Biased Random Walk Research Articles

Replica Bounds by Combinatorial Interpolation for Diluted Spin Systems

In two papers Franz, Leone and Toninelli proved bounds for the free energy of diluted random constraints satisfaction problems, for a Poisson degree distribution [5] and a general distribution [6]. Panchenko and Talagrand [16] simplified the proof and generalized the result of [5] for the Poisson case. We provide a new proof for the general degree distribution case and as a corollary, we obtain new bounds for the size of the largest independent set (also known as hard core model) in a large random regular graph. Our proof uses a combinatorial interpolation based on biased random walks [21] and allows to bypass the arguments in [6] based on the study of the Sherrington-Kirkpatrick (SK) model.

Invariant measures, Hausdorff dimension and dimension drop of some harmonic measures on Galton-Watson trees

We consider infinite Galton-Watson trees without leaves together with i.i.d. random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed in [12], to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\lambda $-biased random walk on Galton-Watson trees, for which the invariant measure and the dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.

On the speed of once-reinforced biased random walk on trees

We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinforcement the ORbRW can be recurrent even when the underlying biased random walk is ballistic. We also prove that, on Galton-Watson trees without leaves, the speed is positive in the transient regime. Finally, we prove that, on regular trees, the speed of the ORbRW is monotone decreasing in the reinforcement parameter when the underlying random walk has high speed, and the reinforcement parameter is small.

Sensitivity of Mixing Times in Eulerian Digraphs

Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, where $m$ is the number of edges and $n$ is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes. We also study exploration and cover times for random walks on Eulerian digraphs and prove universal upper bounds in analogy to the undirected case.

Biased continuous-time random walks for ordinary and equilibrium cases: facilitation of diffusion, ergodicity breaking and ageing.

We examine renewal processes with power-law waiting time distributions (WTDs) and non-zero drift via computing analytically and by computer simulations their ensemble and time averaged spreading characteristics. All possible values of the scaling exponent α are considered for the WTD ψ(t) ∼ 1/t1+α. We treat continuous-time random walks (CTRWs) with 0 < α < 1 for which the mean waiting time diverges, and investigate the behaviour of the process for both ordinary and equilibrium CTRWs for 1 < α < 2 and α > 2. We demonstrate that in the presence of a drift CTRWs with α < 1 are ageing and non-ergodic in the sense of the non-equivalence of their ensemble and time averaged displacement characteristics in the limit of lag times much shorter than the trajectory length. In the sense of the equivalence of ensemble and time averages, CTRW processes with 1 < α < 2 are ergodic for the equilibrium and non-ergodic for the ordinary situation. Lastly, CTRW renewal processes with α > 2-both for the equilibrium and ordinary situation-are always ergodic. For the situations 1 < α < 2 and α > 2 the variance of the diffusion process, however, depends on the initial ensemble. For biased CTRWs with α > 1 we also investigate the behaviour of the ergodicity breaking parameter. In addition, we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements, in the entire range of the WTD exponent α.

Modeling of protein and salt redistribution during drying of a solution from a square cell

Background: Drying of biological fluids and saline biopolymer solutions is an actively researched topic, since the textures of the films remaining on the substrate after evaporation carry information about the state of the organism or biopolymer. Earlier, we have shown that the texture area and the amount of zigzag crystallization patterns depend on the structural state of the biopolymer. Models of flows and particle redistribution in a round sessile drop have been described by multiple authors, but in our experiments the solution completely fills a square cell to its walls, which results in different drying conditions and dynamics. Objectives: Conducting a numerical simulation of liquid evaporation and the corresponding redistribution of BSA and NaCl particles for a square 20×1×20 mm3 cell. Materials and methods: Liquid evaporation was simulated in the OpenFOAM continuous media modeling package using the interThermalPhaseChangeFoam module, and the redistribution of the particle concentrations (BSA, Na+ and Cl–) in the solution was simulated using the method of biased random walk on a discrete Cartesian lattice. Results: According to the obtained results, for most of the drying time, the flows in the liquid are from the corners to the center of the cell and from the diagonals to the walls of the cell, which leads to the accumulation of a significant fraction of the particles near the walls. When the liquid-air interface in the central part of the cell reaches the bottom, the surface tension forces quickly withdraw the solution to the cell walls, although a small amount of liquid can remain as a drop in the center. After complete drying, the majority of BSA and NaCl particles that did not settle at the cell walls were found to be concentrated at a distance of 1-3 mm from the cell walls, in the form of a rounded band. A small amount of BSA is also present in the central part of the cell, while the remainder of the salt is uniformly distributed over the entire area of the cell due to its greater diffusivity. Conclusions: Taking into account the experimental distributions of textures, these results support the hypotheses that textures are not formed in the absence of a biopolymer, and that zigzag patterns are formed in locations with high drying rate.

