Graph Connectivity Research Articles

Exploiting graph neural networks to perform finite-difference time-domain based optical simulations

Having an artificial neural network that solves Maxwell’s equations in a general setting is an intellectual challenge and a great utility. Recently, there have been multiple successful attempts to use artificial neural networks to predict electromagnetic fields, given a specific source and interacting material distribution. However, many of these attempts are limited in domain size and restricted to object shapes similar to the learned ones. Here, we overcome these restrictions by using graph neural networks (GNNs) that adapt the propagation scheme of the finite-difference time-domain (FDTD) method to solve Maxwell’s equations for a distinct time step. GNNs yield a significant advantage, i.e., size invariance, over conventional neural network architectures, such as convolutional or linear neural networks. Once trained, a GNN can work on graphs of arbitrary size and connectivity. This allows us to train them on the propagation procedure of electromagnetic fields on small domain sizes and, finally, expand the domain to an arbitrary scale. Moreover, GNNs can adapt to any material shape and work not only on structured grids, such as FDTD, but also on arbitrary meshes. This work may be seen as the first benchmark for field predictions with graph networks and could be expanded to more complex mesh-based optical simulations, e.g., those based on finite elements.

A spatiotemporal graph wavelet neural network for traffic flow prediction

The traffic flow prediction is fast becoming a key instrument in the transportation system, which has achieved impressive performance for traffic management. The graph neural network plays a critical role in the development of the traffic network management. However, it is worthwhile mentioning that the complexity of road networks and traffic conditions makes it unable to obtain sufficient spatiotemporal information. In view of capturing precise environment characteristics, the context could have a precise effect on the prediction results while previous methods rarely took this into account. Besides, the nonlinear characteristics of the graph neural network are hard to quantify with fine granularity and to eliminate overfitting. To stack these challenges, in this paper, we present a spatiotemporal graph wavelet neural network to improve the ability of representations. Specifically, we introduce the wavelet transforms into the deep learning model according to the strong nonlinear optimization ability. Furthermore, we dig the location and time patterns to evaluate the temporal dependence and the spatial proximity correlation. In addition, we introduce a historical context attention mechanism giving fine-grained historical context grade evaluation to ease the phenomenon of over-smoothing. The experimental results on real-world datasets show that our work gets considerable results compared with the baseline and start-of-the-art models. Moreover, our work has better learning performance by employing the connection and interaction of graphs.

Restricted connectivity of Cayley graph generated by transposition trees

Restricted connectivity of Cayley graph generated by transposition trees

Distributed Interval Consensus of Multiagent Systems With a Pulse Width Modulation Protocol

Rather than the conventional pure consensus, this article proposes a pulsewidth modulation (PWM) protocol for an improved interval consensus of multiagent systems from the distributed perspective. In particular, the consensus of all the agents is achieved within a prescribed interval, which is merely available to partial agents, though. The PWM protocol effectively relieves the running burden of the agents without the additional analogue-to-digital conversion. The modulation periods of different agents are allowed to be heterogeneous. By nominating the agents that have access to the given interval as leaders and others as followers, the proposed distributed PWM protocol introduces a projection operator for the leaders in order for the collective aggregation. In terms of the graph connectivity condition of containing a spanning tree, it is shown that the concerned interval consensus objective is realized by the proposed distributed PWM protocol. Simulations are eventually performed to validate the established theoretical results.

PADLoC: LiDAR-Based Deep Loop Closure Detection and Registration Using Panoptic Attention

A key component of graph-based SLAM systems is the ability to detect loop closures in a trajectory to reduce the drift accumulated over time from the odometry. Most LiDAR-based methods achieve this goal by using only the geometric information, disregarding the semantics of the scene. In this work, we introduce PADLoC for joint loop closure detection and registration in LiDAR-based SLAM frameworks. We propose a novel transformer-based head for point cloud matching and registration, and to leverage panoptic information during training time. In particular, we propose a novel loss function that reframes the matching problem as a classification task for the semantic labels and as a graph connectivity assignment for the instance labels. During inference, PADLoC does not require panoptic annotations, making it more versatile than other methods. Additionally, we show that using two shared matching and registration heads with their source and target inputs swapped increases the overall performance by enforcing forward-backward consistency. We perform extensive evaluations of PADLoC on multiple real-world datasets demonstrating that it achieves state-of-the-art results.

