Tree Graph Research Articles

Research and Application of Power Grid Data Anomaly Tracing Method Based on Time Series Correlation

Abstract Power grid data is a “barometer” that reflects the operational trend of the power grid system, so conducting a traceability study on abnormal data generated by the power grid system is crucial to maintaining the stable operation of the power grid system and preventing further malfunctions. This article proposes a maximum time series correlation ring tracing algorithm based on time series correlation: firstly, for a large amount of time series data collected from measurement points in the power grid system, a correlation coefficient matrix is calculated, and then a time series correlation graph is constructed using graph theory knowledge. The Kruskal algorithm is used to search for the longest spanning tree in the time series correlation graph. Finally, based on the spanning tree, the breadth-first search (BFS) algorithm is further used to obtain the maximum time series correlation ring[1]. By using the time series correlation of each node within the ring and the physical topological relationship between nodes, the abnormal data can be traced back. Distinguish whether the generation of abnormal data is due to a single point failure or system failure. Most of the existing fault tracing techniques are based on machine learning algorithms, which are difficult to widely apply in high-dimensional time series data due to their high complexity.Through experiments on real power grid data, the results verify the effectiveness of this method in anomaly tracing of high-dimensional time series data. Through comparative experiments, this method is superior to machine learning model algorithms in terms of efficiency and stability. At the same time, this method does not require additional calculations of the statistical distribution of the samples to be tested, thus greatly saving computational costs.

Oscillatory differentiation dynamics fundamentally restricts the resolution of pseudotime reconstruction algorithms.

The challenge to understand differentiation and cell lineages in development has resulted in many bioinformatics software tools, notably those working with gene expression data obtained via single-cell RNA sequencing obtained at snapshots in time. Reconstruction methods for trajectories often proceed by dimension reduction, data clustering and then computation of a tree graph in which edges indicate closely related clusters. Cell lineages can then be deduced by following paths through the tree. In the case of multi-potent cells undergoing differentiation, this trajectory reconstruction involves the reconstruction of multiple distinct lineages corresponding to commitment to each of a set of distinct fates. Recent work suggests that there may be cases in which the cell differentiation process involves trajectories that explore, in a dynamic and oscillatory fashion, propensity to differentiate into a number of possible cell fates before commitment finally occurs. Here, we show theoretically that the presence of such oscillations provides intrinsic constraints on the quality and resolution of the trajectory reconstruction process, even for idealized noise-free data. These constraints point to inherent common limitations of current methodologies and serve both to provide additional challenge in the development of software tools and also may help to understand features observed in recent experiments.

The GHP scaling limit of uniform spanning trees of dense graphs

AbstractWe consider dense graph sequences that converge to a connected graphon and prove that the GHP scaling limit of their uniform spanning trees (USTs) is Aldous' Brownian CRT. Furthermore, we are able to extract the precise scaling constant from the limiting graphon. As an example, we can apply this to the scaling limit of the USTs of the Erdös–Rényi sequence for any fixed , and sequences of dense expanders. A consequence of GHP convergence is that several associated quantities of the spanning trees also converge, such as the height, diameter and law of a simple random walk.

On the Enhanced Power Graph of a Semigroup

The enhanced power graph [Formula: see text] of a semigroup [Formula: see text] is a simple graph whose vertex set is [Formula: see text] and two vertices [Formula: see text], [Formula: see text] are adjacent if and only if [Formula: see text], [Formula: see text] for some [Formula: see text], where [Formula: see text] is the subsemigroup generated by [Formula: see text]. In this paper, we first describe the structure of [Formula: see text] for an arbitrary semigroup [Formula: see text], and then discuss the connectedness of [Formula: see text]. Further, we characterize the semigroup [Formula: see text] in the cases when [Formula: see text] is separately a complete, bipartite, regular, tree and null graph. The planarity, together with the minimum degree and independence number, of [Formula: see text] is also investigated. The chromatic number of a spanning subgraph, i.e., the cyclic graph, of [Formula: see text] is proved to be countable. In the final part of this paper, we construct an example of a semigroup [Formula: see text] such that the chromatic number of [Formula: see text] need not be countable.

