Distribution Of Vertices Research Articles

On individual leaf depths of trees

We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of Panholzer and Prodinger, we apply this method to study the depth distributions of individual nodes or leaves in rooted binary trees, plane trees, noncrossing trees, and increasing trees; the height distributions of individual vertices and individual steps in Dyck paths; and the number of diagonals separating two fixed sides of a convex polygon in a triangulation or dissection. We obtain both exact and asymptotic results, which sometimes refine known formulas or provide combinatorial proofs of results from the probability literature.

Rethinking Mesh Watermark: Towards Highly Robust and Adaptable Deep 3D Mesh Watermarking

The goal of 3D mesh watermarking is to embed the message in 3D meshes that can withstand various attacks imperceptibly and reconstruct the message accurately from watermarked meshes. The watermarking algorithm is supposed to withstand multiple attacks, and the complexity should not grow significantly with the mesh size. Unfortunately, previous methods are less robust against attacks and lack of adaptability. In this paper, we propose a robust and adaptable deep 3D mesh watermarking Deep3DMark that leverages attention-based convolutions in watermarking tasks to embed binary messages in vertex distributions without texture assistance. Furthermore, our Deep3DMark exploits the property that simplified meshes inherit similar relations from the original ones, where the relation is the offset vector directed from one vertex to its neighbor. By doing so, our method can be trained on simplified meshes but remains effective on large size meshes (size adaptable) and unseen categories of meshes (geometry adaptable). Extensive experiments demonstrate our method remains efficient and effective even if the mesh size is 190× increased. Under mesh attacks, Deep3DMark achieves 10%∼50% higher accuracy than traditional methods, and 2× higher SNR and 8% higher accuracy than previous DNN-based methods.

Optimal Network Analysis through Vertex Order Coloring of Intuitionistic Fuzzy Graph Operations

Intuitionistic fuzzy graphs (IFGs) are a powerful tool for modeling uncertainty and complex relationships. They offer versatile frameworks for addressing real‐world challenges. In this research, we have introduced intuitionistic fuzzy vertex order coloring (IFVOC) and analyzed the alpha‐strong (), beta‐strong (), and gamma‐strong () vertices through their degree. We explored important theorems based on the types of strong vertices, broadening the scope of our study. We analyzed multiple IFG products to determine the most optimal network based on some important metrics, including the weight and total number of vertices, the chromatic number, and the weight of the graph’s minimum spanning tree. We systematically evaluate different types of IFG products, such as conormal, modular, residue, and maximal, and consider their implications for fuzzy graph structure and connectivity. We have investigated new metrics to measure the presence and importance of vertices in the graph, assessing both their frequency and overall impact. Additionally, we have investigated the impact of product operations on the significance of nodes and the resultant minimum spanning tree, providing insights into the overall robustness and efficiency of each product. We have explored how variations in product operations impact the distribution of vertices and the characteristics of the minimum spanning tree and chromatic number, elucidating the trade‐offs between different product options. We have modeled the product graph as a network to represent complex systems composed of interconnected entities. The outcomes of this research contribute to the advancement of IFG theory and its use in various fields, such as network design, strategic planning, and system analysis.

Impact of bogie fairing configuration on the aerodynamic performance of high-speed train under crosswind

In this paper, the unsteady aerodynamic characteristics of a three-car grouped high-speed train (HST) with different bogie fairing configurations under the crosswind are studied using improved delayed detached eddy simulation (IDDES) at Re = 1.85 × 106. The numerical method results are validated against the previous wind tunnel experiments. There are three cases with original half-fairing configuration (HFC), non-fairing configuration (NFC) and full-fairing configuration (FFC), respectively. The numerical results show that the bogie fairing configuration has significant effects. Compared to the HFC case, the drag force coefficients of the head, middle and tail cars with NFC increase by 53.13%, 14.87% and 27.04%, and corresponding parts of the FFC case decrease by 49.98%, 24.40% and 12.41%, respectively. The FFC case can also optimize the pressure distribution on the windward and leeward sides of the HST, and the flow structure and vertex distribution around the car body are also optimized. In addition, the increasing encirclement of the fairing configurations reduces the maximum lateral flow velocity inside the bogie cavities by 70% for HFC and 50% for FFC compared to the NFC. Overall, the FFC presents better aerodynamic performance than the NFC and HFC cases under crosswind, which is recommended for the design of the HST.

