Sequence Of Vertices Research Articles

Persistent Mayer Dirac

Topological data analysis (TDA) has made significant progress in developing a new class of fundamental operators known as the Dirac operator, particularly in topological signals and molecular representations. However, the current approaches being used are based on the classical case of chain complexes. The present study establishes Mayer Dirac operators based on N-chain complexes. These operators interconnect an alternating sequence of Mayer Laplacian operators, providing a generalization of the classical result . Furthermore, the research presents an explicit formulation of the Laplacian for N-chain complexes induced by vertex sequences on a finite set. Weighted versions of Mayer Laplacian and Dirac operators are introduced to expand the scope and improve applicability, showcasing their effectiveness in capturing physical attributes in various practical scenarios. The study presents a generalized version for factorizing Laplacian operators as an operator's product and its 'adjoint'. Additionally, the proposed persistent Mayer Dirac operators and extensions are applied to biological and chemical domains, particularly in the analysis of molecular structures. The study also highlights the potential applications of persistent Mayer Dirac operators in data science.

Engineering Rings in Network Materials

AbstractNetwork materials can be crystalline or amorphous solids, or even liquids, where typically directional interactions link the building blocks together, resulting in a physical representation of a mathematical object, called a graph or equivalently a network. Rings, which correspond to a cyclic path in the underlying network, consisting of a sequence of vertices and edges, are medium‐range structural motifs in the physical space. This Perspective presents an overview of recent studies, which showcase the importance of rings in the emergence of crystalline order as well as in phase transitions between two liquid phases for certain network materials, comprised of colloidal or molecular building blocks. These studies demonstrate how the selection of ring sizes can be exploited for programming self‐assembly of colloidal open crystals with an underlying network and elucidate rings as a vehicle for entanglement that distinguishes the two liquid phases of different densities involved in liquid–liquid phase transitions of network liquids with local tetrahedral order. In this context, an outlook is presented for engineering rings in network materials composed of colloidal and molecular building blocks, with implications also for metal‐organic frameworks, which have been extensively studied as porous crystals, but, more recently, as network‐forming liquids and glasses as well.

WalkGAN: Network Representation Learning With Sequence-Based Generative Adversarial Networks.

Network representation learning, also known as network embedding, aims to learn the low-dimensional representations of vertices while capturing and preserving the network structure. For real-world networks, the edges that represent some important relationships between the vertices of a network may be missed and may result in degenerated performance. The existing methods usually treat missing edges as negative samples, thereby ignoring the true connections between two vertices in a network. To capture the true network structure effectively, we propose a novel network representation learning method called WalkGAN, where random walk scheme and generative adversarial networks (GAN) are incorporated into a network embedding framework. Specifically, WalkGAN leverages GAN to generate the synthetic sequences of the vertices that sufficiently simulate random walk on a network and further learn vertex representations from these vertex sequences. Thus, the unobserved links between the vertices are inferred with high probability instead of treating them as nonexistence. Experimental results on the benchmark network datasets demonstrate that WalkGAN achieves significant performance improvements for vertex classification, link prediction, and visualization tasks.

The median function of a block graph: Axiomatic characterizations

A profile π=(x1,x2,…,xk) of length k in a connected graph G is a sequence of vertices in G with possible repetitions of vertices. A median x of a profile π in G is a vertex that minimizes the remoteness value, that is, the sum of the distances from x to the elements xi in π is minimized. The median function output the set of medians (denoted as Med(π)) of π, for every profile π in G. It is one of the basic models for the location of a desirable facility in a network. The median function on graphs is well studied. For instance, it has been characterized axiomatically by simple axioms on trees, hypercubes, median graphs, cocktail-party graphs, and complete graphs minus a matching. In this paper, we study the median sets and median function of block graphs, an immediate generalization of trees. We determine the median sets of all types of profiles and we obtain two sets of axiomatic characterizations of the median function on block graphs.

Grundy Total Hop Dominating Sequences in Graphs

Let G = (V (G), E(G)) be an undirected graph with γ(C) ̸= 1 for each component C of G. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Sˆ ={v1, v2, . . . , vk}. Then S is a legal open hop neighborhood sequence if N2G(vi) \Si−1j=1 N2G(vj ) ̸= ∅for every i ∈ {2, . . . , k}. If, in addition, Sˆ is a total hop dominating set of G, then S is a Grundy total hop dominating sequence. The maximum length of a Grundy total hop dominating sequence in a graph G, denoted by γth gr(G), is the Grundy total hop domination number of G. In this paper, we show that the Grundy total hop domination number of a graph G is between the total hop domination number and twice the Grundy hop domination number of G. Moreover, determine values or bounds of the Grundy total hop domination number of some graphs.

