Graph Connectivity Research Articles

Taming Connectedness in Machine-Learning-Based Topology Optimization with Connectivity Graphs

Despite the remarkable advancements in machine learning (ML) techniques for topology optimization, the predicted solutions often overlook the necessary structural connectivity required to meet the load-carrying demands of the resulting designs. Consequently, these predicted solutions exhibit subpar structural performance because disconnected components are unable to bear loads effectively and significantly compromise the manufacturability of the designs.In this paper, we propose an approach to enhance the topological accuracy of ML-based topology optimization methods by employing a predicted dual connectivity graph. We show that the tangency graph of the Maximal Disjoint Ball Decomposition (MDBD), which accurately captures the topology of the optimal design, can be used in conjunction with a point transformer network to improve the connectivity of the design predicted by Generative Adversarial Networks and Convolutional Neural Networks. Our experiments show that the proposed method can significantly improve the connectivity of the final predicted structures. Specifically, in our experiments the error in the number of disconnected components was reduced by a factor of 4 or more without any loss of accuracy. We demonstrate the flexibility of our approach by presenting examples including various boundary conditions (both seen and unseen), domain resolutions, and initial design domains. Importantly, our method can seamlessly integrate with other existing deep learning-based optimization algorithms, utilize training datasets with models using any valid geometric representations, and naturally extend to three-dimensional applications.

Reconstruction and edge reconstruction of triangle-free graphs

The Reconstruction Conjecture due to Kelly and Ulam states that every graph with at least 3 vertices is uniquely determined by its multiset of subgraphs {G−v:v∈V(G)}. Let diam(G) and κ(G) denote the diameter and the connectivity of a graph G, respectively, and let G2:={G:diam(G)=2} and G3:={G:diam(G)=diam(G‾)=3}. It is known that the Reconstruction Conjecture is true if and only if it is true for every 2-connected graph in G2∪G3. Balakumar and Monikandan showed that the Reconstruction Conjecture holds for every triangle-free graph G in G2∪G3 with κ(G)=2. Moreover, they asked whether the result still holds if κ(G)≥3. (If yes, the class of graphs critical for solving the Reconstruction Conjecture is restricted to 2-connected graphs in G2∪G3 which contain triangles.) The case when G∈G3 and κ(G)≥3 was recently confirmed by Devi Priya and Monikandan. In this paper, we further show the Reconstruction Conjecture holds for every triangle-free graph G in G2 with κ(G)=3. We also prove similar results about the Edge Reconstruction Conjecture.

The maximum number of complete multipartite subgraphs in graphs with given circumference or matching number

The circumference c(G) of a graph G is the length of a longest cycle in G and the matching number α′(G) is the maximum size of a matching in G. In 1959, Erdős and Gallai determined the maximum size of graphs with given c(G) or α′(G). Let Kr1,…,rs denote the complete multipartite graph with class sizes r1,…,rs and δ(G) be the minimum degree of G. In this paper, we determine the maximum number of copies of Kr1,…,rs in a 2-connected n-vertex graph G with given c(G) and δ(G)≥k. The corresponding problem for an n-vertex graph G with given α′(G) and δ(G)≥k is also addressed. As a corollary of our main results, we determine the maximum number of copies of Kr1,…,rs in an n-vertex graph with given c(G) or α′(G). In addition, the corresponding problems for graphs with given detour order are solved as well.

Non-Centralised Balance Dispatch Strategy in Waked Wind Farms through a Graph Sparsification Partitioning Approach

A novel non-centralised dispatch strategy is presented for wake redirection to optimise large-scale offshore wind farms operation, creating a balanced control between power production and fatigue thrust loads evenly among the wind turbines. This approach is founded on a graph sparsification partitioning strategy that takes into account the impact of wake propagation. More specifically, the breadth-first search algorithm is employed to identify the subgraph based on the connectivity of the wake direction graph, while the PageRank centrality computation algorithm is utilised to determine and rank scores for the shared turbines’ affiliation with the subgraphs. By doing so, the wind farm is divided into smaller subsets of partitioned turbines, resulting in decoupling. The objective function is then formulated by incorporating penalty terms, specifically the standard deviation of fatigue thrust loads, into the maximum power equation. Meanwhile, the non-centralisation sequential quadratic programming optimisation algorithm is subsequently employed within each partition to determine the control actions while considering the objectives of the respective controllers. Finally, the simulation results of case studies prove to reduce computational costs and improve wind farm power production by balancing accumulated fatigue thrust loads over the operational lifetime as much as possible.

On the bond polytope

Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar (K 5 − e)-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on (K 5 − e)-minor free graphs.

