Path Counting Research Articles

Evaluating current status of network pharmacology for herbal medicine focusing on identifying mechanisms and therapeutic effects.

Network pharmacology has gained significant traction as a tool for identifying the mechanisms and therapeutic effects of herbal medicines. However, despite the usefulness of these approaches, their diversity underscores the critical need for a systematic evaluation to ensure consistency and reliability. We aimed to evaluate the network pharmacological analyses, focusing on identifying the mechanisms and therapeutic effects of herbal medicines. We employed a comprehensive approach involving systematic data retrieval, network construction, and analysis. Herbal compounds and their targets were meticulously extracted from five distinct network pharmacology databases to ensure extensive coverage and high data reliability. Advanced network-based methods were used to identify key herbal targets and predict therapeutic effects, thereby enriching the depth and breadth of the analysis. Experimental validation was performed on prostate cancer models to substantiate the computational predictions. The results of the recapitulating task for known herbal ingredient targets revealed distinct patterns in performance and coverage based on network construction and aggregation methods. We performed the same analysis to identify herbal targets and found that network centrality, path counts, and downweighted path counts had their own pros and cons. By comparing network-based methods, we found that considering the impact on the multiscale interactome yielded the highest accuracy in discriminating known therapeutic effects. Using optimal conditions, we successfully identified new indications for herbal medicines and validated these findings through follow-up in vitro and in vivo experiments. This study presents the first comprehensive and critical evaluation of the current network pharmacology analyses in the field of herbal medicine and provides valuable guidance for continued advances in the elucidation of the mechanisms and therapeutic effects.

MPSA-Conformer-CTC/Attention: A High-Accuracy, Low-Complexity End-to-End Approach for Tibetan Speech Recognition.

This study addresses the challenges of low accuracy and high computational demands in Tibetan speech recognition by investigating the application of end-to-end networks. We propose a decoding strategy that integrates Connectionist Temporal Classification (CTC) and Attention mechanisms, capitalizing on the benefits of automatic alignment and attention weight extraction. The Conformer architecture is utilized as the encoder, leading to the development of the Conformer-CTC/Attention model. This model first extracts global features from the speech signal using the Conformer, followed by joint decoding of these features through CTC and Attention mechanisms. To mitigate convergence issues during training, particularly with longer input feature sequences, we introduce a Probabilistic Sparse Attention mechanism within the joint CTC/Attention framework. Additionally, we implement a maximum entropy optimization algorithm for CTC, effectively addressing challenges such as increased path counts, spike distributions, and local optima during training. We designate the proposed method as the MaxEnt-Optimized Probabilistic Sparse Attention Conformer-CTC/Attention Model (MPSA-Conformer-CTC/Attention). Experimental results indicate that our improved model achieves a word error rate reduction of 10.68% and 9.57% on self-constructed and open-source Tibetan datasets, respectively, compared to the baseline model. Furthermore, the enhanced model not only reduces memory consumption and training time but also improves generalization capability and accuracy.

Shortest path counting in complex networks based on powers of the adjacency matrix.

Complex networks describe a broad range of systems in nature and society. As a fundamental concept of graph theory, the path connecting nodes and edges plays a crucial role in network science, where the computation of shortest path lengths and numbers has garnered substantial focus. It is well known that powers of the adjacency matrix can calculate the number of walks, specifying their corresponding lengths. However, developing methodologies to quantify both the number and length of shortest paths through the adjacency matrix remains a challenge. Here, we extend powers of the adjacency matrix from walks to shortest paths. We address the all-pairs shortest path count problem and propose a fast algorithm based on powers of the adjacency matrix that counts both the number and the length of all shortest paths. Numerous experiments on synthetic and real-world networks demonstrate that our algorithm is significantly faster than the classical algorithms across various network types and sizes. Moreover, we verified that the time complexity of our proposed algorithm significantly surpasses that of the current state-of-the-art algorithms. The superior property of the algorithm allows for rapid calculation of all shortest paths within large-scale networks, offering significant potential applications in traffic flow optimization and social network analysis.

