Multiple Walkers Research Articles

Increasing the Dimensionality of Quantum Walks Using Multiple Walkers

We show that with the addition of multiple walkers, quantum walks on a line can be transformed into lattice graphs of higher dimension. Thus, multi-walker walks can simulate single-walker walks on higher dimensional graphs and vice versa. This exponential complexity opens up new applications for present-day quantum walk experiments. We discuss the applications of such higher-dimensional structures and how they relate to linear optics quantum computing. In particular we show that multi-walker quantum walks are equivalent to the BosonSampling model for linear optics quantum computation proposed by Aaronson & Arkhipov. With the addition of control over phase-defects in the lattice, which can be simulated with entangling gates, asymmetric lattice structures can be constructed which are universal for quantum computation.

Perceiving group behavior: Sensitive ensemble coding mechanisms for biological motion of human crowds.

Many species, including humans, display group behavior. Thus, perceiving crowds may be important for social interaction and survival. Here, we provide the first evidence that humans use ensemble-coding mechanisms to perceive the behavior of a crowd of people with surprisingly high sensitivity. Observers estimated the headings of briefly presented crowds of point-light walkers that differed in the number and headings of their members (i.e., people in differently sized crowds had identical or increasingly variable directions of walking). We found that observers rapidly pooled information from multiple walkers to estimate the heading of a crowd. This ensemble code was precise; observer's perceived the behavior of a crowd better than the behavior of an individual. We also showed that this pooling provided tolerance against crowd variability and may cause a chaotic group to cohere into a unified Gestalt. Sensitive perception of a crowd's behavior required integration of human form and motion, suggesting that the ensemble code was generated in high-level visual areas. Overall, these mechanisms may reflect the prevalence of crowd behavior in nature and a social benefit for perceiving crowds as unified entities.

Cooperative linear cargo transport with molecular spiders

Molecular spiders are nanoscale walkers made with DNA enzyme legs attached to a common body. They move over a surface of DNA substrates, cleaving them and leaving behind product DNA strands, which they are able to revisit. Simple one-dimensional models of spider motion show significant superdiffusive motion when the leg-substrate bindings are longer-lived than the leg-product bindings. This gives the spiders potential as a faster-than-diffusion transport mechanism. However, analysis shows that single-spider motion eventually decays into an ordinary diffusive motion, owing to the ever increasing size of the region of cleaved products. Inspired by cooperative behavior of natural molecular walkers, we propose a symmetric exclusion process model for multiple walkers interacting as they move over a one-dimensional lattice. We show that when walkers are sequentially released from the origin, the collective effect is to prevent the leading walkers from moving too far backwards. Hence, there is an effective outward pressure on the leading walkers that keeps them moving superdiffusively for longer times, despite the growth of the product region. Multi-spider systems move faster and farther than single spiders or systems with multiple simple random walkers.

Assessing the Performance of Metadynamics and Path Variables in Predicting the Binding Free Energies of p38 Inhibitors.

The accurate yet efficient evaluation of the free energy profiles of ligand-target association is a long sought goal in rational drug design. Methods that calculate the free energy along realistic association pathways, such as metadynamics, have been shown to provide reliable profiles, while accounting properly for solvation and target flexibility. However, these approaches usually require prohibitive computational resources and expert human intervention. Here, we show how multiple walkers metadynamics, when performed with optimal path collective variables (PCV), provides in a predetermined amount of computer time an accurate set of free energy profiles for a series of p38 inhibitors. The chosen test set, spanning a wide range of activity, is a challenging benchmark, both for computational methods and for human intuition, as the correct order for the binding affinity cannot be easily guessed. An excellent ranking of the ligands was obtained with minimal human assistance, an important step toward a fully automated pharmaceutical work-flow.

RankCompete: Simultaneous ranking and clustering of information networks

Random walk was first introduced by Karl Pearson in 1905 and has inspired many research works in different fields. In recent years, random walk has been adopted in information network research, for example, ranking and similarity estimation. In this paper, we introduce a new model called RankCompete, which allows multiple random walkers in the same network (existing work mostly focus on random walks of a single category). By introducing the “competition” concept into the random walk framework, our method can fulfill both clustering and ranking tasks simultaneously and thus provide an effective analysis tool for networks. Compared with the traditional network ranking approaches, our new method focuses more on the structure of specialized clusters. Compared with the traditional graph clustering approaches, our new method provides a faster and more intuitive way to group the network nodes. We validate our approach on both bibliography networks and visual information networks, and the results show that our approach can obtain 100% perfect clustering results in clustering the DBLP 20 conferences and outperform the state-of-the-art on the Cora dataset. Furthermore, we show that our method can be effectively used to summarize personal photo collections.