Path description for q-characters of fundamental modules in type C

In this paper, we investigate the behavior of monomials in the q-characters of the fundamental modules over a quantum affine algebra of untwisted type C. As a result, we give simple closed formulae for the q-characters of the fundamental modules in terms of sequences of vertices in R 2 , so-called paths, with an admissible condition. This may be viewed as a type C analog of the path description of q-characters in types A and B due to Mukhin–Young.

Boundary delineation in transrectal ultrasound images for region of interest of prostate

Accurate and robust prostate segmentation in transrectal ultrasound (TRUS) images is of great interest for ultrasound-guided brachytherapy for prostate cancer. However, the current practice of manual segmentation is difficult, time-consuming, and prone to errors. To overcome these challenges, we developed an accurate prostate segmentation framework (A-ProSeg) for TRUS images. The proposed segmentation method includes three innovation steps: (1) acquiring the sequence of vertices by using an improved polygonal segment-based method with a small number of radiologist-defined seed points as prior points; (2) establishing an optimal machine learning-based method by using the improved evolutionary neural network; and (3) obtaining smooth contours of the prostate region of interest using the optimized machine learning-based method. The proposed method was evaluated on 266 patients who underwent prostate cancer brachytherapy. The proposed method achieved a high performance against the ground truth with a Dice similarity coefficient of 96.2% ± 2.4%, a Jaccard similarity coefficient of 94.4% ± 3.3%, and an accuracy of 95.7% ± 2.7%; these values are all higher than those obtained using state-of-the-art methods. A sensitivity evaluation on different noise levels demonstrated that our method achieved high robustness against changes in image quality. Meanwhile, an ablation study was performed, and the significance of all the key components of the proposed method was demonstrated.

An Independent Cascade Model of Graph Burning

Graph burning was introduced to simulate the spreading of news/information/rumors in social networks. The symmetric undirected graph is considered here. That is, vertex u can spread the information to vertex v, and symmetrically vertex v can also spread information to vertex u. When it is modeled as a graph burning process, a vertex can be set on fire directly or burned by its neighbor. Thus, the task is to find the minimum sequence of vertices chosen as sources of fire to burn the entire graph. This problem has been proved to be NP-hard. In this paper, from a new perspective, we introduce a generalized model called the Independent Cascade Graph Burning model, where a vertex v can be burned by one of its burning neighbors u only if the influence that u gives to v is larger than a given threshold β≥0. We determine the graph burning number with this new Independent Cascade Graph Burning model for several graphs and operation graphs and also discuss its upper and lower bounds.

Joint semantic–geometric learning for polygonal building segmentation from high-resolution remote sensing images

As a fundamental task for geographical information updating, 3D city modeling, and other critical applications, the automatic extraction of building footprints from high-resolution remote sensing images has been substantially explored and received increasing attention over recent years. Among different types of building extraction methods, the polygonal segmentation methods produce vector building polygons that are in a more realistic format compared with those obtained from pixel-wise semantic labeling and contour-based methods. However, existing polygonal building segmentation methods usually require a perfect segmentation map and a complex post-processing procedure to guarantee the polygonization quality, or produce inaccurate vertex prediction results that suffer from wrong vertex sequence, self-intersections, fixed vertex quantity, etc. In our previous work, we have proposed a method for polygonal building segmentation from remote sensing images that addresses the above limitations of existing methods. In this paper, we propose PolyCity, which further extends and improves our previous work in terms of the application scenario, methodology design, and experimental results. Our proposed PolyCity contains the following three components: (1) a pixel-wise multi-task network for learning the semantic and geometric information via three tasks, i.e., building segmentation, boundary prediction, and edge orientation prediction; (2) a simple but effective vertex selection module (VSM), which effectively bridges the gap between pixel-wise and graph-based models via transforming the segmentation map into valid polygon vertices; (3) a graph-based vertex refinement network (VRN) for automatically adjusting the coordinates of VSM-generated valid polygon vertices, producing the final building polygons with more precise vertices. Results on three large-scale building extraction datasets demonstrate that our proposed PolyCity generates vector building footprints with more accurate vertices, edges, shapes, etc., achieving significant vertex score improvements while maintaining high segmentation and boundary scores compared with the current state-of-the-art. The code of PolyCity will be released at https://github.com/liweijia/polycity.