View‐graph key‐subset extraction for efficient and robust structure from motion

AbstractStructure from motion (SfM) is used to recover camera poses and the sparse structure of real scenes from multiview images. SfM methods construct a view‐graph from the matching relationships of images. Redundancy and incorrect edges are usually observed in it. Redundancy inhibits the efficiency and incorrect edges result in the misalignment of structures. In addition, the uneven distribution of vertices usually affects the global accuracy. To address these problems, we propose a coarse‐to‐fine approach in which the poses of an extracted key‐subset of images are first computed and then all remaining images are oriented. The core of this approach is view‐graph key‐subset extraction, which not only prunes redundant data and incorrect edges but also obtains properly distributed key‐subset vertices. The extraction approach is based on a replaceability score and an iteration‐update strategy. In this way, only vertices with high SfM importance are preserved in the key‐subset. Different public datasets are used to evaluate our approach. Due to the absence of ground‐truth camera poses in large‐scale datasets, we present new datasets with accurate camera poses and point clouds. The results demonstrate that our approach greatly increases the efficiency of SfM. Furthermore, the robustness and accuracy can be improved.

Reticular-Chemical Approach to Soft-Matter Self-Assembly: Why Are srs and noh Nets Realized in Thermotropics?

Abstract Interface geometry and skeletal graphs are two complementary characterizations of micro-phase separated organizations in soft matter. This paper explores the possibility of analyzing the latter instead of the former to gain insight into aggregation structures. Analyses of the ideality of vertices geometry (closeness to equiangular three-coordination) and the spatial homogeneity of the vertex distribution strongly suggest that the pair of noh nets, recently proposed for thermotropic liquid crystals of rodlike molecules without the interface between counter-chiral domains, is beneficial besides the srs twins if cubic symmetry is presumed as an experimental input.

Investigation of the Properties of First Nearest Neighbors’ Graphs

In this study we present a benchmark of statistical distributions of the first nearest neighbors in random graphs. We consider distribution of such graphs by the number of disconnected fragments, fragments by the number of involved nodes, and nodes by their degrees. The statements about the asymptotic properties of these distributions for graphs of large dimension are proved. The problem under investigation is to estimate the probability of realization of a certain structure of the first nearest neighbors graph depending on the distribution function of distances between the elements of the studied set. It is shown that, up to isomorphism, the graph of the first nearest neighbors does not depend on the distance distribution. This fact makes it possible to conduct numerical experiments on the construction of basic statistics based on a uniform distribution of distances and obtain tabulated data as a result of numerical modeling. We also discuss the approximation of the distribution of graph vertices by degrees, which allows us to estimate the proportion of randomness for a particular structure resulting from clustering elements of a certain set by the nearest neighbor method. The asymptotic analysis of the fragment distribution is discussed.

Vertex transitivity and distance metric of the quad-cube

The quad-cube is a special case of the metacube that itself is derivable from the hypercube. It is amenable to an application as a network topology, especially when the node size exceeds several million. This paper presents the following welcome properties of the graph, relating to its structure: (1) vertex transitivity that facilitates the working of an algorithm meant for a “local” context in the global context as well, and (2) an exact formula for the distance metric, which leads to a precise result on the distance-wise vertex distribution of the graph and an exact formula for the average vertex distance. Remarkably, the vertex distribution of the quad-cube resembles, to a large extent, the vertex distribution of the twin copies of a hypercube. In a parallel study, the author recently reported similar results with respect to the dual-cube (Jha in J Supercomput 78:17758–17775, 2022)

