Multiple Walkers Research Articles

Lambda-ABF: Simplified, Portable, Accurate, and Cost-Effective Alchemical Free-Energy Computation.

We introduce the lambda-Adaptive Biasing Force (lambda-ABF) method for the computation of alchemical free-energy differences. We propose a software implementation and showcase it on biomolecular systems. The method arises from coupling multiple-walker adaptive biasing force with λ-dynamics. The sampling of the alchemical variable is continuous and converges toward a uniform distribution, making manual optimization of the λ schedule unnecessary. Contrary to most other approaches, alchemical free-energy estimates are obtained immediately without any postprocessing. Free diffusion of λ improves orthogonal relaxation compared to fixed-λ thermodynamic integration or free-energy perturbation. Furthermore, multiple walkers provide generic orthogonal space coverage with minimal user input and negligible computational overhead. We show that our high-performance implementations coupling the Colvars library with NAMD and Tinker-HP can address real-world cases including ligand-receptor binding with both fixed-charge and polarizable models, with a demonstrably richer sampling than fixed-λ methods. The implementation is fully open-source, publicly available, and readily usable by practitioners of current alchemical methods. Thanks to the portable Colvars library, lambda-ABF presents a unified user interface regardless of the back-end (NAMD, Tinker-HP, or any software to be interfaced in the future), sparing users the effort of learning multiple interfaces. Finally, the Colvars Dashboard extension of the visual molecular dynamics (VMD) software provides an interactive monitoring and diagnostic tool for lambda-ABF simulations.

Confluence of Random Walks, Interacting Particle Systems, and Distributed Machine Learning: Federated Learning through Crawling over Networks

In this work, we aim to unveil a new class of intermediate FL architectures between centralized and decentralized schemes called “FedCrawl.” FedCrawl takes advantage of benefits of D2D communications similar to decentralized schemes; however, it uses them in a nuanced way. FedCrawl is inspired by web crawlers, which effectively explore the websites to find updated/new content posted on the internet. The cornerstone of FedCrawl is its innovative conceptualization of neural networks (NNs) or other used ML models as autonomous entities, called random walkers, with the capability to move or jump across nodes in the network through peer-to-peer (P2P) or device-to-device (D2D) connections. We introduce five research aspects to study the nuanced intricacies governing random walker behavior in these environments. The first research aspect addresses the interplay between network topology and data distribution, emphasizing the importance of considering both factors for designing efficient random walks in FedCrawl. The second research aspect explores the applicability of node importance metrics in optimizing random walker paths for FedCrawl. We propose a dynamic perception-aware design, discussed in the third research aspect, where transition matrices adapt to the evolving state of random walkers, balancing exploration and exploitation. The fourth research aspect introduces innovative features like skipping, memory look-back, and caching/trailing to enhance random walker performance. Lastly, the fifth research aspect delves into the dynamics of multiple random walkers in networked environments, introducing the concept of multi-pole random walkers. Complementing these five research aspects, we present five conjectures, each introducing novel perspectives and methodologies in the domain of decentralized learning. These conjectures encompass areas such as temperature-based characterization of random walkers and network nodes, dynamic transition matrices, non-Markovian processes, and an evolutionary framework for random walker patterns.

A novel DNAzyme-driven walker-walkers serial amplified electrochemical aptasensor for ultrasensitive detection of cardiac troponin I

Sensitive and accurate detection of Cardiac troponin I (cTnI) plays an instrumental role in the prevention and management of acute myocardial infarction. Herein, a novel electrochemical aptasensor was designed for cTnI ultrasensitive detection based on DNAzyme-driven walker-walkers serial amplified strategy. The Au/NiCo2S4 nanocomposites, which possess excellent conductivity and a high specific surface area, were used to assemble on the modified electrode for the first time. The Hairpin probe I (HPI) and hairpin probe II (HP II) were self-assembled onto an Au/NiCo2S4 electrode surface through the interaction of Au-S bond. In the present of cTnI, the walker-walkers strategy was activated. After the hybridization between cTnI and HP I, the DNA walker shears the specific sequence of HP II and releases triggered DNA. Since the triggered DNA corresponds to the HP I complementary sequence, which could trigger multiple recycling and release DNA walkers. By the joint action of cTnI and triggered DNA, HP I was opened and released more DNA walkers. With the use of cDNA, capturing methylene blue (MB), resulting in the significant electrochemical signal in differential pulse voltammetry (DPV). The proposed strategy can effectively avoid problems of the insufficient distance or limited available driving force that general exist in unipedal DNA walkers. Under optimal conditions, the change of current was linearly related to the negative logarithm of cTnI concentration in the detection range of 0.01–60 ng/mL with a low detection limit of 0.26 pg/mL. Moreover, the proposed aptasensor was successfully applied in detection of human serum samples.