An improved cuckoo search algorithm and its application in vibration fault diagnosis for a hydroelectric generating unit

ABSTRACTThis article presents an improved cuckoo search (ICS) algorithm based on the parametric adaptation mechanism and non-uniform mutation. In the ICS algorithm, the biased random walk operator is modified to take advantage of the neighbourhood information of the current solution, and then the control parameters of step size, discovery probability and scaling factor are directly integrated into the optimized problems. Meanwhile, the non-uniform mutation operation is used to adaptively tune the search step of the current optimal solution. To evaluate the feasibility and efficiency of the presented algorithm, three groups of benchmark test functions are employed to perform the validation analysis. Subsequently, to diagnose the vibration fault of a hydroelectric generating unit, a combinational model is built, which combines ICS with a back-propagation neural network (ICSBP). The experimental results indicate that ICS is competitive on the optimization problems, and the ICSBP approach can effectively improve the accuracy of vibration fault diagnosis for a hydroelectric generating unit.

Organs on chip approach: a tool to evaluate cancer -immune cells interactions

In this paper we discuss the applicability of numerical descriptors and statistical physics concepts to characterize complex biological systems observed at microscopic level through organ on chip approach. To this end, we employ data collected on a microfluidic platform in which leukocytes can move through suitably built channels toward their target. Leukocyte behavior is recorded by standard time lapse imaging. In particular, we analyze three groups of human peripheral blood mononuclear cells (PBMC): heterozygous mutants (in which only one copy of the FPR1 gene is normal), homozygous mutants (in which both alleles encoding FPR1 are loss-of-function variants) and cells from ‘wild type’ donors (with normal expression of FPR1). We characterize the migration of these cells providing a quantitative confirmation of the essential role of FPR1 in cancer chemotherapy response. Indeed wild type PBMC perform biased random walks toward chemotherapy-treated cancer cells establishing persistent interactions with them. Conversely, heterozygous mutants present a weaker bias in their motion and homozygous mutants perform rather uncorrelated random walks, both failing to engage with their targets. We next focus on wild type cells and study the interactions of leukocytes with cancerous cells developing a novel heuristic procedure, inspired by Lyapunov stability in dynamical systems.

Multitarget search on complex networks: A logarithmic growth of global mean random cover time.

We investigate multitarget search on complex networks and derive an exact expression for the mean random cover time that quantifies the expected time a walker needs to visit multiple targets. Based on this, we recover and extend some interesting results of multitarget search on networks. Specifically, we observe the logarithmic increase of the global mean random cover time with the target number for a broad range of random search processes, including generic random walks, biased random walks, and maximal entropy random walks. We show that the logarithmic growth pattern is a universal feature of multi-target search on networks by using the annealed network approach and the Sherman-Morrison formula. Moreover, we find that for biased random walks, the global mean random cover time can be minimized, and that the corresponding optimal parameter also minimizes the global mean first passage time, pointing towards its robustness. Our findings further confirm that the logarithmic growth pattern is a universal law governing multitarget search in confined media.

Exploring activity-driven network with biased walks

We investigate the concurrent dynamics of biased random walks and the activity-driven network, where the preferential transition probability is in terms of the edge-weighting parameter. We also obtain the analytical expressions for stationary distribution and the coverage function in directed and undirected networks, all of which depend on the weight parameter. Appropriately adjusting this parameter, more effective search strategy can be obtained when compared with the unbiased random walk, whether in directed or undirected networks. Since network weights play a significant role in the diffusion process.

A time-varying biased random walk approach to human growth

Growth and development are dominated by gene-environment interactions. Many approaches have been proposed to model growth, but most are either descriptive or describe population level phenomena. We present a random walk-based growth model capable of predicting individual height, in which the growth increments are taken from time varying distributions mimicking the bursting behaviour of observed saltatory growth. We derive analytic equations and also develop a computational model of such growth that takes into account gene-environment interactions. Using an independent prospective birth cohort study of 190 infants, we predict height at 6 years of age. In a subset of 27 subjects, we adaptively train the model to account for growth between birth and 1 year of age using a Bayesian approach. The 5-year predicted heights compare well with actual data (measured height = 0.838*predicted height + 18.3; R2 = 0.51) with an average error of 3.3%. In one patient, we also exemplify how our growth prediction model can be used for the early detection of growth deficiency and the evaluation of the effectiveness of growth hormone therapy.