Fast and accurate local ancestry inference with Recomb-Mix.

The availability of large genotyped cohorts brings new opportunities for revealing the high-resolution genetic structure of admixed populations via local ancestry inference (LAI), the process of identifying the ancestry of each segment of an individual haplotype. Though current methods achieve high accuracy in standard cases, LAI is still challenging when reference populations are more similar (e.g., intra-continental), when the number of reference populations is too numerous, or when the admixture events are deep in time, all of which are increasingly unavoidable in large biobanks. Here, we present a new LAI method, Recomb-Mix. Recomb-Mix integrates the elements of existing methods of the site-based Li and Stephens model and introduces a new graph collapsing trick to simplify counting paths with the same ancestry label readout. Through comprehensive benchmarking on various simulated datasets, we show that Recomb-Mix is more accurate than existing methods in diverse sets of scenarios while being competitive in terms of resource efficiency. We expect that Recomb-Mix will be a useful method for advancing genetics studies of admixed populations.

Trace the evolution path of pharmaceutical wastewater treatment technology based on a patent perspective

ABSTRACT This paper discusses the research hotspots and future development trends of pharmaceutical wastewater treatment technology, in order to provide valuable reference and guidance to promote the development of technology. First, the search path counting (SPC) algorithm is utilized to identify the global main path to perform a preliminary analysis of the technology evolution path from a macro perspective. Second, according to the life cycle, S-curve divides the time windows of the temporal themes and uses the Latent Dirichlet Allocation (LDA) modeling to identify the themes under each time window. Then, a visualized Sankey diagram is generated to determine the evolutionary relationship, which can be supplemented with the paths of technological evolution from the micro point of view. Finally, six research hotspots of pharmaceutical wastewater treatment technology were found, which are compounding agents, advanced oxidation technology, nanomaterials, artificial intelligence (AI) technology, membrane pollution prevention, and various combinations of wastewater treatment technology. The study also predicted its future development direction.

Node and edge centrality based failures in multi-layer complex networks

Multi-layer complex networks (MLCN) appears in various domains, such as, transportation, supply chains, etc. Failures in MLCN can lead to major disruptions in systems. Several research have focussed on different kinds of failures, such as, cascades, their reasons and ways to avoid them. This paper considers failures in a specific type of MLCN where the lower layer provides services to the higher layer without cross layer interaction, typical of a computer network. A three layer MLCN is constructed with the same set of nodes where each layer has different characteristics, the bottom most layer is Erdos–Renyi (ER) random graph with shortest path hop count among the nodes as gaussian, the middle layer is ER graph with higher number of edges from the previous, and the top most layer is preferential attachment graph with even higher number of edges. Both edge and node failures are considered. Failures happen with decreasing order of centralities of edges and nodes in static batch mode and when the centralities change dynamically with progressive failures. Emergent pattern of three key parameters, namely, average shortest path length (ASPL), total shortest path count (TSPC) and total number of edges (TNE) for all the three layers after node or edge failures are studied. Extensive simulations show that all but one parameters show definite degrading patterns. Surprising, ASPL for the middle layer starts showing a chaotic behaviour beyond a certain point for all types of failures.

A Study on Different Types of Blackhole and Wormhole Attack in MANET

Ad-hoc mobile networking (MANET) is an important technology in wireless networks with mobile nodes. Where nodes are collaborate with each other in a distributed manner and to provide multiple communications between sources and destination node. In general, the basic assumption of MANET is that each node is a trusted node. However, in a real case, there are some untrusted nodes that misbehave and attack the network like a black hole and wormhole. Where black hole nodes attract all traffic, providing false information about small hop count path to destination and with very sequence number to destinations. In wormhole attack, attacker collect data packets at one place and tunnels this data packet to another place in the network. For this attacker will give false information to the source node that the attacker node has the shortest path between sender and receiver. Source node will eventually choose that path for transmission. Once the path is established, both the nodes will drop all the incoming and outgoing packets and this causes denial of service attack, using wormhole attacks in this paper, we will discuss black hole and worm hole attack and discuss the different techniques which is used to detect and prevent these two attacks by different researchers.