Inferring routing preferences from user-generated trajectories using a compression criterion

The optimal path between two vertices in a graph depends on the optimization objective, which is often defined as a weighted sum of multiple criteria. When integrating two criteria, their relative importance is expressed with a balance factor α. We present a new approach for inferring α from trajectories. The core of our approach is a compression algorithm that requires a graph G representing a transportation network, two edge costs modeling routing criteria, and a path P in G representing the trajectory. It yields a minimum subsequence S of the sequence of vertices of P and a balance factor α, such that the path P can be fully reconstructed from S, G, its edge costs, and α. By minimizing the size of S over α, we learn the balance factor that corresponds best to the user's routing preferences. In an evaluation with crowd-sourced cycling trajectories, we weigh the usage of official signposted cycle routes against other routes. More than 50% of the trajectories can be segmented into five optimal sub-paths or less. Almost 40% of the trajectories indicate that the cyclist is willing to take a detour of 50% over the geodesic shortest path to use an official cycle path.

Graphs with equal Grundy domination and independence number

The Grundy domination number, γgr(G), of a graph G is the maximum length of a sequence (v1,v2,…,vk) of vertices in G such that for every i∈{2,…,k}, the closed neighborhood N[vi] contains a vertex that does not belong to any closed neighborhood N[vj], where j<i. It is well known that the Grundy domination number of any graph G is greater than or equal to the upper domination number Γ(G), which is in turn greater than or equal to the independence number α(G). In this paper, we initiate the study of the class of graphs G with Γ(G)=γgr(G) and its subclass consisting of graphs G with α(G)=γgr(G). We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs G with Γ(G)=γgr(G) and present large families of such graphs.

Sequential locality of graphs and its hypothesis testing

The adjacency matrix is the most fundamental and intuitive object in graph analysis that is useful not only mathematically but also for visualizing the structures of graphs. Because the appearance of an adjacency matrix is critically affected by the ordering of rows and columns, or vertex ordering, statistical assessment of graphs together with their vertex sequences is important in identifying the characteristic structures of graphs. In this paper, we propose a hypothesis testing framework that assesses how locally vertices are connected to each other along a specified vertex sequence, which provides a statistical foundation for an optimization problem called envelope reduction or minimum linear arrangement. The proposed tests are particularly suitable for moderately small data and formulated based on a combinatorial approach and a block model with intrinsic vertex ordering.

Order based algorithms for the core maintenance problem on edge-weighted graphs

The cohesive subgraph k-core is a maximal connected subgraph with the minimum degree δ≥k of a simple graph, where integer k≥0. Define the core number of a vertex w as the maximum k such that w is contained in a k-core. The core decomposition problem which is calculating the core numbers of all vertices in static graphs, and the core maintenance problem which is updating the core numbers in dynamic graphs are our main concern. Although, core numbers can be updated by the core decomposition algorithms, only a small part of vertices' core numbers have changed after the change of a graph. Thus, it is necessary to update core numbers locally to reduce the cost. In this paper, we study the core maintenance problem on edge-weighted graphs by using the vertex sequence k-order which is ordered by the order that the core decomposition algorithm removes vertices. We design the core maintenance algorithms for inserting one edge at a time and the method of updating the k-order, which reduce the searching range and the time cost evidently. For the removing case, we use the existing subcore algorithm to do the core maintenance and modify it with the method of updating k-order we design. Finally, we do extensive experiments to evaluate the effectiveness and the efficiency of our algorithms, which shows that the order based algorithm has a better performance than the existing for the core maintenance on edge-weighted graphs.

Grundy Hop Domination in Graphs

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. Let S = (v1, v2, · · · , vk) be a sequence of distinct vertices of G and let Sˆ = {v1, v2, . . . , vk}. Then S is a legal closed hop neighborhood sequence of G if N2 G[vi]\∪i−1j=1N 2 G[vj ] ̸= ∅ for each i ∈ {2, · · · , k}. If, in addition, Sˆ is a hop dominating set of G, then S is called a Grundy hop dominating sequence.The maximum length of a Grundy hop dominating sequence in a graph G, denoted by γ hgr(G), is called the Grundy hop domination number of G. In this paper, we determine some (extreme) values for the Grundy hop domination number. It is pointed out that the Grundy hop domination number is at least equal to the hop domination. Bounds for the Grundy hop domination numbersof some graphs resulting from some binary operations of two graphs are also obtained.

Space-efficient random walks on streaming graphs

Graphs in many applications, such as social networks and IoT, are inherently streaming, involving continuous additions and deletions of vertices and edges at high rates. Constructing random walks in a graph, i.e., sequences of vertices selected with a specific probability distribution, is a prominent task in many of these graph applications as well as machine learning (ML) on graph-structured data. In a streaming scenario, random walks need to constantly keep up with the graph updates to avoid stale walks and thus, performance degradation in the downstream tasks. We present Wharf, a system that efficiently stores and updates random walks on streaming graphs. It avoids a potential size explosion by maintaining a compressed, high-throughput, and low-latency data structure. It achieves (i) the succinct representation by coupling compressed purely functional binary trees and pairing functions for storing the walks, and (ii) efficient walk updates by effectively pruning the walk search space. We evaluate Wharf, with real and synthetic graphs, in terms of throughput and latency when updating random walks. The results show the high superiority of Wharf over inverted index- and tree-based baselines.