Harmonic measure for biased random walk in a supercritical Galton–Watson tree

We consider random walks $\lambda $-biased towards the root on a Galton–Watson tree, whose offspring distribution $(p_{k})_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<\lambda <m$, the loop-erased trajectory of the biased random walk defines the $\lambda $-harmonic ray, whose law is the $\lambda $-harmonic measure on the boundary of the Galton–Watson tree. We answer a question of Lyons, Pemantle and Peres (In Classical and Modern Branching Processes (Minneapolis, MN, 1994) (1997) 223–237 Springer) by showing that the $\lambda $-harmonic measure has a.s. strictly larger Hausdorff dimension than the visibility measure, which is the harmonic measure corresponding to the simple forward random walk. We also prove that the average number of children of the vertices along the $\lambda $-harmonic ray is a.s. bounded below by $m$ and bounded above by $m^{-1}\sum k^{2}p_{k}$. Moreover, at least for $0<\lambda \leq 1$, the average number of children of the vertices along the $\lambda $-harmonic ray is a.s. strictly larger than that of the $\lambda $-biased random walk trajectory. We observe that the latter is not monotone in the bias parameter $\lambda $.

Measuring site fidelity and spatial segregation within animal societies.

Summary Animals often display a marked tendency to return to previously visited locations that contain important resources, such as water, food, or developing brood that must be provisioned. A considerable body of work has demonstrated that this tendency is strongly expressed in ants, which exhibit fidelity to particular sites both inside and outside the nest. However, thus far many studies of this phenomena have taken the approach of reducing an animal's trajectory to a summary statistic, such as the area it covers.Using both simulations of biased random walks, and empirical trajectories from individual rock ants, Temnothorax albipennis, we demonstrate that this reductive approach suffers from an unacceptably high rate of false negatives.To overcome this, we describe a site‐centric approach which, in combination with a spatially‐explicit null model, allows the identification of the important sites towards which individuals exhibit statistically significant biases.Using the ant trajectories, we illustrate how the site‐centric approach can be combined with social network analysis tools to detect groups of individuals whose members display similar space‐use patterns.We also address the mechanistic origin of individual site fidelity; by examining the sequence of visits to each site, we detect a statistical signature associated with a self‐attracting walk – a non‐Markovian movement model that has been suggested as a possible mechanism for generating individual site fidelity.

Escape regimes of biased random walks on Galton\u2013Watson trees

We study biased random walk on subcritical and supercritical Galton–Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballistic phase.

Unsupervised ensemble ranking of terms in electronic health record notes based on their importance to patients

BackgroundAllowing patients to access their own electronic health record (EHR) notes through online patient portals has the potential to improve patient-centered care. However, EHR notes contain abundant medical jargon that can be difficult for patients to comprehend. One way to help patients is to reduce information overload and help them focus on medical terms that matter most to them. Targeted education can then be developed to improve patient EHR comprehension and the quality of care. ObjectiveThe aim of this work was to develop FIT (Finding Important Terms for patients), an unsupervised natural language processing (NLP) system that ranks medical terms in EHR notes based on their importance to patients. MethodsWe built FIT on a new unsupervised ensemble ranking model derived from the biased random walk algorithm to combine heterogeneous information resources for ranking candidate terms from each EHR note. Specifically, FIT integrates four single views (rankers) for term importance: patient use of medical concepts, document-level term salience, word co-occurrence based term relatedness, and topic coherence. It also incorporates partial information of term importance as conveyed by terms’ unfamiliarity levels and semantic types. We evaluated FIT on 90 expert-annotated EHR notes and used the four single-view rankers as baselines. In addition, we implemented three benchmark unsupervised ensemble ranking methods as strong baselines. ResultsFIT achieved 0.885 AUC-ROC for ranking candidate terms from EHR notes to identify important terms. When including term identification, the performance of FIT for identifying important terms from EHR notes was 0.813 AUC-ROC. Both performance scores significantly exceeded the corresponding scores from the four single rankers (P<0.001). FIT also outperformed the three ensemble rankers for most metrics. Its performance is relatively insensitive to its parameter. ConclusionsFIT can automatically identify EHR terms important to patients. It may help develop future interventions to improve quality of care. By using unsupervised learning as well as a robust and flexible framework for information fusion, FIT can be readily applied to other domains and applications.