Logarithms of Catalan Generating Functions: A Combinatorial Approach

We analyze the combinatorics behind the operation of taking the logarithm of the generating function $G_k$ for $k^\text{th}$ generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of tree graphs. Using explicit bijections, we are able to recover known closed expressions for the coefficients of $\log G_k$ by purely combinatorial means of enumeration. The non-algebraic proof easily generalizes to higher powers $\log^a G_k$, $a\geq 2$.

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

Complexity of Some Types of Cyclic Snake Graphs

The number of spanning trees in graphs (networks) is a crucial invariant, and it is also an important measure of the reliability of a network. Spanning trees are special subgraphs of a graph that have several important properties. First, <I>T</I> must span <I>G</I>, which means it must contain every vertex in graph <I>G</I>, if <I>T</I> is a spanning tree of graph <I>G</I>. <I>T</I> needs to be a subgraph of <I>G</I>, second. Stated differently, any edge present in <I>T</I> needs to be present in <I>G </I>as well. Third, <I>G</I> is the same as <I>T</I> if each edge in <I>T</I> is likewise present in <I>G</I>. In path-finding algorithms like Dijkstra's shortest path algorithm and A* search algorithm, spanning trees play an essential part. In those approaches, spanning trees are computed as component components. Protocols for network routing also take advantage of it. In numerous techniques and applications, minimum spanning trees are highly beneficial. Computer networks, electrical grids, and water networks all frequently use them. They are also utilized in significant algorithms like the min-cut max-flow algorithm and in graph issues like the travelling salesperson problem. In this paper, we use matrix analysis and linear algebra techniques to obtain simple formulas for the number of spanning trees of certain kinds of cyclic snake graphs.

Recognizing LBFS trees of bipartite graphs

The graph searches Breadth First Search (BFS) and Depth First Search (DFS) and the spanning trees constructed by them are some of the most basic concepts in algorithmic graph theory. BFS trees are first-in trees, i.e., every vertex is connected to its first visited neighbor. DFS trees are last-in trees, i.e., every vertex is connected to the last visited neighbor before it. The problem whether a given spanning tree can be the first-in tree or last-in tree of a graph search ordering was introduced in the 1980s and has been studied for several graph searches and graph classes. Here, we consider the problem of deciding whether a given spanning tree of a bipartite graph can be a first-in tree or a last-in tree of the Lexicographic Breadth First Search (LBFS), a special variant of BFS that is commonly used in graph algorithms. We show that the recognition of both first-in trees and last-in trees of LBFS is NP-hard even if the start vertex of the search ordering is fixed and the height of the tree is four. We prove that the bound on the height is tight (unless P=NP) by showing that for all spanning trees of bipartite graphs with height smaller than four we can solve both search tree recognition problems of LBFS in polynomial time. Finally, we give a linear-time algorithm that solves both problems for chordal bipartite graphs and fixed start vertices.

Closeness centrality reconstruction of tree graphs

Closeness centrality reconstruction of tree graphs

StructCoder: Structure-Aware Transformer for Code Generation

There has been a recent surge of interest in automating software engineering tasks using deep learning. This article addresses the problem of code generation, in which the goal is to generate target code given source code in a different language or a natural language description. Most state-of-the-art deep learning models for code generation use training strategies primarily designed for natural language. However, understanding and generating code requires a more rigorous comprehension of the code syntax and semantics. With this motivation, we develop an encoder-decoder Transformer model in which both the encoder and decoder are explicitly trained to recognize the syntax and dataflow in the source and target codes, respectively. We not only make the encoder structure aware by leveraging the source code’s syntax tree and dataflow graph, but we also support the decoder in preserving the syntax and dataflow of the target code by introducing two novel auxiliary tasks: Abstract Syntax Tree (AST) path prediction and dataflow prediction. To the best of our knowledge, this is the first work to introduce a structure-aware Transformer decoder that models both syntax and dataflow to enhance the quality of generated code. The proposed StructCoder model achieves state-of-the-art performance on code translation and text-to-code generation tasks in the CodeXGLUE benchmark and improves over baselines of similar size on the APPS code generation benchmark. Our code is publicly available at https://github.com/reddy-lab-code-research/StructCoder/ .