A high sensitivity Cherenkov detector for prompt gamma timing and time imaging

We recently proposed a new approach for the real-time monitoring of particle therapy treatments with the goal of achieving high sensitivities on the particle range measurement already at limited counting statistics. This method extends the Prompt Gamma (PG) timing technique to obtain the PG vertex distribution from the exclusive measurement of particle Time-Of-Flight (TOF). It was previously shown, through Monte Carlo simulation, that an original data reconstruction algorithm (Prompt Gamma Time Imaging) allows to combine the response of multiple detectors placed around the target. The sensitivity of this technique depends on both the system time resolution and the beam intensity. At reduced intensities (Single Proton Regime—SPR), a millimetric proton range sensitivity can be achieved, provided the overall PG plus proton TOF can be measured with a 235 ps (FWHM) time resolution. At nominal beam intensities, a sensitivity of a few mm can still be obtained by increasing the number of incident protons included in the monitoring procedure. In this work we focus on the experimental feasibility of PGTI in SPR through the development of a multi-channel, Cherenkov-based PG detector with a targeted time resolution of 235 ps (FWHM): the TOF Imaging ARrAy (TIARA). Since PG emission is a rare phenomenon, TIARA design is led by the concomitant optimisation of its detection efficiency and Signal to Noise Ratio (SNR). The PG module that we developed is composed of a small PbF_{2} crystal coupled to a silicon photoMultiplier to provide the time stamp of the PG. This module is currently read in time coincidence with a diamond-based beam monitor placed upstream the target/patient to measure the proton time of arrival. TIARA will be eventually composed of 30 identical modules uniformly arranged around the target. The absence of a collimation system and the use of Cherenkov radiators are both crucial to increase the detection efficiency and the SNR, respectively. A first prototype of the TIARA block detector was tested with 63 MeV protons delivered from a cyclotron: a time resolution of 276 ps (FWHM) was obtained, resulting in a proton range sensitivity of 4 mm at 2sigma with the acquisition of only 600 PGs. A second prototype was also evaluated with 148 MeV protons delivered from a synchro-cyclotron obtaining a time resolution below 167 ps (FWHM) for the gamma detector. Moreover, using two identical PG modules, it was shown that a uniform sensitivity on the PG profiles would be achievable by combining the response of gamma detectors uniformly distributed around the target. This work provides the experimental proof-of-concept for the development of a high sensitivity detector that can be used to monitor particle therapy treatments and potentially act in real-time if the irradiation does not comply to treatment plan.

Loose Cores and Cycles in Random Hypergraphs

Inspired by the study of loose cycles in hypergraphs, we define the loose core in hypergraphs as a structurewhich mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial random hypergraph $H^r(n,p)$, the order of the loose core undergoes a phase transition at a certain critical threshold and determine this order, as well as the number of edges, asymptotically in the subcritical and supercritical regimes.
 Our main tool is an algorithm called CoreConstruct, which enables us to analyse a peeling process for the loose core. By analysing this algorithm we determine the asymptotic degree distribution of vertices in the loose core and in particular how many vertices and edges the loose core contains. As a corollary we obtain an improved upper bound on the length of the longest loose cycle in $H^r(n,p)$.

Community Evolution Prediction Based on Multivariate Feature Sets and Potential Structural Features

The current study on community evolution prediction ignores the problem of internal community topology characteristics and does not take feature sets extraction into account. Therefore, the MF-PSF (Multivariate Feature sets and Potential Structural Features) method based on multivariate feature sets and potential structural features for community evolution prediction is proposed in this paper. Firstly, the multivariate feature sets are built from four aspects: community core node features, community structural features, community sequential features and community behavior features. Secondly, the community’s potential structural characteristics based on DeepWalk and spectral propagation theories are extracted, and the overall community’s internal structural characteristics and vertex distribution are analyzed. Finally, the community’s multivariate structural features and potential structural features are merged to predict community evolution events, and the importance of each feature in the process of evolutionary prediction is discussed. The experimental results show that compared with other community evolution prediction methods, the MF-PSF prediction method not only provides a foundation for analyzing the influence of various feature sets on predicted events, but it also effectively improves the accuracy of evolution prediction.