The cubic vertices of solid minimal bricks

A 3-connected graph is a brick if, for any two vertices u and v, G−u−v has a perfect matching. A brick G is minimal if G−e is not a brick for every edge e of G. Norine and Thomas [J. Combin. Theory Ser. B, 96(4) (2006), pp. 505-513.] conjectured that there exists α>0 such that every minimal brick G contains at least α|V(G)| cubic vertices.A brick is solid if for any two disjoint odd cycles C1 and C2, G−V(C1∪C2) has no perfect matching. In this paper, by proving that every solid minimal brick G has at least 2|V(G)|/5 cubic vertices, we obtain the first non-trivial class of graphs that support the above conjecture.

AB/Push-Pull method for distributed optimization in time-varying directed networks

In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linearly to the optimal solution over a time-varying directed network for a constant stepsize when the agent's cost function is smooth and strongly convex. The linear convergence of the method has been shown in [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], where the multi-step consensus contraction parameters for row- and column-stochastic mixing matrices are not directly related to the underlying graph structure, and the explicit range for the stepsize value is not provided. With respect to [F. Saadatniaki, R. Xin, and U.A. Khan, Decentralized optimization over time-varying directed graphs with row and column-stochastic matrices, IEEE Trans. Autom. Control 65(11) (2020), pp. 4769–4780], the novelty of this work is twofold: (1) we establish the one-step consensus contraction for both row- and column-stochastic mixing matrices with the contraction parameters given explicitly in terms of the graph diameter and other graph properties; and (2) we provide explicit upper bounds for the stepsize value in terms of the properties of the cost functions, the mixing matrices, and the graph connectivity structure.

Multiscale graph neural network autoencoders for interpretable scientific machine learning

The goal of this work is to address two limitations in autoencoder-based models: latent space interpretability and compatibility with unstructured meshes. This is accomplished here with the development of a novel graph neural network (GNN) autoencoding architecture with demonstrations on complex fluid flow applications. To address the first goal of interpretability, the GNN autoencoder achieves reduction in the number nodes in the encoding stage through an adaptive graph reduction procedure. This reduction procedure essentially amounts to flowfield-conditioned node sampling and sensor identification, and produces interpretable latent graph representations tailored to the flowfield reconstruction task in the form of so-called masked fields. These masked fields allow the user to (a) visualize where in physical space a given latent graph is active, and (b) interpret the time-evolution of the latent graph connectivity in accordance with the time-evolution of unsteady flow features (e.g. recirculation zones, shear layers) in the domain. To address the goal of unstructured mesh compatibility, the autoencoding architecture utilizes a series of multi-scale message passing (MMP) layers, each of which models information exchange among node neighborhoods at various lengthscales. The MMP layer, which augments standard single-scale message passing with learnable coarsening operations, allows the decoder to more efficiently reconstruct the flowfield from the identified regions in the masked fields. Analysis of latent graphs produced by the autoencoder for various model settings are conducted using unstructured snapshot data sourced from large-eddy simulations in a backward-facing step (BFS) flow configuration with an OpenFOAM-based flow solver at high Reynolds numbers.

Wheeled Robot Formation Control under Uncertainty Including Adaptive Rejection of Harmonic Disturbances with Arbitrary Frequencies

This work addresses an adaptive regulation problem for formation control of uncertain unicycle-like dynamical robots. The control goal is to drive a finite number of robots to a prescribed formation despite the presence of multi-sinusoidal input disturbances of unknown frequencies and despite uncertainties in the unicycle model, (mass and inertia’s are unknown). Numerical Simulations were done to illustrate the viability of the suggested strategy. The only measurements available for control are the orientation of the unicycles and the relative positions and velocities according to a given connectivity graph.

Deep orientated distance-transform network for geometric-aware centerline detection

The detection of structure centerlines from imaging data plays a crucial role in the understanding, application and further analysis of many diverse problems, such as road mapping, crack detection, medical imaging and biometric identification. In each of these cases, pixel-wise segmentation is not sufficient to understand and quantify overall graph structure and connectivity without further processing that can lead to compound error. We thus require a method for automatic extraction of graph representations of patterning. In this paper, we propose a novel Deep Orientated Distance-transform Network (DODN), which predicts the centerline map and an orientated distance map, comprising orientation and distance in relation to the centerline and allowing exploitation of its geometric properties. This is refined by jointly modeling the relationship between neighboring pixels and connectivity to further enhance the estimated centerline and produce a graph of the structure. The proposed approach is evaluated on a diverse range of problems, including crack detection, road mapping and superficial vein centerline detection from infrared/ color images, improving over the state-of-the-art by 2.1%, 10.9% and 17.3%/ 4.6% respectively in terms of quality, demonstrating its generalizability and performance in a wide range of mapping problems.