A novel topic clustering algorithm based on graph neural network for question topic diversity

In community question answering, many questions have no topic labeling or the topic labeling is very diverse, which has become the biggest obstacle to building the bridge between users and posts. Topic clustering methods could alleviate this issue. However, existing research employed words as topic representation units and could not express topic semantic relevance. In this paper, we propose a novel Topic Clustering framework based on the Graph Neural Network (called TCGNN) to alleviate topic diversity in Community Question Answering. Firstly, we separately consider the relationship representation of existing topics and unlabeled topics. For manually labeled topics, we count the frequency of topics in community questions and construct a topic co-occurrence matrix to represent the topic relation. For unmarked topics, we extract the core phrases from community questions and employ them to indicate the topics of questions. Then, we transform the topic co-occurrence matrix into a topic relation graph, optimizing the topic relevance and improving presentation efficiency. Next, we employ a graph neural network for embedding the topic connection graph and get the vector representation of each topic. Finally, an improved K-mean method is proposed for topic clustering based on the distance of topic vectors. Additionally, we briefly discuss the extended effect of topic clustering methods in other domains (bibliographic information and reviews). In the literature we have, it is a primary work that conders topic clustering in multiple situations and offers innovative cogitation to apply graph neural networks in topic clustering. Our experiment compared prevalent clustering methods and some combination methods of text representation and graph embedding. The outcome of experiments on four extensive and varied datasets (Stack Overflow, DBLP, Yelp, and Zhihu) illustrate that TCGNN leads the prevalent baseline in Entropy and Purity.

Disease identification method based on graph features between pulse cycles

The pulse signal at the wrist communicates a range of cardiovascular information and reflects a person’s state of health. In recent years pulse signals have been used extensively in the field of disease identification. As a typical quasi-cyclic physiological signal, many researchers have simply used single-cycle pulse signals for disease diagnosis, ignoring the influence of inter-cycle differences on the diagnosis. In this paper, we propose a disease identification method based on graph features between pulse cycles, to focus on the differences in pulse cycles brought about by the disease. The pulse was pre-processing using a Sliding Window, followed by the construction of the correlation coefficient matrix, the setting of a threshold and the construction of the connection graph to obtain the graph features; the Multiscale Permutation Entropy method was used to extract the pulse’s entropy features. The ReliefF algorithm is then used to select a total of 18 features and a classification algorithm is used to distinguish between patients and healthy people. Both the self-constructed dataset and the open dataset from the PLA 211 Hospital had high recognition rates for the experiments. The accuracy of identifying hypertension was 94.16% with a precision of 90.41%; the accuracy of identifying chronic diseases was 97.23% with a precision of 97.81%. The experimental results from the two datasets demonstrate both the validity of the difference between cycles for disease classification and the applicability of the method proposed in this paper.

On k-restricted connectivity of direct product of graphs

Let [Formula: see text] be a graph. A vertex-cut [Formula: see text] of [Formula: see text] is said to be k-restricted if every component of [Formula: see text] has at least [Formula: see text] vertices, and cyclic if [Formula: see text] has at least two components which contain a cycle. The minimum cardinality over all [Formula: see text]-restricted vertex-cuts of [Formula: see text] is called the k-restricted connectivity of [Formula: see text] and is denoted by [Formula: see text]. Also the minimum cardinality over all cyclic vertex-cuts of [Formula: see text] is called the cyclic connectivity of graph [Formula: see text] and is denoted by [Formula: see text]. In this paper, the [Formula: see text]-restricted connectivity and cyclic connectivity of the direct product of two graphs [Formula: see text] and [Formula: see text] is obtained for some [Formula: see text], where [Formula: see text] is a complete graph, and [Formula: see text] is a complete graph, a complete bipartite graph or a cycle.