PageRank asymptotics on directed preferential attachment networks

We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted power law hypothesis. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.

Machine learning-based event recognition in SiFi Compton camera imaging for proton therapy monitoring

Objective. Online monitoring of dose distribution in proton therapy is currently being investigated with the detection of prompt gamma (PG) radiation emitted from a patient during irradiation. The SiPM and scintillation Fiber based Compton Camera (SiFi-CC) setup is being developed for this aim. Approach. A machine learning approach to recognize Compton events is proposed, reconstructing the PG emission profile during proton therapy. The proposed method was verified on pseudo-data generated by a Geant4 simulation for a single proton beam impinging on a polymethyl methacrylate (PMMA) phantom. Three different models including the boosted decision tree (BDT), multilayer perception (MLP) neural network, and k-nearest neighbour (k-NN) were trained using 10-fold cross-validation and then their performances were assessed using the receiver operating characteristic (ROI) curves. Subsequently, after event selection by the most robust model, a software based on the List-Mode Maximum Likelihood Estimation Maximization (LM-MLEM) algorithm was applied for the reconstruction of the PG emission distribution profile. Main results. It was demonstrated that the BDT model excels in signal/background separation compared to the other two. Furthermore, the reconstructed PG vertex distribution after event selection showed a significant improvement in distal falloff position determination. Significance. A highly satisfactory agreement between the reconstructed distal edge position and that of the simulated Compton events was achieved. It was also shown that a position resolution of 3.5 mm full width at half maximum (FWHM) in distal edge position determination is feasible with the proposed setup.

Unsupervised ground filtering of airborne-based 3D meshes using a robust cloth simulation

• This work proposes a primitive-oriented cloth simulation method for ground filtering of large-scale 3D meshes from airborne platforms. • This method is not affected by low and unbalanced density distribution of mesh vertices. • This method does not require a coherent noise-free 3D mesh. • This work presents a labeled dataset for ground filtering on 3D meshes. Airborne platforms have been improved in the past decade to provide geographic information systems (GISs) with large-scale 3D geographical information. Objectification of such information organized in meshes is a significant challenge for 3D GISs. The ground filtering of 3D meshes is a key step in meeting this challenge, however, its accuracy is highly affected by negative blunders and unbalanced vertex density. This paper proposes a novel method for differentiating ground geometric primitives from realistic 3D meshes based on a cloth simulation filter. Within the method, the fall of a piece of cloth is simulated on a flipped 3D mesh, and the stationary shape of the cloth is considered to be the fitted ground. Utilizing the spatial continuity of meshes, a collision detection based on bounding volume hierarchy is introduced, making the results independent of vertex density. Further, a collision correction based on the scan line and ray casting is proposed to make it applicable to data with negative blunders. The method is assessed quantitatively and visually over several datasets with different vertex densities, scenes, and noise distributions. Results demonstrate that it is a robust method suitable for different landscapes and is not impacted by vertex density and noise.

Average scattering entropy for periodic, aperiodic and random distribution of vertices in simple quantum graphs

This work deals with the average scattering entropy of quantum graphs. We explore this concept in several distinct scenarios that involve periodic, aperiodic and random distribution of vertices of distinct degrees. In particular, we compare distinct situations to see how they behave as we change the arrangements of vertices and the topology and geometry of the proposed structures. The results show that the average scattering entropy may depend on the number of vertices, and on the topological and geometrical disposition of vertices and edges of the quantum graph. In this sense, it can be seen as another tool to be used to explore geometric and topological effects of current interest for quantum systems.

Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations

We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric setting. The weak formulations are presented in terms of the generating curves of the axisymmetric surfaces. The proposed numerical methods are based on piecewise linear parametric finite elements. The constructed fully practical schemes satisfy the conservation of the enclosed volume. In addition, we prove the unconditional stability and consider the distribution of vertices for the discretized schemes. The introduced methods are implicit and the resulting nonlinear systems of equations can be solved very efficiently and accurately via the Newton's iterative method. Numerical results are presented to show the accuracy and efficiency of the introduced schemes for computing the considered axisymmetric geometric flows.

Online Learning Bipartite Matching with Non-stationary Distributions

Online bipartite matching has attracted wide interest since it can successfully model the popular online car-hailing problem and sharing economy. Existing works consider this problem under either adversary setting or i.i.d. setting. The former is too pessimistic to improve the performance in the general case; the latter is too optimistic to deal with the varying distribution of vertices. In this article, we initiate the study of the non-stationary online bipartite matching problem, which allows the distribution of vertices to vary with time and is more practical. We divide the non-stationary online bipartite matching problem into two subproblems, the matching problem and the selecting problem, and solve them individually. Combining Batch algorithms and deep Q-learning networks, we first construct a candidate algorithm set to solve the matching problem. For the selecting problem, we use a classical online learning algorithm, Exp3, as a selector algorithm and derive a theoretical bound. We further propose CDUCB as a selector algorithm by integrating distribution change detection into UCB. Rigorous theoretical analysis demonstrates that the performance of our proposed algorithms is no worse than that of any candidate algorithms in terms of competitive ratio. Finally, extensive experiments show that our proposed algorithms have much higher performance for the non-stationary online bipartite matching problem comparing to the state-of-the-art.

Symmetric Random Walks on Three Half-Cubes

We study random walks on the vertices of three non-isomorphic halfcubes obtained from a cube by a plane cut through its center. Starting from a particular vertex (called the origin), at each step a particle moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the means and the standard deviations of the number of steps needed to: (1) return to origin, (2) visit all vertices, and (3) return to origin after visiting all vertices. We also find (4) the probability distribution of the last vertex visited

Depth first exploration of a configuration model

We introduce an algorithm that constructs a random graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that the renormalized contour process of the Depth First Search Tree has a deterministic limiting profile that we identify. The proof goes through a detailed analysis of the evolution of the empirical degree distribution of unexplored vertices. This evolution is driven by an infinite system of differential equations which has a unique and explicit solution. As a byproduct, we deduce the existence of a macroscopic simple path and get a lower bound on its length.

Digital and automatic design of free-form single-layer grid structures

Grid shells have been widely used in various long-span public buildings, and many of them are defined over free-form surfaces with complex boundaries. This emphasizes the importance of general and digitalised grid generation and optimization methods in the initial design stage to achieve visually sound grid shells. In this paper, a framework is presented for the development of a digital tool and to generate regular and fluent grids for structural design over free-form surfaces, especially those with complex boundaries. Both triangular and quadrilateral grid generation are addressed. To generate regular and fluent grids for free-form surfaces, a simple yet practical framework is proposed based on a spring-mass model. Firstly, an initial casual quadrilateral grid is tiled on the surface based on surface discretization and mesh parameterization. Secondly, the distribution of the initial grid vertices is adjusted by a dynamic relaxation procedure, assuming the grid as a spring-mass system. Thirdly, the grid vertices corresponding to the adjusted particles in the equilibrium state are then reconnected to produce a grid with a predefined pattern (triangular or quadrilateral). Finally, the generated grid is relaxed with the spring-mass model, alongside additional geometric operations including grid size adjustment and filtering techniques, to further improve the grid regularity and fluency. As part of its contribution, this paper also broadens the application scope of the fluency index, which can be used to quantitatively evaluate the suitability of a given triangular or quadrilateral grid for architectural and structural expression. Examples are presented and show that the proposed framework is effective for the triangular and quadrilateral grid generation over various surfaces and to optimize the resulted grids along complex boundaries. The method proposed can be useful for rapid design and performance evaluation of free-form grid structures.