Improved reweighting protocols for variationally enhanced sampling simulations with multiple walkers.

In molecular dynamics simulations utilizing enhanced-sampling techniques, reweighting is a central component for recovering the targeted ensemble averages of the "unbiased" system by calculating and applying a bias-correction function c(t). We present enhanced reweighting protocols for variationally enhanced sampling (VES) simulations by exploiting a recent reweighting method, originally introduced in the metadynamics framework [Giberti et al. J. Chem. Theory Comput., 2020, 16, 100-107], which was modified and extended to multiple-walker simulations: these may be implemented either as "independent" walkers (associated with one unique correction function per walker) or "cooperative" ones that all share one correction function, which is the hitherto only explored option. When each case is combined with the two possibilities of determining c(t) by time integration up to either t or over the entire simulation period , altogether four reweighting options result. Their relative merits were assessed by well-tempered VES simulations of two model problems: locating the free-energy difference between two metastable molecular conformations of the N-acetyl-L-alanine methylamide dipeptide, and the recovery of an a priori known distribution when one water molecule in the liquid phase is perturbed by a periodic free-energy function. The most rapid convergence occurred for large cooperative walkers, regardless of the upper integration limit, but integrating up to t proved advantageous for small walker ensembles. That novel reweighting method compared favorably to the standard VES reweighting, as well as to current state-of-the-art reweighting options introduced for metadynamics simulations that estimate c(t) by integration over the collective variables. For further gains in computational speed and accuracy, we also introduce analytical solutions for c(t), as well as offering further insight into its features by approximative analytical expressions in the "high-temperature" regime.

Following Other People’s Footsteps: A Contextual-Attraction Effect Induced by Biological Motion

Our visual system is bombarded with numerous social interactions that form intangible social bonds among people, as exemplified by synchronized walking in crowds. Here, we investigated whether these perceived social bonds implicitly intrude on visual perception and induce a contextual effect. Using multiple point-light walkers and a classical contextual paradigm, we tested 72 college-age adults across six experiments and found that the perceived direction of the central walker was attracted toward the direction of the surrounding walkers. The observed contextual-attraction effect occurred even when the surrounding walkers differed from the central walker in gender and walking speed but disappeared when they were asynchronously presented or replaced by inanimate motion. Strikingly, this contextual-attraction effect partially persisted in the context of local motion rather than static figures. These findings, in contrast to the typical contextual-repulsion effect, lend support for the distinctiveness of perceived social bonds on contextual modulation and suggest a specialized contextual mechanism tuned to social factors.

Recursive Sparse Representation for Identifying Multiple Concurrent Occupants Using Floor Vibration Sensing

In this paper, we present a multiple concurrent occupant identification approach through footstep-induced floor vibration sensing. Identification of human occupants is useful in a variety of indoor smart structure scenarios, with applications in building security, space allocation, and healthcare. Existing approaches leverage sensing modalities such as vision, acoustic, RF, and wearables, but are limited due to deployment constraints such as line-of-sight requirements, sensitivity to noise, dense sensor deployment, and requiring each walker to wear/carry a device. To overcome these restrictions, we use footstep-induced structural vibration sensing. Footstep-induced signals contain information about the occupants' unique gait characteristics, and propagate through the structural medium, which enables sparse and passive identification of indoor occupants. The primary research challenge is that multiple-person footstep-induced vibration responses are a mixture of structurally-codependent overlapping individual responses with unknown timing, spectral content, and mixing ratios. As such, it is difficult to determine which part of the signal corresponds to each occupant. We overcome this challenge through a recursive sparse representation approach based on cosine distance that identifies each occupant in a footstep event in the order that their signals are generated, reconstructs their portion of the signal, and removes it from the mixed response. By leveraging sparse representation, our approach can simultaneously identify and separate mixed/overlapping responses, and the use of the cosine distance error function reduces the influence of structural codependency on the multiple walkers' signals. In this way, we isolate and identify each of the multiple occupants' footstep responses. We evaluate our approach by conducting real-world walking experiments with three concurrent walkers and achieve an average F1 score for identifying all persons of 0.89 (1.3x baseline improvement), and with a 10-person "hybrid" dataset (simulated combination of single-walker real-world data), we identify 2, 3, and 4 concurrent walkers with a trace-level accuracy of 100%, 93%, and 73%, respectively, and observe as much as a 2.9x error reduction over a naive baseline approach.