Interarrival Times of Message Propagation on Directed Networks

One of the challenges in fighting cybercrime is to understand the dynamics of message propagation on botnets, networks of infected computers used to send viruses, unsolicited commercial emails (SPAM) or denial of service attacks. We map this problem to the propagation of multiple random walkers on directed networks and we evaluate the inter-arrival time distribution between successive walkers arriving at a target. We show that the temporal organization of this process, which models information propagation on unstructured peer to peer networks, has the same features as SPAM reaching a single user. We study the behavior of the message inter-arrival time distribution on three different network topologies using two different rules for sending messages. In all networks the propagation is not a pure Poisson process. It shows universal features on Poissonian networks and a more complex behavior on scale free networks. Results open the possibility to indirectly learn about the process of sending messages on networks with unknown topologies, by studying inter-arrival times at any node of the network.

Vection can be induced without explicit motion signal using backscroll illusion

We demonstrated that vection is induced by a motion stimuli that does not have an explicit, bottom-up motion component. The stimulus motion used in this experiment was animation movie clips of walking people, with no positional changes within the stimulus field. There were no low-level motion signals in the direction of gait. The results indicate that strong vection was observed under optimal stimuli conditions, that is, large visual field and multiple walkers. These results suggest that vection can be elicited solely by motion signals extracted at relatively higher levels within the visual system. This is the first report that a pure high-level motion related to “implied motion” induces vection.

Interarrival times of message propagation on directed networks

One of the challenges in fighting cybercrime is to understand the dynamics of message propagation on botnets, networks of infected computers used to send viruses, unsolicited commercial emails (SPAM) or denial of service attacks. We map this problem to the propagation of multiple random walkers on directed networks and we evaluate the interarrival time distribution between successive walkers arriving at a target. We show that the temporal organization of this process, which models information propagation on unstructured peer to peer networks, has the same features as SPAM reaching a single user. We study the behavior of the message interarrival time distribution on three different network topologies using two different rules for sending messages. In all networks the propagation is not a pure Poisson process. It shows universal features on Poissonian networks and a more complex behavior on scale free networks. Results open the possibility to indirectly learn about the process of sending messages on networks with unknown topologies, by studying interarrival times at any node of the network.

Free-energy landscapes from adaptively biased methods: Application to quantum systems

Several parallel adaptive biasing methods are applied to the calculation of free-energy pathways along reaction coordinates, choosing as a difficult example the double-funnel landscape of the 38-atom Lennard-Jones cluster. In the case of classical statistics, the Wang-Landau and adaptively biased molecular-dynamics (ABMD) methods are both found efficient if multiple walkers and replication and deletion schemes are used. An extension of the ABMD technique to quantum systems, implemented through the path-integral MD framework, is presented and tested on Ne38 against the quantum superposition method.

Adaptively biased molecular dynamics: An umbrella sampling method with a time‐dependent potential

AbstractWe discuss an adaptively biased molecular dynamics (ABMD) method for the computation of a free energy surface for a set of reaction coordinates. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential. It is characterized by a small number of control parameters and an O(t) numerical cost with simulation time t. The method naturally allows for extensions based on multiple walkers and replica exchange mechanism. The workings of the method are illustrated with a number of examples, including sugar puckering, and free energy landscapes for polymethionine and polyproline peptides, and for a short β‐turn peptide. ABMD has been implemented into the latest version (Case et al., AMBER 10; University of California: San Francisco, 2008) of the AMBER software package and is freely available to the simulation community. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Network formation determined by the diffusion process of random walkers

We studied the diffusion process of random walkers in networks formed by their traces. This model considers the rise and fall of links determined by the frequency of transports of random walkers. In order to examine the relation between the formed network and the diffusion process, a situation in which multiple random walkers start from the same vertex is investigated. The difference in diffusion rate of random walkers according to the difference in dimension of the initial lattice is very important for determining the time evolution of the networks. For example, complete subgraphs can be formed on a one-dimensional lattice while a graph with a power-law vertex degree distribution is formed on a two-dimensional lattice. We derived some formulae for predicting network changes for the 1D case, such as the time evolution of the size of nearly complete subgraphs and conditions for their collapse. The networks formed on the 2D lattice are characterized by the existence of clusters of highly connected vertices and their life time. As the life time of such clusters tends to be small, the exponent of the power-law distribution changes from γ ≃ 1–2 to γ ≃ 3.

Adaptively biased molecular dynamics for free energy calculations

We present an adaptively biased molecular dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters and an O(t) numerical cost with molecular dynamics time t. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.

Statistics of cellular signal transduction as a race to the nucleus by multiple random walkers in compartment/phosphorylation space.