A multilayered urban tree dataset of point clouds, quantitative structure and graph models

The significance of urban trees in promoting human health and well-being has been amplified by urbanization and the climate change effects. Simultaneously, advancements in remote sensing techniques have enhanced the opportunities for studying urban trees. The TreeML-Data has been compiled to support these efforts. It consists of labelled point clouds of 40 scanning projects of streets in Munich, 3,755 leaf-off (scans in winter) point clouds of individual trees, quantitative structure models (QSM), tree structure measurements, and tree graph structure models of these trees. The dataset offers valuable data for generating and evaluating models in various scientific disciplines, which include remote sensing, computer vision, machine learning, urban forestry, urban ecosystem, green architecture, and graph analysis. To ensure its quality, the tree structure measurements and QSM have been crosschecked. For instance, the tree diameter at breast height (DBH) in the sample dataset exhibits a deviation of approximately 1.5 cm (4.3%) when compared to manual measurements. In conclusion, the quality checks confirm its reliability for subsequent studies when compared to manual measurements.

On [formula omitted]-stable locally checkable problems parameterized by mim-width

In this paper we continue the study of locally checkable problems under the framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on graphs of bounded mim-width. We study which restrictions on a locally checkable problem are necessary in order to be able to solve it efficiently on graphs of bounded mim-width. To this end, we introduce the concept of d-stability of a check function. The related locally checkable problems contain large classes of problems, among which we can mention, for example, LCVP problems. We give an algorithm showing that these problems are XP when parameterized by the mim-width of a given binary decomposition tree of the input graph , that is, that they can be solved in polynomial time given a binary decomposition tree of bounded mim-width. We explore the relation between d-stable locally checkable problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke, 2022), and show that both frameworks model the same family of problems. We include a list of concrete examples of d-stable locally checkable problems whose complexity on graphs of bounded mim-width was open so far.

极大外平面图中树的 Anti-Ramsey 数

极大外平面图中树的 Anti-Ramsey 数

A Language and Its Dimensions: Intrinsic Dimensions of Language Fractal Structures

The present paper introduces a novel object of study, a language fractal structure; we hypothesize that a set of embeddings of all n‐grams of a natural language constitutes a representative sample of this fractal set. (We use the term Hailonakea to refer to the sum total of all language fractal structures, over all n). The paper estimates intrinsic (genuine) dimensions of language fractal structures for the Russian and English languages. To this end, we employ methods based on (1) topological data analysis and (2) a minimum spanning tree of a data graph for a cloud of points considered (Steele theorem). For both languages, for all n, the intrinsic dimensions appear to be noninteger values (typical for fractal sets), close to 9 for both of the Russian and English language.

Ramsey numbers for a large tree versus multiple copies of complete graphs of different sizes

Ramsey numbers for a large tree versus multiple copies of complete graphs of different sizes

A Novel Personalized Learning Path Mining Based on Steiner Tree and Knowledge Graph

Modern society has accumulated great amounts of knowledge since human start learning and communicating. With knowledge’s continuously thriving and getting refined, knowledge itself has developed into a complex system with a huge number of content and structure features. Together with the development of science and technology, the ways of people learning have been changed greatly. Under the pressure of high-paced life and workplace competition, more and more people are in demand of personalized learning paths, in order to achieve their learning target as fast as possible. A personized and scientific learning path is the key to effective learning grogram. However, studies on algorithms of optimal learning path discovery are not comprehensively developed, especially on its performance and application. Therefore, in this article, we propose a new algorithm, Portfolio ST, which is improved from Steiner Tree, for personalized and optimal learning path mining. At first, this paper constructs the knowledge graph of finance, then builds learner models by profiling learners’ owned and aimed knowledge based on the knowledge network. At last, according to the knowledge structure of a person’s learner model, the most personized and shortest learning path is calculated by the Portfolio ST. After the experiment compared with two baseline algorithms, the Portfolio ST algorithm is proved to have the best performance and scalability, by changing owned/aimed/overall knowledge respectively, which is also the best algorithm of learning path mining that can achieve the global optimal among these three algorithms.