Connection number topological aspect for backbone DNA networks.

The present study investigates the complex topological characteristics of DNA networks, with a specific emphasis on the innovative metric known as Connection Number (CN) as a key factor in determining network structure. The Connection Number, represented as CN(v) for a vertex v, measures the count of unique paths that link v to every other vertex in the network. By employing rigorous mathematical modeling and analysis techniques, we are able to reveal the profound implications of CN (complex networks) in characterizing the structural robustness and dynamics of information flow within DNA networks. The study of how the theory of graphs and chemicals interact is known as chemical graph theory. This paper, computing the hyper Zagreb connection index, augmented connection index, inverse sum connection index, harmonic connection index, symmetric division connection index, geometric arithmetic connection index, and atom bond connectivity connection index, of two significant types of backbone DNA and Barycentric subdivision of backbone DNA networks. Direct method computation is used to produce these Connection-based topological descriptors.

Research on the identification and formation mechanism of the main path of digital technology diffusion: Empirical evidence from China

The diffusion of digital technology has brought a subversive impact on the technological development of various industries, but the diffusion path and formation mechanism of digital technology in various industries have not been clear. In this paper, the patent data of 97 Chinese national economic industries from 2000 to 2019 were used to construct the citation network, and the Search Path Count (SPC) and critical path algorithm were used to identify the main path of digital technology diffusion among industries. This paper analyzes the formation mechanism of the main path of digital technology diffusion from four dimensions of technology characteristics by using exponential random graph model (ERGM). Through the identification of the diffusion main paths, it is found that the 13 digital technology diffusion main paths are disconnected and involve significant differences in technology fields, core technologies, major cities and institutions, possibly due to the influence of industry boundaries and industry heterogeneity. The results of the exponential random graph model show that technology combinatorial opportunity and technology proximity have significant positive effects on the formation of diffusion main path, while technical knowledge diversity and technical cooperative potential have significant negative effects on the formation of diffusion main path. In addition, the endogenous network structure variables have a significant positive impact on the formation of the main path of digital technology diffusion. These findings reveal the current trend and direction of China's digital technology development, which provides an important basis for technology developers and policy makers to choose the path of digital technology development, and provides effective guidance for traditional industries to grasp the development opportunities brought by digital technology.

Tropical descendant invariants with line constraints

AbstractVia correspondence theorems, rational log Gromov–Witten invariants of the plane can be computed in terms of tropical geometry. For many cases, there exists a range of algorithms to compute tropically: for instance, there are (generalised) lattice path counts and floor diagram techniques. So far, the cases for which there exist algorithms do not extend to non‐stationary rational descendant log Gromov–Witten invariants, that is, those where Psi‐conditions do not have to be matched up with the evaluation of a point. The case of rational descendant log Gromov–Witten invariants satisfying point conditions (without Psi‐conditions) and one Psi‐condition of any power combined with a line plays a particularly important role, because it shows up in mirror symmetry as contributions to coefficients of the ‐function. We provide recursive formulas to compute those numbers via tropical methods. Our method is inspired by the tropical proof of the WDVV equations. We also extend our study to counts involving two lines, both paired up with a Psi‐condition, appearing with power 1.

Exploring the effects of directional antennas on the secure connectivity and hop count of secure multi-hop ad-hoc wireless networks