Diameter and Connectivity of Finite Simple Graphs

Diameter and Connectivity of Finite Simple Graphs

A polygonal buildings aggregation method considering obstacle elements and visual clarity

A polygonal building aggregation method is proposed in this study to address the shortcomings of existing aggregation methods that mostly focus on maintaining a few features, ignoring the requirement for a clear expression. First, visually conglutinated buildings were clustered considering obstacle elements. Unreasonable triangles, starting from the marginal triangles, were removed to extract the visual conglutinate areas. Bridging modes were designed and local shape features were considered to bridge the visual conglutinate areas. Then, the contours of the aggregated results were established through a minimum loop search on the path connection graph. An experiment was conducted using the data of OpenStreetMap from Nanjing, and the comparative experimental results showed that the proposed method preserved the aggregated results of right-angled and locally shaped features. The advantages of improving the visual clarity of the results were verified by introducing degrees of visual conglutination and complexity.

Path eccentricity of graphs

Let G be a connected graph. The eccentricity of a path P, denoted by eccG(P), is the maximum distance from P to any vertex in G. In the Central path (CP) problem, our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph G as a parameter. Let pe(G) denote the value of eccG(P) for a central path P of G. We obtain tight upper bounds for pe(G) in some graph classes. We show that pe(G)≤1 on biconvex graphs and design an algorithm that finds such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe(G) on arbitrary graphs and k-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that, on bipartite permutation graphs, a longest path is also a central path. Furthermore, for any such class of graphs, we exhibit a superclass that does not satisfy this property.

Influence of the level of connectivity on the members of the Octahedron and X-Octahedron families of tensegrities

Tensegrity structures based on topological patterns have been developed greatly in recent years. Tensegrities obtained from the same pattern are said to belong to a family. Two examples of tensegrity families are the Octahedron and the X-Octahedron, whose members are composed of rhombic and X-rhombic cells, respectively, which are collected in three groups. The general connectivity pattern of both families consists of three levels of connectivity. This work analyzes the influence of the reduction of the level of connectivity on the members of both families. The connection graphs corresponding to different levels of connectivity are defined based on the new concept of “twin tensegrities”. Analytical computations have been performed to determine the force:length ratios that satisfy equilibrium, stability, and super-stability conditions. In addition, the mathematical sequence that follows the ratio between the force:length ratio of struts and cables of the X-Octahedron family that leads to a super-stable equilibrium configuration is presented. The new tensegrities obtained in this work also belong to the Octahedron and X-Octahedron families and could have promising engineering applications such as modular constructions.

Independence numbers of polyhedral graphs

A polyhedral graph is a 3-connected planar graph. We find the least possible order p(k,a) of a polyhedral graph containing a k-independent set of size a for all positive integers k and a. In the case k=1 and a even, we prove that the extremal graphs are exactly the vertex-face (radial) graphs of maximal planar graphs.

Graph structure reforming framework enhanced by commute time distance for graph classification

As a graph data mining task, graph classification has high academic value and wide practical application. Among them, the graph neural network-based method is one of the mainstream methods. Most graph neural networks (GNNs) follow the message passing paradigm and can be called Message Passing Neural Networks (MPNNs), achieving good results in structural data-related tasks. However, it has also been reported that these methods suffer from over-squashing and limited expressive power. In recent years, many works have proposed different solutions to these problems separately, but none has yet considered these shortcomings in a comprehensive way. After considering these several aspects comprehensively, we identify two specific defects: information loss caused by local information aggregation, and an inability to capture higher-order structures. To solve these issues, we propose a plug-and-play framework based on Commute Time Distance (CTD), in which information is propagated in commute time distance neighborhoods. By considering both local and global graph connections, the commute time distance between two nodes is evaluated with reference to the path length and the number of paths in the whole graph. Moreover, the proposed framework CTD-MPNNs (Commute Time Distance-based Message Passing Neural Networks) can capture higher-order structural information by utilizing commute paths to enhance the expressive power of GNNs. Thus, our proposed framework can propagate and aggregate messages from defined important neighbors and model more powerful GNNs. We conduct extensive experiments using various real-world graph classification benchmarks. The experimental performance demonstrates the effectiveness of our framework. Codes are released on https://github.com/Haldate-Yu/CTD-MPNNs.

The diagnostic performance of machine learning based on resting-state functional magnetic resonance imaging data for major depressive disorders: a systematic review and meta-analysis.

Machine learning (ML) has been widely used to detect and evaluate major depressive disorder (MDD) using neuroimaging data, i.e., resting-state functional magnetic resonance imaging (rs-fMRI). However, the diagnostic efficiency is unknown. The aim of the study is to conduct an updated meta-analysis to evaluate the diagnostic performance of ML based on rs-fMRI data for MDD. English databases were searched for relevant studies. The Quality Assessment of Diagnostic Accuracy Studies (QUADAS-2) was used to assess the methodological quality of the included studies. A random-effects meta-analytic model was implemented to investigate the diagnostic efficiency, including sensitivity, specificity, diagnostic odds ratio (DOR), and area under the curve (AUC). Regression meta-analysis and subgroup analysis were performed to investigate the cause of heterogeneity. Thirty-one studies were included in this meta-analysis. The pooled sensitivity, specificity, DOR, and AUC with 95% confidence intervals were 0.80 (0.75, 0.83), 0.83 (0.74, 0.82), 14.00 (9, 22.00), and 0.86 (0.83, 0.89), respectively. Substantial heterogeneity was observed among the studies included. The meta-regression showed that the leave-one-out cross-validation (loocv) (sensitivity: p < 0.01, specificity: p < 0.001), graph theory (sensitivity: p < 0.05, specificity: p < 0.01), n > 100 (sensitivity: p < 0.001, specificity: p < 0.001), simens equipment (sensitivity: p < 0.01, specificity: p < 0.001), 3.0T field strength (Sensitivity: p < 0.001, specificity: p = 0.04), and Beck Depression Inventory (BDI) (sensitivity: p = 0.04, specificity: p = 0.06) might be the sources of heterogeneity. Furthermore, the subgroup analysis showed that the sample size (n > 100: sensitivity: 0.71, specificity: 0.72, n < 100: sensitivity: 0.81, specificity: 0.79), the different levels of disease evaluated by the Hamilton Depression Rating Scale (HDRS/HAMD) (mild vs. moderate vs. severe: sensitivity: 0.52 vs. 0.86 vs. 0.89, specificity: 0.62 vs. 0.78 vs. 0.82, respectively), the depression scales in patients with comparable levels of severity. (BDI vs. HDRS/HAMD: sensitivity: 0.86 vs. 0.87, specificity: 0.78 vs. 0.80, respectively), and the features (graph vs. functional connectivity: sensitivity: 0.84 vs. 0.86, specificity: 0.76 vs. 0.78, respectively) selected might be the causes of heterogeneity. ML showed high accuracy for the automatic diagnosis of MDD. Future studies are warranted to promote the potential use of these classification algorithms in clinical settings.

On the strong connectivity of the 2-Engel graphs of almost simple groups

Abstract The Engel graph of a finite group 𝐺 is a directed graph encoding the pairs of elements in 𝐺 satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of almost simple groups are strongly connected. In this paper, we give a refinement to this analysis.

A Benchmark Study on Knowledge Graphs Enrichment and Pruning Methods in the Presence of Noisy Relationships

In the past few years, knowledge graphs (KGs), as a form of structured human intelligence, have attracted considerable research attention from academia and industry. In this very active field of study, a widely explored problem is that of link prediction, the task of predicting whether two nodes should be connected, based on node attributes and local or global graph connectivity properties. The state of the art in this area is represented by techniques based on graph embeddings. However, KGs, especially those acquired using automated or partly automated techniques, are often riddled with noise, e.g., wrong relationships, which makes the problem of link deletion as important as that of link prediction. In this paper, we address three main research questions. The first is about the true effectiveness of different knowledge graph embedding models under the presence of an increasing number of wrong links. The second is to asses if methods that can predict unknown relationships effectively, work equally well in recognizing incorrect relations. The third is to verify if there are systems robust enough to maintain primacy in all experimental conditions. To answer these research questions, we performed a systematic benchmark study in which the experimental setting includes ten state-of-the-art models, three common KG datasets with different structural properties and three downstream tasks: the widely explored tasks of link prediction and triple classification, and the less popular task of link deletion. Comparative studies often yield contradictory results, where the same systems score better or worse depending on the experimental context. In our work, in order to facilitate the discovery of clear performance patterns and their interpretation, we select and/or aggregate performance data to highlight each specific comparison dimension: dataset complexity, type of task, category of models, and robustness against noise.

On the connectivity and the diameter of betweenness-uniform graphs

Betweenness centrality is a centrality measure based on the overall amount of shortest paths passing through a given vertex. A graph is betweenness-uniform if all its vertices have the same betweenness centrality. We study the properties of betweenness-uniform graphs. In particular, we show that every connected betweenness-uniform graph is either a cycle or a 3-connected graph. Also, we show that betweenness uniform graphs of high maximal degree have small diameter.