A time­varying biased random walk model of growth: Application to height from birth to childhood

A time­varying biased random walk model of growth: Application to height from birth to childhood

Active Cargo Positioning from Actin-Polarity Sensing by Small Myosin Assemblies

Molecular motors navigate the cytoskeleton to position vesicles and organelles at specific locations in the cell. Cytoskeletal filaments assemble into parallel, antiparallel or disordered networks, providing a complex environment that constrains active transport properties. Using surface micro-patterns of nucleation-promoting factors to control the geometry of actin polymerization, we studied in vitro the interplay between the actin-network architecture and cargo transport by small myosin assemblies. With two parallel nucleation lines, we produced an antiparallel network of overlapping filaments. We found that 200-nm beads coated with processive myosin-5 motors displayed directed movements towards the midline of the pattern, where the net polarity of the actin network was null, and accumulated there. The bead distribution was dictated by the spatial profiles of bead velocity and diffusion coefficient, indicating that a diffusion-drift process was at work. Interestingly, beads coated with skeletal heavy mero-myosin-2 motors showed a similar behavior. However, although velocity gradients were sharper with myosin 2, the much larger bead diffusion observed with this non-processive motor resulted in less precise positioning. Strikingly, bead positioning did not depend on the spacing between the nucleation lines. Our observations are well described by a three-state model of bead transport, in which active beads locally sense the net polarity of the filament network by frequently detaching from and reattaching to the filaments. A stochastic sequence of processive runs and diffusive searches results in a biased random walk with an effective drift velocity and diffusion coefficient. Positioning relies on spatial gradients of the net actin polarity, as well as on the run length of the cargo in the attached state. Altogether, our results on a minimal acto-myosin system demonstrate the key role played by the actin-network architecture on motor transport.

Census: fast, scalable and robust data aggregation in MANETs

This paper describes Census, a protocol for data aggregation and statistical counting in MANETs. Census operates by circulating a set of tokens in the network using biased random walks such that each node is visited by at least one token. The protocol is structure-free so as to avoid high messaging overhead for maintaining structure in the presence of node mobility. It biases the random walks of tokens so as to achieve fast cover time; the bias involves short albeit multi-hop gradients that guide the tokens towards hitherto unvisited nodes. Census thus achieves a cover time of O(N) and message overhead of $$O(N\,log(N))$$ where N is the number of nodes. Notably, it enjoys scalability and robustness, which we demonstrate via simulations in networks ranging from 100 to 4000 nodes under different network densities and mobility models. We also observe a speedup by a factor of k when k different tokens are used ( $$1 \le k \le \sqrt{N}$$ ).

Multiple-predators-based capture process on complex networks**Project supported by the National Natural Science Foundation of China (Grant No. 61304154), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20133219120032), the Postdoctoral Science Foundation of China (Grant No. 2013M541673), and China Postdoctoral Science Special Foundation (Grant No. 2015T80556).

The predator/prey (capture) problem is a prototype of many network-related applications. We study the capture process on complex networks by considering multiple predators from multiple sources. In our model, some lions start from multiple sources simultaneously to capture the lamb by biased random walks, which are controlled with a free parameter α. We derive the distribution of the lamb’s lifetime and the expected lifetime 〈T〉. Through simulation, we find that the expected lifetime drops substantially with the increasing number of lions. Moreover, we study how the underlying topological structure affects the capture process, and obtain that locating on small-degree nodes is better than on large-degree nodes to prolong the lifetime of the lamb. The dense or homogeneous network structures are against the survival of the lamb. We also discuss how to improve the capture efficiency in our model.

Bayesian microsaccade detection.

Microsaccades are high-velocity fixational eye movements, with special roles in perception and cognition. The default microsaccade detection method is to determine when the smoothed eye velocity exceeds a threshold. We have developed a new method, Bayesian microsaccade detection (BMD), which performs inference based on a simple statistical model of eye positions. In this model, a hidden state variable changes between drift and microsaccade states at random times. The eye position is a biased random walk with different velocity distributions for each state. BMD generates samples from the posterior probability distribution over the eye state time series given the eye position time series. Applied to simulated data, BMD recovers the “true” microsaccades with fewer errors than alternative algorithms, especially at high noise. Applied to EyeLink eye tracker data, BMD detects almost all the microsaccades detected by the default method, but also apparent microsaccades embedded in high noise—although these can also be interpreted as false positives. Next we apply the algorithms to data collected with a Dual Purkinje Image eye tracker, whose higher precision justifies defining the inferred microsaccades as ground truth. When we add artificial measurement noise, the inferences of all algorithms degrade; however, at noise levels comparable to EyeLink data, BMD recovers the “true” microsaccades with 54% fewer errors than the default algorithm. Though unsuitable for online detection, BMD has other advantages: It returns probabilities rather than binary judgments, and it can be straightforwardly adapted as the generative model is refined. We make our algorithm available as a software package.