Characterization of 2-arc-transitive partial cubes

Let Γ be a graph with its automorphism group G and s(⩾0) an integer. We call a vertex sequence (v0,v1,…,vs) of length s+1 of Γ an s-arc if two consecutive vertices are adjacent and vi≠vi+2 for 0⩽i⩽s−2. Γ is called s-arc-transitive if G acts on its s-arc set transitively. If Γ is s-arc-transitive, but not (s+1)-arc-transitive, we call it an s-transitive graph. Γ is a partial cube if Γ can be embedded into a hypercube isometrically. In this paper, we characterized non-trivial regular 2-arc-transitive partial cubes as three classes of graphs: hypercubes, doubled Odd graphs and even cycles. As a corollary, we prove that there exist no regular s-transitive partial cubes for s⩾4 and characterized regular s-transitive partial cubes for s=2 or 3.

Structural parameterizations of Tracking Paths problem

Given a graph G with source and destination vertices s,t∈V(G) respectively, Tracking Paths asks for a minimum set of vertices T⊆V(G), such that the sequence of vertices restricted to T encountered in each simple path from s to t is unique. The problem was proven NP-hard [3] and was found to admit a quadratic kernel when parameterized by the size of the desired solution [8]. Following recent trends, for the first time, we study Tracking Paths with respect to structural parameters of the input graph, parameters that measure how far the input graph is, from an easy instance. We prove that Tracking Paths admits fixed-parameter tractable (FPT) algorithms when parameterized by the following parameters: (i) size of a 2-dominating set, (ii) size of cluster vertex deletion set, and (iii) size of split deletion set. En route we also look at Vertex Cover parameterized by the size of edge clique cover and show it fixed-parameter tractable.

Improving Network Representation Learning via Dynamic Random Walk, Self-Attention and Vertex Attributes-Driven Laplacian Space Optimization.

Network data analysis is a crucial method for mining complicated object interactions. In recent years, random walk and neural-language-model-based network representation learning (NRL) approaches have been widely used for network data analysis. However, these NRL approaches suffer from the following deficiencies: firstly, because the random walk procedure is based on symmetric node similarity and fixed probability distribution, the sampled vertices’ sequences may lose local community structure information; secondly, because the feature extraction capacity of the shallow neural language model is limited, they can only extract the local structural features of networks; and thirdly, these approaches require specially designed mechanisms for different downstream tasks to integrate vertex attributes of various types. We conducted an in-depth investigation to address the aforementioned issues and propose a novel general NRL framework called dynamic structure and vertex attribute fusion network embedding, which firstly defines an asymmetric similarity and h-hop dynamic random walk strategy to guide the random walk process to preserve the network’s local community structure in walked vertex sequences. Next, we train a self-attention-based sequence prediction model on the walked vertex sequences to simultaneously learn the vertices’ local and global structural features. Finally, we introduce an attributes-driven Laplacian space optimization to converge the process of structural feature extraction and attribute feature extraction. The proposed approach is exhaustively evaluated by means of node visualization and classification on multiple benchmark datasets, and achieves superior results compared to baseline approaches.

On the length of L-Grundy sequences

An L-sequence of a graph G is a sequence of distinct vertices S=(v1,…,vk) such that N[vi]∖∪j=1i−1N(vj)≠0̸. The length of a longest L-sequence is called the L-Grundy domination number, denoted γgrL(G). In this paper, we prove γgrL(G)≤n(G)−δ(G)+1, which was conjectured by Brešar, Gologranc, Henning, and Kos. We also prove some initial results about characteristics of n-vertex graphs satisfying γgrL(G)=n.

Constant amortized time enumeration of Eulerian trails

In this paper, we consider enumeration problems for edge-distinct and vertex-distinct Eulerian trails. Two Eulerian trails are said to be edge-distinct if the edge sequences are not identical, and they are said to be vertex-distinct if the vertex sequences are not identical. To solve these problems, we propose optimal enumeration algorithms that run in O(N+m) total time, where N is the number of solutions and m is the number of edges in an input connected graph. The proposed algorithms are based on the reverse search technique introduced by [Avis and Fukuda, DAM 1996], and the push-out amortization technique introduced by [Uno, WADS 2015].