Video object segmentation via random walks on two-frame graphs comprising superpixels

We propose a novel video object segmentation method employing random walkers to travel on graphs constructed on two consecutive frames. First, we estimate the initial foreground and background distributions by minimising an energy function that incorporates the stationary distributions of the random walks. The random walkers frequently travel between similar nodes of the graph constructed on two adjacent frames, which enables the incorporation of the inter-frame information into the energy function effectively and elegantly. Then, we refine the initial results by simulating the movements of multiple random walkers. We process the sequence in a recursive manner, which naturally propagates the previous segmentation labels to the subsequent frames. Additionally, we develop a strategy for adjusting the superpixel number using region similarity and the average Frobenius norm of optical flow gradient. This strategy can improve performance significantly. Furthermore, we discuss the feature selection problem in the method to select a more effective feature representation. Extensive and comparable experiments on Segtrack and Segtrack v2 demonstrate that the proposed algorithm yields higher performance than several recent state-of-the-art approaches.

Two-dimensional quantum walks of correlated photons

Quantum walks in an elaborately designed graph are a powerful tool for simulating physical and topological phenomena, constructing novel quantum algorithms, and realizing universal quantum computing. Integrated photonics technology has emerged as a versatile platform for implementing a variety of quantum information tasks and as a promising candidate for performing large-scale quantum walks. Both extending physical dimensions and involving more particles will increase the complexity of the evolving systems. Pioneering studies have demonstrated a single particle walking on two-dimensional lattices and multiple walkers interfering on a one-dimensional structure. However, multiple particles evolving in a genuine two-dimensional space in a scalable fashion has remained a vacancy for nearly 10 years. We present a genuine two-dimensional quantum walk with correlated photons on a triangular photonic lattice, which is mapped to a 37 × 37 high-dimensional state space. The genuine two-dimensional quantum walk breaks through the physical restrictions of single-particle evolution, allowing for the encoding of information in large spaces and construction of high-dimensional graphs, which are beneficial for quantum information processing. Between the chip and the two-dimensional fanout interface, site-by-site addressing enables simultaneous detection of over 600 nonclassical interferences and observation of quantum correlations that violate a classical limit by 57 standard deviations. Our implementation provides a paradigm for multi-photon quantum walks in a two-dimensional configuration on a large scale, paving the way for practical quantum simulation and computation beyond the classical regime.

Independent Nontargeted Parallel Cascade Selection Molecular Dynamics (Ino-PaCS-MD) to Enhance the Conformational Sampling of Proteins.

Biological functions are related to long-time protein dynamics (rare events) that are induced over microseconds. Such protein dynamics can be investigated using molecular dynamics (MD) simulations. However, the detection of rare events remains challenging using conventional MD (cMD) since the accessible timescales of cMD are shorter than those of the biological functions. Recently, the parallel cascade selection MD (PaCS-MD) has been proposed to detect such rare events, wherein transition paths are generated between a given reactant and product. As an extension, the nontargeted PaCS-MD (nt-PaCS-MD) has been proposed to predict the transition paths without requiring reference to any product. Thus, as a further extension, we herein propose independent nt-PaCS-MD, namely, Ino-PaCS-MD, wherein multiple walkers are launched from a set of different starting configurations. Each walker repeats a cycle of restarting short-time MD simulations from configurations with high potentials for making transitions to neighboring metastable states. To further enhance the sampling ability, Ino-PaCS-MD temporarily stops the conformational search and periodically resets the starting configurations so that they are uniformly distributed in a conformational subspace, thereby preventing a given protein from being trapped in one of the metastable states. As a demonstration, Ino-PaCS-MD successfully detects rare events of a maltose-binding protein as open-close transitions with a nanosecond-order simulation time, although a microsecond-order cMD simulation failed to detect these rare events, showing the high sampling efficiency of Ino-PaCS-MD.

Improved Sampling and Free Energy Estimates for Antibiotic Permeation through Bacterial Porins.

Antibiotics enter into bacterial cells via protein channels that serve as low-energy pathways through the outer membrane, which is otherwise impenetrable. Insights into the molecular mechanisms underlying the transport processes are vital for the development of effective antibacterials. A much-desired prerequisite is an accurate and reproducible determination of free energy surfaces for antibiotic translocation, enabling quantitative and meaningful comparisons of permeation mechanisms for different classes of antibiotics. Inefficient sampling along the orthogonal degrees of freedom, for example, in umbrella sampling and metadynamics approaches, is however a key limitation affecting the accuracy and the convergence of free energy estimates. To overcome this limitation, two sampling methods have been employed in the present study that, respectively, combine umbrella sampling and metadynamics-style biasing schemes with temperature acceleration for improved sampling along orthogonal degrees of freedom. As a model for the transport of bulky solutes, the ciprofloxacin-OmpF system has been selected. The well-tempered metadynamics approach with multiple walkers is compared with its "temperature-accelerated" variant in terms of improvements in sampling and convergence of free energy estimates. We find that the inclusion of collective variables governing solute degrees of freedom and solute-water interactions within the sampling scheme largely alleviates sampling issues. Concerning improved sampling and convergence of free energy estimates from independent simulations, the temperature-accelerated sliced sampling approach that combines umbrella sampling with temperature-accelerated molecular dynamics performs even better as shown for the ciprofloxacin-OmpF system.

(Invited) Single-Walled Carbon Nanotubes As Optical Probes for Imaging and Biosensing

Single-walled carbon nanotubes (SWCNT) are exceptionally photostable fluorophores that emit in the near-infrared range. Their novel physical and chemical properties make them particularly appealing as optical probes for microscopy and spectroscopy. The π-electron rich side wall of SWCNT allows various biopolymers to be conjugated through π-π stacking, enabling novel applications in biology and medicine. In particular, oligonucleotides (DNA or RNA) have been widely used to functionalize SWCNT due to its specific base-pairing properties and programmability. Here we demonstrate the use of DNA-SWCNT hybrid nanomaterial in fundamental biophysics research and biomolecular sensing applications. In the first system, we show SWCNT can be used as a photo-stable imaging probe for optically tracking DNA walker's transport characteristics. DNA walkers are synthetic nanosystems that exploit the structural specificity and functional diversity of oligonucleotides to mimic intracellular protein motors. We integrated a DNA walker system onto SWCNT and demonstrated transportation of quantum dots along SWCNT track. We used localization microscopy and single-particle tracking to record the displacement time series of multiple walkers, which is then used to build the statistical distribution of speed, displacement and walking turnover time. The statistical characteristics of the system help extract time averaged and ensemble averaged walker kinetics, which provide biochemical and biophysical insights about the system. In the second system, we demonstrate using aptamer-SWCNT as a novel sensor architecture for biomolecular detection. We non-covalently functionalized SWCNT with porphyrin binding aptamers, which retains aptamer's target specificity and nanotube's optical property. We discovered the photoluminescence of aptamer-SWCNT is selectively quenched upon binding to heme, an iron-containing porphyrin present in diverse biological processes. The aptamer-SWCNT complex also serves as a quencher for the intrinsic fluorescence of porphyrins. We propose the two optical transduction signatures are induced by specific binding of aptamer to heme and non-specific adsorption of other porphyrin species on the side wall of SWCNT. Using these two properties, we developed a sensing mechanism for four porphyrin species present in the biosynthetic pathway, and demonstrated multiplexed porphyrin detection in blood plasma. Our work shows that SWCNT is a useful optical material for both fundamental biomolecular engineering research and biomedical applications.

Quantum walks on a programmable two-dimensional 62-qubit superconducting processor.

Quantum walks are the quantum mechanical analog of classical random walks and an extremely powerful tool in quantum simulations, quantum search algorithms, and even for universal quantum computing. In our work, we have designed and fabricated an 8-by-8 two-dimensional square superconducting qubit array composed of 62 functional qubits. We used this device to demonstrate high-fidelity single- and two-particle quantum walks. Furthermore, with the high programmability of the quantum processor, we implemented a Mach-Zehnder interferometer where the quantum walker coherently traverses in two paths before interfering and exiting. By tuning the disorders on the evolution paths, we observed interference fringes with single and double walkers. Our work is a milestone in the field, bringing future larger-scale quantum applications closer to realization for noisy intermediate-scale quantum processors.

Five-Site Model for Brownian Dynamics Simulations of a Molecular Walker in Three Dimensions.

Motor proteins play an important role in many biological processes and have inspired the development of synthetic analogues. Molecular walkers, such as kinesin, dynein, and myosin V, fulfill a diverse set of functions including transporting cargo along tracks, pulling molecules through membranes, and deforming fibers. The complexity of molecular motors and their environment makes it difficult to model the detailed dynamics of molecular walkers over long time scales. In this work, we present a simple, three-dimensional model for a molecular walker on a bead-spring substrate. The walker is represented by five spherically symmetric particles that interact through common intermolecular potentials and can be simulated efficiently in Brownian dynamics simulations. The movement of motor protein walkers entails energy conversion through ATP hydrolysis while artificial motors typically rely on a local conversion of energy supplied through external fields. We model energy conversion through rate equations for mechanochemical states that couple positional and chemical degrees of freedom and determine the walker conformation through interaction potential parameters. We perform Brownian dynamics simulations for two scenarios: In the first, the model walker transports cargo by walking on a substrate whose ends are fixed. In the second, a tethered motor pulls a mobile substrate chain against a variable force. We measure relative displacements and determine the effects of cargo size and retarding force on the efficiency of the walker. We find that, while the efficiency of our model walker is less than for the biological system, our simulations reproduce trends observed in single-molecule experiments on kinesin. In addition, the model and simulation method presented here can be readily adapted to biological and synthetic systems with multiple walkers.

Mean encounter times for multiple random walkers on networks

We introduce a general approach for the study of the collective dynamics of noninteracting random walkers on connected networks. We analyze the movement of R independent (Markovian) walkers, each defined by its own transition matrix. By using the eigenvalues and eigenvectors of the R independent transition matrices, we deduce analytical expressions for the collective stationary distribution and the average number of steps needed by the random walkers to start in a particular configuration and reach specific nodes the first time (mean first-passage times), as well as global times that characterize the global activity. We apply these results to the study of mean first-encounter times for local and nonlocal random walk strategies on different types of networks, with both synchronous and asynchronous motion.

Graph Degree Heterogeneity Facilitates Random Walker Meetings

Various graph algorithms have been developed with multiple random walks, the movement of several independent random walkers on a graph. Designing an efficient graph algorithm based on multiple random walks requires investigating multiple random walks theoretically to attain a deep understanding of their characteristics. The first meeting time is one of the important metrics for multiple random walks. The first meeting time on a graph is defined by the time it takes for multiple random walkers to meet at the same node in a graph. This time is closely related to the rendezvous problem, a fundamental problem in computer science. The first meeting time of multiple random walks has been analyzed previously, but many of these analyses have focused on regular graphs. In this paper, we analyze the first meeting time of multiple random walks in arbitrary graphs and clarify the effects of graph structures on expected values. First, we derive the spectral formula of the expected first meeting time on the basis of spectral graph theory. Then, we examine the principal component of the expected first meeting time using the derived spectral formula. The clarified principal component reveals that (a)the expected first meeting time is almost dominated by $n/(1+d_{\rm std}^2/d_{\rm avg}^2)$ and (b)the expected first meeting time is independent of the starting nodes of random walkers, where $n$ is the number of nodes of the graph. $d_{\rm avg}$ and $d_{\rm std}$ are the average and the standard deviation of weighted node degrees, respectively. The characteristic(a) is useful for understanding the effect of the graph structure on the first meeting time. According to the revealed effect of graph structures, the variance of the coefficient $d_{\rm std}/d_{\rm avg}$(degree heterogeneity) for weighted degrees facilitates the meeting of random walkers.

QSW_MPI: A framework for parallel simulation of quantum stochastic walks

QSW_MPI is a Python package developed for time-series simulation of continuous-time quantum stochastic walks. This model allows for the study of Markovian open quantum systems in the Lindblad formalism, including a generalisation of the continuous-time random walk and continuous-time quantum walk. Consisting of a Python interface accessing parallelised Fortran libraries utilising sparse data structures, QSW_MPI is scalable to massively parallel computers, which makes possible the simulation of a wide range of walk dynamics on directed and undirected graphs of arbitrary complexity. Program summaryProgram Title: QSW_MPICPC Library link to program files:http://dx.doi.org/10.17632/r8jsx4xxgv.1Licensing provisions: GPLv3Programming language: Python 3 + Fortran 2003External routines/libraries: NumPy [1], SciPy [2], MPI for Python [3], h5py[4]Nature of problem: QSW_MPI provides a framework for the simulation of quantum stochastic walks on arbitrary graphs (directed/undirected, weighted/unweighted).Solution method: A parallel distributed-memory implementation of the matrix exponential via a truncated Taylor series expansion with scaling and squaring [5].Additional comments including restrictions and unusual features: QSW_MPI will provide support for the simulation of multiple quantum walkers in a future version.