Directional antennas can increase the received power in specific directions while reducing it in other directions. Utilizing this unique feature, we configure directional antennas at legitimate users to obtain twofold benefits for secure multi-hop ad-hoc wireless networks (AHWNs), i.e., enhancing the secure connectivity among legitimate users and lowering the hop count of the multi-hop paths. For the quantitative evaluation of these benefits, we first use a stochastic geometry graph to model the multi-hop AHWNs under the coexistence of eavesdroppers. Then, we introduce a simulation-based analysis method comprised of two algorithms: (a) the first algorithm provides the connectivity and the hop count of the shortest path between random source–destination pairs; (b) the second algorithm utilizes the results of the first one to return the average path connectivity, the probability mass function (pmf) of the hop count distribution, and the average hop count. This simulation-based analysis method helps to exhaustively study the behaviors of connectivity, hop count distribution, average hop count, and their mutual relationship in different scenarios. To the best of our knowledge, this paper is the first research work that studies the effects of directional antennas equipped at legitimate users on both secure connectivity and hop count of secure AHWNs in four combinations of two types of practical directional antennas, i.e., uniform circular array (UCA) antenna and uniform linear array (ULA) antennas, and two beamforming schemes, i.e., centralized and randomized beamforming. Moreover, comparisons with the case of using omnidirectional antennas are also provided. From the obtained simulation results, we demonstrate that using directional antennas does not always benefit the secure connectivity and hop count of secure AHWNs, i.e., only UCA-random gives higher secure connectivity and lower hop count than the omnidirectional antenna among four combinations of antenna type — beamforming scheme. Moreover, in the cases of high-density legitimate users and low-density eavesdroppers, lowering the average hop count is more remarkable than improving secure connectivity. The critical findings in this paper are valuable for deploying secure multi-hop AHWNs with directional antennas in practice.

Parameterised Counting in Logspace

Logarithmic space-bounded complexity classes such as textbf{L} and textbf{NL} play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators textbf{para}_{textbf{W}} and textbf{para}_beta for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators textbf{para}_{textbf{W}} and textbf{para}_beta by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, textbf{para}_{{textbf{W}}[1]} and textbf{para}_{beta {textbf{tail}}}. Then, we consider counting versions of all four operators and apply them to the class textbf{L} . We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is #textbf{para}_{beta {textbf{tail}}}textbf{L} -hard and can be written as the difference of two functions in #textbf{para}_{beta {textbf{tail}}}textbf{L} . These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #textbf{para}_{beta {textbf{tail}}}textbf{L} under parameterised logspace parsimonious reductions coincides with #textbf{para}_beta textbf{L} . In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research.

To the path counting on graphs in phase space

It was argued before that the Hilbert space of quantum harmonic oscillators can be realized in terms of particular trees. We generalize this approach for the Wigner functions defined on the phase space. Along this way, we derive a new representation for the Laguerre function in terms of canonical partition function of the statistical model defined on particular trees with loops.

Flagged Schur Polynomial Duality via a Lattice Path Bijection

This paper proves an identity between flagged Schur polynomials, giving a duality between row flags and column flags. This identity generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric function duality theorem due to Aitken. As corollaries we obtain the lifts of the binomial determinant duality theorem to \(q\)-binomial coefficients and to symmetric polynomials. Our method is a path counting argument on a novel lattice generalising that used by Gessel and Viennot.

Does Information Infrastructure Promote Low-Carbon Development? Evidence from the "Broadband China" Pilot Policy.

While information infrastructure has remarkably boosted global economic prosperity in the last several decades, how it propels low-carbon development has failed to draw enough attention. Based on panel data from 284 cities in China from 2006 to 2019, this study used the "Broadband China" pilot policy as an exogenous event to examine the impact of information infrastructure on carbon emission intensity. We found the following: (1) The "Broadband China" pilot policy significantly reduced carbon emission intensity, which held true in a series of robustness tests. (2) Promoting the development of the service sector, encouraging innovation activities, and fostering low-carbon lifestyles are the influential mechanisms by which information infrastructure reduced carbon emission intensity. (3) The population size, administration rank, marketization, industrialization, and informatization base significantly strengthened the reduction effect of information infrastructure on carbon emission intensity, while the disparity in human capital does not cause an evident difference. This paper's findings reveal a counting path through which improving information infrastructure advances low-carbon and sustainable growth.

