer walk, is endowed with memory-controlled diffusion, responsible for the model's long-range correlations, and is characterized by a rich variety of diffusive regimes. The importance of this model is that superdiffusion arises due not to memory per se, but rather also due to loss of memory. The recently reported numerically and analytically estimated values for the Hurst exponent are hereby reviewed. We report the finding of two, previously overlooked, phases, namely, evanescent log-periodic diffusion and log-periodic diffusion with escape, both with Hurst exponent H = 1/2. In the former, the log-periodicity gets damped, whereas in the latter the first moment diverges. These phases further enrich the already intricate phase diagram. The results are discussed in the context of phase transitions, aging phenomena, and symmetry breaking.

Superdiffusion in a non-Markovian random walk model with a Gaussian memory profile

Most superdiffusive Non-Markovian random walk models assume that correlations are maintained at all time scales, e. g., fractional Brownian motion, Levy walks, the Elephant walk and Alzheimer walk models. In the latter two models the random walker can always "remember" the initial times near t = 0. Assuming jump size distributions with finite variance, the question naturally arises: is superdiffusion possible if the walker is unable to recall the initial times? We give a conclusive answer to this general question, by studying a non-Markovian model in which the walker's memory of the past is weighted by a Gaussian centered at time t/2, at which time the walker had one half the present age, and with a standard deviation sigma t which grows linearly as the walker ages. For large widths we find that the model behaves similarly to the Elephant model, but for small widths this Gaussian memory profile model behaves like the Alzheimer walk model. We also report that the phenomenon of amnestically induced persistence, known to occur in the Alzheimer walk model, arises in the Gaussian memory profile model. We conclude that memory of the initial times is not a necessary condition for generating (log-periodic) superdiffusion. We show that the phenomenon of amnestically induced persistence extends to the case of a Gaussian memory profile.

Shape anisotropy of a single random-walk polymer

Random walks have been used to describe a wide variety of systems ranging from cell colonies to polymers. Sixty-five years ago, Kuhn [Kuhn, W. (1934) Kolloid-Z. 68, 2–11] made the prediction, backed later by computer simulations, that the overall shape of a random-walk polymer is aspherical, yet no experimental work has directly tested Kuhn's general idea and subsequent computer simulations. By using fluorescence microscopy, we monitored the conformation of individual, long, random-walk polymers (fluorescently labeled DNA molecules) at equilibrium. We found that a polymer most frequently adopts highly extended, nonfractal structures with a strongly anisotropic shape. The ensemble-average ratio of the lengths of the long and short axes of the best-fit ellipse of the polymer was much larger than unity.

A new betweenness centrality measure based on an algorithm for ranking the nodes of a network

We propose and discuss a new centrality index for urban street patterns represented as networks in geographical space. This centrality measure, that we call ranking-betweenness centrality, combines the idea behind the random-walk betweenness centrality measure and the idea of ranking the nodes of a network produced by an adapted PageRank algorithm. We initially use a PageRank algorithm in which we are able to transform some information of the network that we want to analyze into numerical values. Numerical values summarizing the information are associated to each of the nodes by means of a data matrix. After running the adapted PageRank algorithm, a ranking of the nodes is obtained, according to their importance in the network. This classification is the starting point for applying an algorithm based on the random-walk betweenness centrality. A detailed example of a real urban street network is discussed in order to understand the process to evaluate the ranking-betweenness centrality proposed, performing some comparisons with other classical centrality measures.

The emergence of order in random walk resource discovery protocols

While others have attempted to determine, by way of mathematical formulae, optimal resource duplication strategies for random walk protocols, this paper is concerned with studying the emergent effects of dynamic resource propagation and replication. In particular, we show, via modelling and experimentation, that under any given decay (purge) rate the number of nodes that have knowledge of particular resource converges to a fixed point or a limit cycle. We also show that even for high rates of decay - that is, when few nodes have knowledge of a particular resource - the number of hops required to find that resource is small.

Molecular phase space transport in water:non-stationary random walk model

Molecular transport in phase space is crucial for chemical reactions because it defines how pre-reactive molecular configurations are found during the time evolution of the system. Using Molecular Dynamics (MD) simulated atomistic trajectories we test the assumption of the normal diffusion in the phase space for bulk water at ambient conditions by checking the equivalence of the transport to the random walk model. Contrary to common expectations we have found that some statistical features of the transport in the phase space differ from those of the normal diffusion models. This implies a non-random character of the path search process by the reacting complexes in water solutions. Our further numerical experiments show that a significant long period of non-stationarity in the transition probabilities of the segments of molecular trajectories can account for the observed non-uniform filling of the phase space. Surprisingly, the characteristic periods in the model non-stationarity constitute hundreds of nanoseconds, that is much longer time scales compared to typical lifetime of known liquid water molecular structures (several picoseconds).

On Multi-Dimensional Random Walk Models Approximating Symmetric Space-Fractional Diffusion Processes

Mathematics Subject Classification: 26A33, 47B06, 47G30, 60G50, 60G52, 60G60.