Jamming of multiple persistent random walkers in arbitrary spatial dimension

We consider the persistent exclusion process in which a set of persistent random walkers interact via hard-core exclusion on a hypercubic lattice in d dimensions. We work within the ballistic regime whereby particles continue to hop in the same direction over many lattice sites before reorienting. In the case of two particles, we find the mean first-passage time to a jammed state where the particles occupy adjacent sites and face each other. This is achieved within an approximation that amounts to embedding the one-dimensional system in a higher-dimensional reservoir. Numerical results demonstrate the validity of this approximation, even for small lattices. The results admit a straightforward generalization to dilute systems comprising more than two particles. A self-consistency condition on the validity of these results suggest that clusters may form at arbitrarily low densities in the ballistic regime, in contrast to what has been found in the diffusive limit.

Footstep detection in urban seismic data with a convolutional neural network

Seismic data for studying the near surface have historically been extremely sparse in cities, limiting our ability to understand small-scale processes, locate small-scale geohazards, and develop earthquake hazard microzonation at the scale of buildings. In recent years, distributed acoustic sensing (DAS) technology has enabled the use of existing underground telecommunications fibers as dense seismic arrays, requiring little manual labor or energy to maintain. At the Fiber-Optic foR Environmental SEnsEing array under Pennsylvania State University, we detected weak slow-moving signals in pedestrian-only areas of campus. These signals were clear in the 1 to 5 Hz range. We verified that they were caused by footsteps. As part of a broader scheme to remove and obscure these footsteps in the data, we developed a convolutional neural network to detect them automatically. We created a data set of more than 4000 windows of data labeled with or without footsteps for this development process. We describe improvements to the data input and architecture, leading to approximately 84% accuracy on the test data. Performance of the network was better for individual walkers and worse when there were multiple walkers. We believe the privacy concerns of individual walkers are likely to be highest priority. Community buy-in will be required for these technologies to be deployed at a larger scale. Hence, we should continue to proactively develop the tools to ensure city residents are comfortable with all geophysical data that may be acquired.

Hierarchical parallelization of divide-and-conquer density functional tight-binding molecular dynamics and metadynamics simulations.

Massively parallel divide-and-conquer density functional tight-binding (DC-DFTB) molecular dynamics and metadynamics simulations are efficient approaches for describing various chemical reactions and dynamic processes of large complex systems via quantum mechanics. In this study, DC-DFTB simulations were combined with multi-replica techniques. Specifically, multiple walkers metadynamics, replica exchange molecular dynamics, and parallel tempering metadynamics methods were implemented hierarchically into the in-house Dcdftbmd program. Test simulations in an aqueous phase of the internal rotation of formamide and conformational changes of dialanine showed that the newly developed extensions increase the sampling efficiency and the exploration capabilities in DC-DFTB configuration space.

Searching efficiency of multiple walkers on the weighted networks

The weighted networks are realistic forms of networks, where a weight is attached to each link. In order to better study searching efficiency of multiple walkers we extend the study on the binary networks to the weighted networks. In this paper, the main aim is to measure the searching efficiency of multiple walkers on the weighted networks. Firstly, we review the theoretical foundations related to our study. Secondly, we introduce the heterogeneous mean-field (HMF) theory and the annealed network approach on the weighted networks. Then, we deduce the analysis formula of mean first parallel passage time (MFPPT). Finally, we study the global mean first parallel passage time (GMFPPT) and compare it to the searching efficiency of a single walker previously studied. The obtained result shows that the GMFPPT follows a uniform power law with the number of walkers. The key of this paper is to apply the heterogeneous mean-field (HMF) theory and the annealed weighted network approach to replace the weighted uncorrelated networks with the weighted fully connected networks and then construct a general probability transfer matrix on the weighted fully connected networks.