Reversibility of link prediction and its application to epidemic mitigation

Current link prediction strategies are about finding new probable strong relations to establish or weak ones to remove. An interesting strategy is utilizing link prediction to prioritize the edges in the network and finding newly probable established relations. In this paper we will introduce and explain RLP, reverse link prediction, as a new paradigm, and use popular basic scoring methods including CN, JC, AA, RA, and PA, as its core to examine. The test cases are nine datasets. Half of them are contact networks in different levels from personal contact to aviation, and another half is for covering different test situations. After reviewing the edge removal based epidemic mitigation methods, we show that RLP can be used to decrease the epidemics spreading speed as a general method with various link prediction algorithms, and here in this paper, preferential attachment (PA) has the best results overall. But the results heavily depend on the nature of the examined networks: regular, scale-free or small-world. We also propose an easy to understand criteria, path count, for comparing the efficacy of epidemics mitigation methods. RLP can be extended to use other link prediction scoring methods in various types of graphs as well.

Congruence for Lattice Path Models with Filter Restrictions and Long Steps

We derive a path counting formula for a two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves the problem of finding an explicit formula for multiplicities of modules in tensor product decomposition of T(1)⊗N for Uq(sl2) with divided powers, where q is a root of unity. Combinatorial treatment of this problem calls for the definition of congruence of regions in lattice path models, properties of which are explored in this paper.

Combinatorics of generalized Dyck and Motzkin paths.

We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of steps of each kind at each vertical coordinate. A class of generalized compositions of the integer path length emerges in the analysis.

Data structures for categorical path counting queries

Consider an ordinal tree T on n nodes, each of which is assigned a category from an alphabet [σ]={1,2,…,σ}. We preprocess the tree T in order to support categorical path counting queries, which ask for the number of distinct categories occurring on the path in T between two query nodes x and y. For this problem, we propose a linear-space data structure with query time O(nlg⁡lg⁡σlg⁡w), where w=Ω(lg⁡n) is the word size in the word-RAM. As shown in our proof, from the assumption that matrix multiplication cannot be solved in time polynomially faster than cubic (with only combinatorial methods), our result is optimal, save for polylogarithmic speed-ups. For a trade-off parameter 1≤t≤n, we propose an O(n+n2t2)-word, O(tlg⁡lg⁡σlg⁡w) query time data structure. We also consider c-approximate categorical path counting queries, which must return an approximation to the number of distinct categories occurring on the query path, by counting each such category at least once and at most c times. We describe a linear-space data structure that supports 2-approximate categorical path counting queries in O(lg⁡n/lg⁡lg⁡n) time.Next, we generalize the categorical path counting queries to weighted trees. Here, a query specifies two nodes x,y and an orthogonal range Q. The answer to thus formed categorical path range counting query is the number of distinct categories occurring on the path from x to y, if only the nodes with weights falling inside Q are considered. We propose an O(nlg⁡lg⁡n+(n/t)4)-word data structure with O(tlg⁡lg⁡n) query time, or an O(n+(n/t)4)-word data structure with O(tlgϵ⁡n) query time. For an appropriate choice of the trade-off parameter t, this implies a linear-space data structure with O(n3/4lgϵ⁡n) query time. We then extend the approach to the trees weighted with vectors from [n]d, where d is a constant integer greater than or equal to 2. We present a data structure with O(nlgd−1+ϵ⁡n+(n/t)2d+2) words of space and O(tlgd−1⁡n(lg⁡lg⁡n)d−2) query time. For an O(n⋅polylogn)-space solution, one thus has O(n2d+12d+2⋅polylogn) query time.The inherent difficulty revealed by the lower bound we proved motivated us to consider data structures based on sketching. In unweighted trees, we propose a sketching data structure to solve the approximate categorical path counting problem which asks for a (1±ϵ)-approximation (i.e. within 1±ϵ of the true answer) of the number of distinct categories on the given path, with probability 1−δ, where 0<ϵ,δ<1 are constants. The data structure occupies O(n+ntlg⁡n) words of space, for the query time of O(tlg⁡n). For trees weighted with d-dimensional weight vectors (d≥1), we propose a data structure with O((n+ntlg⁡n)lgd⁡n) words of space and O(tlgd+1⁡n) query time.All these problems generalize the corresponding categorical range counting problems in Euclidean space Rd+1, for respective d, by replacing one of the dimensions with a tree topology.

The research and progress of the enumeration of lattice paths

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed.