Graph Distance Research Articles

The extremal average distance of cubic graphs

AbstractThe average distance of a simple connected graph is the average of the distances between all pairs of vertices in . We prove that for a connected cubic graph on vertices, , if ; and , if . Furthermore, all extremal graphs attaining the upper bounds are characterized, and they have the maximum possible diameter. The result solves a question of Plesník and proves a conjecture of Knor, Škrekovski, and Tepeh on the average distance of cubic graphs. The proofs use graph transformations and structural graph analysis.

On [formula omitted]-packing colourings of distance graphs [formula omitted] and [formula omitted

For a non-decreasing sequence of positive integers S=(s1,s2,…), the S-packing chromatic numberχS(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into subsets Xi, i∈{1,2,…,k}, where vertices in Xi are pairwise at distance greater than Si. By an infinite distance graph with distance set D we mean a graph with vertex set Z in which two vertices i,j are adjacent whenever |i−j|∈D. In this paper we investigate the S-packing chromatic number of infinite distance graphs with distance set D={1,t}, t≥2, and D={1,2,t}, t≥3, for sequences S having all elements from {1,2}.

A data-driven model for production prediction of strongly heterogeneous reservoir under uncertainty

Production forecasting plays an important role in the reservoir development process. This task is challenging due to complex spatial and temporal correlations (e.g., inter-well interference and forward-backward linkages temporally). Most existing work tried to solve this problem by mining the impact of historical production conditions on the future production from the perspective of time series analysis. While such works gain satisfactory prediction accuracy, they cannot take the effect of well interference into account. In this study, we revisit the problem of production prediction from strongly heterogeneous reservoirs, aiming to improve the prediction accuracy by leveraging the breakthroughs of deep learning techniques. A new data-driven model is proposed to predict the production of each oil well, where the key novelty is to mine the inter-well interference relationships from the graph perspective. First, based on four constructed inter-well relationship graphs (well pattern structure graph, distance graph, similarity graph, and injection-production relationship graph), a Multi-Graph Convolutional Network (MGCN) is built to capture the spatial interactions of the wells. Then, a Long Short-Term Memory (LSTM) network is used to capture the complex temporal correlations. Finally, to make the model consider more production influencing factors, a Fully Connected Network (FCN) is utilized to fuse spatiotemporal correlations with other production influencing factors to predict production. Besides, considering the uncertainty of the data, an uncertainty evaluation method for predicting production is developed using Latin hypercube sampling and Monte Carlo methods. For evaluation and verification, the proposed model is applied to two case studies of numerical models and actual oilfields. The data of the actual oilfield are preprocessed by the proposed ensemble empirical mode decomposition (EEMD) component threshold denoising method. The optimal hyperparameters in the model are determined by Bayesian optimization algorithm. A series of experiments demonstrated that the proposed method has minimal prediction error and can predict the production with different probabilities, which can reduce the development risk.

Chromatic coloring of distance graphs, III

A graph G(Z, D) with vertex set Z is called an integer distance graph if its edge set is obtained by joining two elements of Z by an edge whenever their absolute difference is a member of D. When D = P or D ⊆ P where P is the set of all prime numbers then we call it a prime distance graph. After establishing the chromatic number of G(Z, P ) as four, Eggleton has classified the collection of graphs as belonging to class i if the chromatic number of G(Z, D) is i. The problem of characterizing the family of graphs belonging to class i when D is of any given size is open for the past few decades. As coloring a prime distance graph is equivalent to producing a prime distance labeling for vertices of G, we have succeeded in giving a prime distance labeling for certain class of all graphs considered here. We have proved that if D = {2, 3, 5, 7, 7th prime, 10th prime, 13th prime, 16th prime, (7 + j)th prime, ..., (4 + j)th prime for any s ∈ N}, then there exists a prime distance graph with distance set D in class 4 and if D = {2, 3, 5, 4th prime, 6th prime, 8th prime, (4 + j)th prime, ..., (2 + j)th prime for any s ∈ N} then there exists a prime distance graphs with distance set D in class 3. Further, we have also obtained some more interesting results that are either general or existential such as a) If D is a specific sequence of integers in arithmetic progression then there exist a prime distance graph with distance set D, b) If G is any prime distance graph in class i for 1 ≤ i ≤ 4 then G × K2 is also a prime distance graph in the respective class i, c) A countable union of disjoint copies of prime distance graph is again a prime distance graph, d) The Middle/Total graph of a path on n vertices is a prime distance graph. In addition we also provide a new different proof for establishing a fact that all cycles are prime distance graph.

Local Multiset Dimension of Amalgamation Graphs

\textbf{Background}: One of the topics of distance in graphs is the resolving set problem. Suppose the set $W=\{s_1,s_2,…,s_k\}\subset V(G)$, the vertex representations of $\in V(G)$ is $r_m(x|W)=\{d(x,s_1),d(x,s_2),…,d(x,s_k)\}$, where $d(x,s_i)$ is the length of the shortest path of the vertex $x$ and the vertex in $W$ together with their multiplicity. The set $W$ is called a local $m$-resolving set of graphs $G$ if $r_m (v|W)\neq r_m (u|W)$ for $uv\in E(G)$. The local $m$-resolving set having minimum cardinality is called the local multiset basis and its cardinality is called the local multiset dimension of $G$, denoted by $md_l(G)$. Thus, if $G$ has an infinite local multiset dimension and then we write $md_l (G)=\infty$. \\ \textbf{Methods}: This research is pure research with exploration design. There are several stages in this research, namely we choose the special graph which is operated by amalgamation and the set of vertices and edges of amalgamation of graphs; determine the set $W\subset V(G)$; determine the vertex representation of two adjacent vertices in $G$; and prove the theorem.\\ \textbf{Results}: The results of this research are an upper bound of local multiset dimension of the amalgamation of graphs namely $md_l(Amal(G,v,m))\leq m.md_l(G)$ and their exact value of local multiset dimension of some families of graphs namely $md_l(Amal(P_n,v,m))=1$, $md_l(Amal(K_n,v,m))=\infty$, $md_l(Amal(W_n,v,m))=m. md_l(W_n)$, $md_l(Amal(F_n,v,m))=m. md_l(F_n)$ for $d(v)=n$, $md_l(Amal(F_n,v,m))=m. \lfloor\frac{n}{4}\rfloor$. \\ \textbf{Conclusions}: We have found the upper bound of a local multiset dimension. There are some graphs which attain the upper bound of local multiset dimension namely wheel graphs\\

On a linear fused Gromov-Wasserstein distance for graph structured data

We present a framework for embedding graph structured data into a vector space, taking into account node features and structures of graphs into the optimal transport (OT) problem. Then we propose a novel distance between two graphs, named LinearFGW, defined as the Euclidean distance between their embeddings. The advantages of the proposed distance are twofold: 1) it takes into account node features and structures of graphs for measuring the dissimilarity between graphs in a kernel-based framework, 2) it is more efficient for computing a kernel matrix than pairwise OT-based distances, particularly fused Gromov-Wasserstein [1], making it possible to deal with large-scale data sets. Our theoretical analysis and experimental results demonstrate that our proposed distance leads to an increase in performance compared to the existing state-of-the-art graph distances when evaluated on graph classification and clustering tasks.

Performance analysis of distance-1 distributed algorithms for admission control under the 2-hop interference model

If each node in a wireless network has information about only its 1-hop neighborhood, then what are the limits to performance? This problem is considered for wireless networks where each communication link has a minimum bandwidth quality-of-service (QoS) requirement. Links in the same vicinity contend for the shared wireless medium. The conflict graph captures which pairs of links interfere with each other and depends on the MAC protocol. In IEEE 802.11 MAC protocol-based networks, when communication between nodes i and j takes place, the neighbors of both i and j remain silent. This model of interference is called the 2-hop interference model because the distance in the network graph between any two links that can be simultaneously active is at least 2. In the admission control problem studied in the present paper, the objective is to determine, using only localized information, whether a given set of flow rates is feasible. While distance-d distributed algorithms have been analyzed for the 1-hop interference model, an open problem in the literature is to extend these results to the K-hop interference model, and the present work initiates the generalization to the K-hop interference model.We show that the centralized version of the problem is NP-hard and then investigate distributed, low-complexity solutions for this problem. We propose a distributed algorithm for this problem where each node has information about only its 1-hop neighborhood. The worst-case performance of the distributed algorithm, i.e. the largest factor by which the performance of this distributed algorithm is away from that of an optimal, centralized algorithm, is analyzed. Lower and upper bounds on the suboptimality of the distributed algorithm are obtained, and both bounds are shown to be tight. The exact worst-case performance is obtained for some ring topologies. The performance of the distance-1 distributed algorithm is compared with that of the row constraints, and these two distributed algorithms are shown to be incomparable.

Bin-scanning: Segmentation of X-ray CT volume of binned parts using Morse skeleton graph of distance transform

X-ray CT scanners, due to the transmissive nature of X-rays, have enabled the non-destructive evaluation of industrial products, even inside their bodies. In light of its effectiveness, this study introduces a new approach to accelerate the inspection of many mechanical parts with the same shape in a bin. The input to this problem is a volumetric image (i.e., CT volume) of many parts obtained by a single CT scan. We need to segment the parts in the volume to inspect each of them; however, random postures and dense contacts of the parts prohibit part segmentation using traditional template matching. To address this problem, we convert both the scanned volumetric images of the template and the binned parts to simpler graph structures and solve a subgraph matching problem to segment the parts. We perform a distance transform to convert the CT volume into a distance field. Then, we construct a graph based on Morse theory, in which graph nodes are located at the extremum points of the distance field. The experimental evaluation demonstrates that our fully automatic approach can detect target parts appropriately, even for a heap of 50 parts. Moreover, the overall computation can be performed in approximately 30 min for a large CT volume of approximately 2000×2000×1000 voxels.

On the hardness of the minimum distance problem of quantum codes

We study the hardness of the problem of finding the distance of quantum error-correcting codes. The analogous problem for classical codes is known to be NP-hard, even in approximate form. For quantum codes, various problems related to decoding are known to be NP-hard, but the hardness of the distance problem has not been studied before. In this work, we show that finding the minimum distance of stabilizer quantum codes exactly or approximately is NP-hard. This result is obtained by reducing the classical minimum distance problem to the quantum problem, using the CWS framework for quantum codes, which constructs a quantum code using a classical code and a graph. A main technical tool used for our result is a lower bound on the so-called graph state distance of 4-cycle free graphs. In particular, we show that for a 4-cycle free graph <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> , its graph state distance is either δ or δ + 1, where δ is the minimum vertex degree of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> . Due to a well-known reduction from stabilizer codes to CSS codes, our results also imply that finding the minimum distance of CSS codes is also NP-hard.

DPGCN Model: A Novel Fault Diagnosis Method for Marine Diesel Engines Based on Imbalanced Datasets

The class imbalance problem is prevalent in the condition monitoring (CM) data of marine diesel engines. That significantly deteriorates the diagnostic performance of a data-driven fault diagnosis. In this article, a novel fault diagnosis method named graph convolutional network (GCN) based on the distance and probability topological graphs (DPGCN) model is proposed for solving the problem of imbalanced classification. First, the collected CM dataset is transformed into two topological graphs—the distance topological graph and the probability topological graph—by exploring the similarity between samples from the linear and nonlinear relationships. Graph learning is then introduced to learn and extract the correlation between adjacent samples in addition to the samples’ own features, which provides more topological information for this imbalanced classification task. After multilayer graph learning, two feature embeddings based on samples’ features and topological graphs are learned, and a self-attention mechanism substantially fuses these two embeddings into one combination embedding to derive deeper correlation information for the fault diagnosis. The collected dataset of a marine diesel engine shows that the proposed DPGCN model can improve the classification accuracy and stability under the imbalanced datasets.

Complex Robotic Manipulation via Graph-Based Hindsight Goal Generation.

Reinforcement learning algorithms, such as hindsight experience replay (HER) and hindsight goal generation (HGG), have been able to solve challenging robotic manipulation tasks in multigoal settings with sparse rewards. HER achieves its training success through hindsight replays of past experience with heuristic goals but underperforms in challenging tasks in which goals are difficult to explore. HGG enhances HER by selecting intermediate goals that are easy to achieve in the short term and promising to lead to target goals in the long term. This guided exploration makes HGG applicable to tasks in which target goals are far away from the object's initial position. However, the vanilla HGG is not applicable to manipulation tasks with obstacles because the Euclidean metric used for HGG is not an accurate distance metric in such an environment. Although, with the guidance of a handcrafted distance grid, grid-based HGG can solve manipulation tasks with obstacles, a more feasible method that can solve such tasks automatically is still in demand. In this article, we propose graph-based hindsight goal generation (G-HGG), an extension of HGG selecting hindsight goals based on shortest distances in an obstacle-avoiding graph, which is a discrete representation of the environment. We evaluated G-HGG on four challenging manipulation tasks with obstacles, where significant enhancements in both sample efficiency and overall success rate are shown over HGG and HER. Videos can be viewed at https://videoviewsite.wixsite.com/ghgg.

Distance evolutions in growing preferential attachment graphs

We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent τ∈(2,3), sample two vertices ut, vt uniformly at random when the graph has t vertices and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in t′≥t, called the distance evolution. We show that there is a tight strip around the function 4loglog(t)−log(log(t′/t)∨1)|log(τ−2)|∨2 that the distance evolution never leaves with high probability as t tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a nonnegative random variable L.

Incompleteness of graph neural networks for points clouds in three dimensions

Graph neural networks (GNN) are very popular methods in machine learning and have been applied very successfully to the prediction of the properties of molecules and materials. First-order GNNs are well known to be incomplete, i.e. there exist graphs that are distinct but appear identical when seen through the lens of the GNN. More complicated schemes have thus been designed to increase their resolving power. Applications to molecules (and more generally, point clouds), however, add a geometric dimension to the problem. The most straightforward and prevalent approach to construct graph representation for molecules regards atoms as vertices in a graph and draws a bond between each pair of atoms within a chosen cutoff. Bonds can be decorated with the distance between atoms, and the resulting ‘distance graph NNs’ (dGNN) have empirically demonstrated excellent resolving power and are widely used in chemical ML, with all known indistinguishable configurations being resolved in the fully-connected limit, which is equivalent to infinite or sufficiently large cutoff. Here we present a counterexample that proves that dGNNs are not complete even for the restricted case of fully-connected graphs induced by 3D atom clouds. We construct pairs of distinct point clouds whose associated graphs are, for any cutoff radius, equivalent based on a first-order Weisfeiler-Lehman (WL) test. This class of degenerate structures includes chemically-plausible configurations, both for isolated structures and for infinite structures that are periodic in 1, 2, and 3 dimensions. The existence of indistinguishable configurations sets an ultimate limit to the expressive power of some of the well-established GNN architectures for atomistic machine learning. Models that explicitly use angular or directional information in the description of atomic environments can resolve this class of degeneracies.

Graph attention mechanism based reinforcement learning for multi-agent flocking control in communication-restricted environment

To solve the poor performance of reinforcement learning (RL) in the multi-agent flocking cooperative control under the communication-restricted environments, we propose a multi-agent cooperative RL (MACRL) method based on the equivalent characteristics of the agents in the flocking task. A distance graph attention (GAT) mechanism is introduced into the policy network of the proposed MACRL to change the agent’s attention weights related to neighbors and reduce the influence of remote neighbors which have poor communication quality on the agent’s behavioral decision-making. Furthermore, a distance GAT-based MACRL (DGAT-MACRL) algorithm is proposed for multi-agent flocking control in the communication-restricted environment. The simulation results show that the proposed flocking algorithm has good adaptability to the communication delay and the communication distance constraint environments. The flocking control effect is significantly better than other RL-based flocking algorithms and traditional flocking algorithms. In addition, the good experimental performance has also confirmed that the proposed DGAT-MACRL is an effective solution to the problem of how to improve the adaptability of traditional RL to the flocking control system with a dynamic scale. Our algorithm provides a novel and practical algorithm offering an effective method for the cooperative task accomplished by multi-agent in non-ideal environments.

An Improved Density Peak Clustering Algorithm for Multi-Density Data.

Density peak clustering is the latest classic density-based clustering algorithm, which can directly find the cluster center without iteration. The algorithm needs to determine a unique parameter, so the selection of parameters is particularly important. However, for multi-density data, when one parameter cannot satisfy all data, clustering often cannot achieve good results. Moreover, the subjective selection of cluster centers through decision diagrams is often not very convincing, and there are also certain errors. In view of the above problems, in order to achieve better clustering of multi-density data, this paper improves the density peak clustering algorithm. Aiming at the selection of parameter dc, the K-nearest neighbor idea is used to sort the neighbor distance of each data, draw a line graph of the K-nearest neighbor distance, and find the global bifurcation point to divide the data with different densities. Aiming at the selection of cluster centers, the local density and distance of each data point in each data division is found, a γ map is drawn, the average value of the γ height difference is calculated, and through two screenings the largest discontinuity point is found to automatically determine the cluster center and the number of cluster centers. The divided datasets are clustered by the DPC algorithm, and then the clustering results are perfected and integrated by using the cluster fusion rules. Finally, a variety of experiments are designed from various perspectives on various artificial simulated datasets and UCI real datasets, which demonstrate the superiority of the F-DPC algorithm in terms of clustering effect, clustering quality, and number of samples.

Shape-aware stochastic neighbor embedding for robust data visualisations

BackgroundThe t-distributed Stochastic Neighbor Embedding (t-SNE) algorithm has emerged as one of the leading methods for visualising high-dimensional (HD) data in a wide variety of fields, especially for revealing cluster structure in HD single-cell transcriptomics data. However, t-SNE often fails to correctly represent hierarchical relationships between clusters and creates spurious patterns in the embedding. In this work we generalised t-SNE using shape-aware graph distances to mitigate some of the limitations of the t-SNE. Although many methods have been recently proposed to circumvent the shortcomings of t-SNE, notably Uniform manifold approximation (UMAP) and Potential of heat diffusion for affinity-based transition embedding (PHATE), we see a clear advantage of the proposed graph-based method.ResultsThe superior performance of the proposed method is first demonstrated on simulated data, where a significant improvement compared to t-SNE, UMAP and PHATE, based on quantitative validation indices, is observed when visualising imbalanced, nonlinear, continuous and hierarchically structured data. Thereafter the ability of the proposed method compared to the competing methods to create faithfully low-dimensional embeddings is shown on two real-world data sets, the single-cell transcriptomics data and the MNIST image data. In addition, the only hyper-parameter of the method can be automatically chosen in a data-driven way, which is consistently optimal across all test cases in this study.ConclusionsIn this work we show that the proposed shape-aware stochastic neighbor embedding method creates low-dimensional visualisations that robustly and accurately reveal key structures of high-dimensional data.

Solutions of Detour Distance Graph Equations.

Graph theory is a useful mathematical structure used to model pairwise relations between sensor nodes in wireless sensor networks. Graph equations are nothing but equations in which the unknown factors are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. In this paper, we solved some graph equations of detour two-distance graphs, detour three-distance graphs, detour antipodal graphs involving with the line graphs.

Eigenvalue localization and Geršgorin disc-related problems on distance and distance-related matrices of graphs

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph [Formula: see text] are denoted by [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] is the diagonal matrix of vertex transmission. Ger[Formula: see text]gorin discs for any [Formula: see text] square matrix [Formula: see text] are the discs [Formula: see text], where [Formula: see text]. The famous Ger[Formula: see text]gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger[Formula: see text]gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger[Formula: see text]gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger[Formula: see text]gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger[Formula: see text]gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

On a Combinatorial Approach to Studying the Steiner Diameter of a Graph and Its Line Graph

In 1989, Chartrand, Oellermann, Tian and Zou introduced the Steiner distance for graphs. This is a natural generalization of the classical graph distance concept. Let Γ be a connected graph of order at least 2, and S\V(Γ). Then, the minimum size among all the connected subgraphs whose vertex sets contain S is the Steiner distancedΓ(S) among the vertices of S. The Steiner k-eccentricity ek(v) of a vertex v of Γ is defined by ek(v)=max{dΓ(S)|S\V(Γ),|S|=k,andv∈S}, where n and k are two integers, with 2≤k≤n, and the Steiner k-diameter of Γ is defined by sdiamk(Γ)=max{ek(v)|v∈V(Γ)}. In this paper, we present an algorithm to derive the Steiner distance of a graph; in addition, we obtain a relationship between the Steiner k-diameter of a graph and its line graph. We study various properties of the Steiner diameter through a combinatorial approach. Moreover, we characterize graph Γ when sdiamk(Γ) is given, and we determine sdiamk(Γ) for some special graphs. We also discuss some of the applications of Steiner diameter, including one in education networks.

Localization game for random geometric graphs

The localization game is a two player combinatorial game played on a graph G=(V,E). The cops choose a set of vertices S1⊆V with |S1|=k. The robber then chooses a vertex v∈V whose location is hidden from the cops, but the cops learn the graph distance between the current position of the robber and the vertices in S1. If this information is sufficient to locate the robber, the cops win immediately; otherwise the cops choose another set of vertices S2⊆V with |S2|=k, and the robber may move to a neighboring vertex. The new distances to the robber are presented, and if the cops can deduce the new location of the robber based on all information they accumulated thus far, then they win; otherwise, a new round begins. If the robber has a strategy to avoid being captured, then she wins. The localization number is defined to be the smallest integer k so that the cops win the game. In this paper we determine the localization number (up to poly-logarithmic factors) of the random geometric graph G∈G(n,r) slightly above the connectivity threshold.