Monte Carlo Random Walk Simulations Based on Distributed Order Differential Equations with Applications to Cell Biology

Mathematics Subject Classification: 65C05, 60G50, 39A10, 92C37

A Random Walk down Main Street

US suburbs have often been characterized by their relatively low walk accessibility compared to more urban environments, and US urban environments have been characterized by low walk accessibility compared to cities in other countries. Lower overall density in the suburbs implies that activities, if spread out, would have a greater distance between them. But why should activities be spread out instead of developed contiguously? This brief research note builds a positive model for the emergence of contiguous development along “Main Street” to illustrate the trade-offs that result in the built environment we observe. It then suggests some policy interventions to place a “thumb on the scale” to choose which parcels will develop in which sequence to achieve socially preferred outcomes.

Testing the random walk hypothesis with variance ratio statistics

Mestrado em Finanças

Random walk with restart over dynamic graphs

Random Walk with Restart (RWR) is an appealing measure of proximity between nodes based on graph structures. Since real graphs are often large and subject to minor changes, it is prohibitively expensive to recompute proximities from scratch. Previous methods use LU decomposition and degree reordering heuristics, entailing O(|V|^3) time and O(|V|^2) memory to compute all (|V|^2) pairs of node proximities in a static graph. In this paper, a dynamic scheme to assess RWR proximities is proposed: (1) For unit update, we characterize the changes to all-pairs proximities as the outer product of two vectors. We notice that the multiplication of an RWR matrix and its transition matrix, unlike traditional matrix multiplications, is commutative. This can greatly reduce the computation of all-pairs proximities from O(|V|^3) to O(|delta|) time for each update without loss of accuracy, where |delta| (<<|V|^2) is the number of affected proximities. (2) To avoid O(|V|^2) memory for all pairs of outputs, we also devise efficient partitioning techniques for our dynamic model, which can compute all pairs of proximities segment-wisely within O(l|V|) memory and O(|V|/l) I/O costs, where 1<=l<=|V| is a user-controlled trade-off between memory and I/O costs. (3) For bulk updates, we also devise aggregation and hashing methods, which can discard many unnecessary updates further and handle chunks of unit updates simultaneously. Our experimental results on various datasets demonstrate that our methods can be 1–2 orders of magnitude faster than other competitors while securing scalability and exactness.

A linear programming approach for estimating the structure of a sparse linear genetic network from transcript profiling data

Background: A genetic network can be represented as a directed graph in which a node corresponds to a gene and a directed edge specifies the direction of influence of one gene on another. The reconstruction of such networks from transcript profiling data remains an important yet challenging endeavor. A transcript profile specifies the abundances of many genes in a biological sample of interest. Prevailing strategies for learning the structure of a genetic network from high-dimensional transcript profiling data assume sparsity and linearity. Many methods consider relatively small directed graphs, inferring graphs with up to a few hundred nodes. This work examines large undirected graphs representations of genetic networks, graphs with many thousands of nodes where an undirected edge between two nodes does not indicate the direction of influence, and the problem of estimating the structure of such a sparse linear genetic network (SLGN) from transcript profiling data. Results: The structure learning task is cast as a sparse linear regression problem which is then posed as a LASSO (l1-constrained fitting) problem and solved finally by formulating a Linear Program (LP). A bound on the Generalization Error of this approach is given in terms of the Leave-One-Out Error. The accuracy and utility of LP-SLGNs is assessed quantitatively and qualitatively using simulated and real data. The Dialogue for Reverse Engineering Assessments and Methods (DREAM) initiative provides gold standard data sets and evaluation metrics that enable and facilitate the comparison of algorithms for deducing the structure of networks. The structures of LP-SLGNs estimated from the INSILICO1, INSILICO2 and INSILICO3 simulated DREAM2 data sets are comparable to those proposed by the first and/or second ranked teams in the DREAM2 competition. The structures of LP-SLGNs estimated from two published Saccharomyces cerevisae cell cycle transcript profiling data sets capture known regulatory associations. In each S. cerevisiae LP-SLGN, the number of nodes with a particular degree follows an approximate power law suggesting that its degree distributions is similar to that observed in real-world networks. Inspection of these LP-SLGNs suggests biological hypotheses amenable to experimental verification. Conclusion: A statistically robust and computationally efficient LP-based method for estimating the topology of a large sparse undirected graph from high-dimensional data yields representations of genetic networks that are biologically plausible and useful abstractions of the structures of real genetic networks. Analysis of the statistical and topological properties of learned LP-SLGNs may have practical value; for example, genes with high random walk betweenness, a measure of the centrality of a node in a graph, are good candidates for intervention studies and hence integrated computational – experimental investigations designed to infer more realistic and sophisticated probabilistic directed graphical model representations of genetic networks. The LP-based solutions of the sparse linear regression problem described here may provide a method for learning the structure of transcription factor networks from transcript profiling and transcription factor binding motif data.

Distribution of the angle of rotation plane random walks

We study the probability distribution of the angle by which the tangent to the trajectory rotates in the course of a plane random walk. It is shown that the determination of this distribution function can be reduced to an integral equation, which can be rigorously transformed into a differential equation of Hill's type. We derive the asymptotic distribution for very long walks.