Modular chromatic number on inflated graphs of some tree graphs

Modular chromatic number on inflated graphs of some tree graphs

Drought responses in Coffea arabica as affected by genotype and phenophase. I – leaf distribution and branching

Summary In Coffea arabica, there is a small genetic distance between wild and bred genotypes. However, coffee genotypes express differential acclimation to multiple drought cycles, allowing them to successfully deal with water-limiting conditions. We hypothesized that bred coffee cultivars have a plant structure less sensitive to drought than wild genotypes. Plant and leaf architecture were analyzed over the coffee strata of two cultivars (Iapar 59 and Catuaí 99) and two wild Ethiopia accessions (‘E083’ and ‘E027’) grown under rainfed conditions and irrigation. During two consecutive productive years, evaluations were taken at leaf and berry expansion (BE1 and BE2) and harvest (BH1 and BH2) phenophases. The plant canopy was divided into up to four strata of 40 cm of thickness. Topological and geometric coding of coffee trees was performed in three botanical scales – metamers, branches, and plants in multiscale tree graphs (MTGs), following the VPlants modeling platform. Leaf and branch area per plant increased with tree structure development, being always significantly higher in irrigated than in rainfed plants over all phenophases. The individual leaf area was the least sensitive to water regime in Catuaí 99, while the 2nd order axis elevation – angle in relation to horizontal plane, ranging from 0° to 90° – of bred cultivars was less sensitive to drought than in ‘E083’. This finding partially corroborated our hypothesis that orchestrated reprograming of leaf/branch responses over the vertical plant profile were less sensitive to water availability in cultivars than in wild accessions. Leaves of 2nd to 4th-order branching were roughly plagiophile, while the 1st-order leaves were classified as extremophiles. When the coffee leaves were planophile, irrespective of genotype, this pattern was found at the lowest, 1st plant stratum, and the newest developed 4th stratum. Such responses were not obligatorily related to water regime, similar to branch elevation – with exception of ‘E083’, very sensitive to drought. Taken together, our data suggest that the leaf and branch elevations in C. arabica were more influenced by light distribution through the canopy profile – i.e., self-shading – than by water availability.

A global model-agnostic rule-based XAI method based on Parameterized Event Primitives for time series classifiers.

Time series classification is a challenging research area where machine learning and deep learning techniques have shown remarkable performance. However, often, these are seen as black boxes due to their minimal interpretability. On the one hand, there is a plethora of eXplainable AI (XAI) methods designed to elucidate the functioning of models trained on image and tabular data. On the other hand, adapting these methods to explain deep learning-based time series classifiers may not be straightforward due to the temporal nature of time series data. This research proposes a novel global post-hoc explainable method for unearthing the key time steps behind the inferences made by deep learning-based time series classifiers. This novel approach generates a decision tree graph, a specific set of rules, that can be seen as explanations, potentially enhancing interpretability. The methodology involves two major phases: (1) training and evaluating deep-learning-based time series classification models, and (2) extracting parameterized primitive events, such as increasing, decreasing, local max and local min, from each instance of the evaluation set and clustering such events to extract prototypical ones. These prototypical primitive events are then used as input to a decision-tree classifier trained to fit the model predictions of the test set rather than the ground truth data. Experiments were conducted on diverse real-world datasets sourced from the UCR archive, employing metrics such as accuracy, fidelity, robustness, number of nodes, and depth of the extracted rules. The findings indicate that this global post-hoc method can improve the global interpretability of complex time series classification models.

Broadcast Domination in Line Graphs of Trees

A dominating broadcast of a graph G is a function f : V ( G ) → { 0 , 1 , 2 , … , diam ( G ) } such that f ( v ) ⩽ e ( v ) for all v ∈ V ( G ) , where e ( v ) is the eccentricity of v , and for every vertex u ∈ V ( G ) , there exists a vertex v with f ( v ) > 0 and d ( u , v ) ⩽ f ( v ) . The cost of f is ∑ v ∈ V ( G ) f ( v ) . The minimum of costs over all the dominating broadcasts of G is called the broadcast domination number γ b ( G ) of G . A graph $G$ is said to be radial if γ b ( G ) = rad ( G ) . In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph L ( G ) of G , in terms of γ b ( G ) , and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees T for which L ( T ) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the i -subdivision graph of K 1 , n and a subclass of caterpillars are radial. Also, we show that γ b ( L ( C ) ) = γ ( L ( C ) ) for any caterpillar C .