Cellular signal transduction often involves a reaction network of phosphorylation and transport events arranged with a ladder topology. If we keep track of the location of the phosphate groups describing an abstract state space, a simple model of signal transduction involving enzymes can be mapped on to a problem of how multiple biased random walkers compete to reach their target in the nucleus yielding a signal. Here, the first passage time probability and the survival probability for multiple walkers can be used to characterize the response of the network. The statistics of the first passage through the network has an asymmetric distribution with a long tail arising from the hierarchical structure of the network. This distribution implies a significant difference between the mean and the most probable signal transduction time. The response patterns for various external inputs generated by our model agree with recent experiments. In addition, the model predicts that there is an optimal phosphorylation enzyme concentration for rapid signal transduction.

Dynamics of congestion transition triggered by multiple walkers on complex networks

The congestion transition triggered by multiple walkers walking along the shortest path on complex networks is numerically investigated. These networks are composed of nodes that have a finite capacity in analogy to the buffer memory of a computer. It is found that a transition from free-flow phase to congestion phase occurs at a critical walker density fc, which varies for complex networks with different topological structures. The dynamic pictures of congestion for networks with different topological structures show that congestion on scale-free networks is a percolation process of congestion clusters, while the dynamics of congestion transition on non-scale-free networks is mainly a process of nucleation.

Efficient Reconstruction of Complex Free Energy Landscapes by Multiple Walkers Metadynamics

Recently, we have introduced a new method, metadynamics, which is able to sample rarely occurring transitions and to reconstruct the free energy as a function of several variables with a controlled accuracy. This method has been successfully applied in many different fields, ranging from chemistry to biophysics and ligand docking and from material science to crystal structure prediction. We present an important development that speeds up metadynamics calculations by orders of magnitude and renders the algorithm much more robust. We use multiple interacting simulations, walkers, for exploring and reconstructing the same free energy surface. Each walker contributes to the history-dependent potential that, in metadynamics, is an estimate of the free energy. We show that the error on the reconstructed free energy does not depend on the number of walkers, leading to a fully linear scaling algorithm even on inexpensive loosely coupled clusters of PCs. In addition, we show that the accuracy and stability of the method are much improved by combining it with a weighted histogram analysis. We check the validity of our new method on a realistic application.

Probabilistic Cellular Automaton for Random Walkers

A cellular automaton of multiple random walkers is proposed to simulate d -dimensional many-body interacting lattice gas system. The fully parallel dynamics of the random walkers in d -dimensional hyper cubic lattice system is defined by introducing a simple probabilistic local rule set which prohibits the multiple occupancy of the walkers on a lattice site and keeps the conservation of the total number at any time. An equation which essentially governs the dynamics is derived by constructing a Boltzmann transport equation. An expression for the diffusion constant is obtained analytically and compared with the simulation results.

Cellular Automaton Model for Random Walkers.

A new stochastic discrete space-time model of multiple random walkers is proposed to simulate one-and two-dimensional many body interacting lattice gas systems. A very simple rule set which prohibits the multiple occupancy of lattice sites by the random walkers while conserving their number is used to define the dynamics of the walkers. We found a linear time dependence for the mean squared end-to-end distances of the two-dimensional random walkers system. Extensions of the model to a wide variety of diffusive systems are noted.

Single and multiple random walks on random lattices: Excitation trapping and annihilation simulations

Random walk simulations of exciton trapping and annihilation on binary and ternary lattices are presented. Single walker visitation efficiencies for ordered and random binary lattices are compared. Interacting multiple random walkers on binary and ternary random lattices are presented in terms of trapping and annihilation efficiencies that are related to experimental observables. A master equation approach, based on Monte Carlo cluster distributions, results in a nonclassical power relationship between the exciton annihilation rate and the exciton density.

Stochastic flows, reaction ? Diffusion processes, and morphogenesis

Recently, anexact procedure has been introduced [C. A. Walsh and J. J. Kozak,Phys. Rev. Lett.47:1500 (1981)] for calculating the expected walk length 〈n〉 for a walker undergoing random displacements on a finite or infinite (periodic)d-dimensional lattice with traps (reactive sites). The method (which is based on a classification of the symmetry of the sites surrounding the central deep trap and a coding of the fate of the random walker as it encounters a site of given symmetry) is applied here to several problems in lattice statistics for each of whichexact results are presented. First, we assess the importance of lattice geometry in influencing the efficiency of reaction-diffusion processes in simple and multiple trap systems by reporting values of 〈n〉 for square (cubic) versus hexagonal lattices ind=2, 3. We then show how the method may be applied to variable-step (distance-dependent) walks for a single walker on a given lattice and also demonstrate the calculation of the expected walk length 〈n〉 for the case of multiple walkers. Finally, we make contact with recent discussions of “mixing” by showing that the degree of chaos associated with flows in certain lattice systems can be calibrated by monitoring the lattice walks induced by the Poincare map of a certain parabolic function.