An integrable matrix NLS equation on star graph and symmetry-dependent connection conditions of vertex

An integrable matrix NLS equation on star graph and symmetry-dependent connection conditions of vertex

Integrating multi-type aberrations from DNA and RNA through dynamic mapping gene space for subtype-specific breast cancer driver discovery.

Driver event discovery is a crucial demand for breast cancer diagnosis and therapy. In particular, discovering subtype-specificity of drivers can prompt the personalized biomarker discovery and precision treatment of cancer patients. Still, most of the existing computational driver discovery studies mainly exploit the information from DNA aberrations and gene interactions. Notably, cancer driver events would occur due to not only DNA aberrations but also RNA alternations, but integrating multi-type aberrations from both DNA and RNA is still a challenging task for breast cancer drivers. On the one hand, the data formats of different aberration types also differ from each other, known as data format incompatibility. On the other hand, different types of aberrations demonstrate distinct patterns across samples, known as aberration type heterogeneity. To promote the integrated analysis of subtype-specific breast cancer drivers, we design a "splicing-and-fusing" framework to address the issues of data format incompatibility and aberration type heterogeneity simultaneously. To overcome the data format incompatibility, the "splicing-step" employs a knowledge graph structure to connect multi-type aberrations from the DNA and RNA data into a unified formation. To tackle the aberration type heterogeneity, the "fusing-step" adopts a dynamic mapping gene space integration approach to represent the multi-type information by vectorized profiles. The experiments also demonstrate the advantages of our approach in both the integration of multi-type aberrations from DNA and RNA and the discovery of subtype-specific breast cancer drivers. In summary, our "splicing-and-fusing" framework with knowledge graph connection and dynamic mapping gene space fusion of multi-type aberrations data from DNA and RNA can successfully discover potential breast cancer drivers with subtype-specificity indication.

Properly 3-contractible edges in a minimally 3-connected graph

Let G be a 3-connected graph. An edge e such that G−e is 3-connected is called a “3-removable edge”, and G is said to be “minimally 3-connected” if G has no 3-removable edge. An edge e of G is said to be a “3-contractible edge” if its contraction results in a 3-connected graph. A 3-contractible edge is said to be “properly 3-contractible” if it is contained in no triangle. Tutte's wheel theorem says that every minimally 3-connected graph with no properly 3-contractible edge is a wheel. Let E⁎(G) denote the set of edges of G which are contained in no triangle and let Ec⁎(G) denote the set of properly 3-contractible edges of G. Let U(i)(G) denote the set of degree 3 vertices x of G such that the subgraph of G induced by the neighborhood of x has i edges.In this paper, we prove that for a minimally 3-connected graph G, the following conditions (1), (2) and (3) are equivalent: (1) G is a wheel, (2) E⁎(G)=∅ and (3) U(0)(G)=U(1)(G)=∅, and that for a minimally 3-connected graph G, (I) we have |Ec⁎(G)|≥12|U(1)(G)|+|U(0)(G)|, and (II) we have |Ec⁎(G)|≥12(|E⁎(G)|+3) if G is not a wheel.We show the sharpness of the inequalities (I) and (II).

Joint Multi-Grained Popularity-Aware Graph Convolution Collaborative Filtering for Recommendation

Graph Convolution Networks (GCNs), with their efficient ability to capture high-order connectivity in graphs, have been widely applied in recommender systems. Stacking multiple neighbor aggregation is the major operation in GCNs. It implicitly captures popularity features because the number of neighbor nodes reflects the popularity of a node. However, existing GCN-based methods ignore a universal problem: users' sensitivity to item popularity is differentiated, but the neighbor aggregations in GCNs actually fix this sensitivity through Graph Laplacian Normalization, leading to suboptimal personalization. In this work, we propose to model multi-grained popularity features and jointly learn them together with high-order connectivity, to match the differentiation of user preferences exhibited in popularity features. Specifically, we develop a Joint Multi-grained Popularity-aware Graph Convolution Collaborative Filtering model, short for JMP-GCF, which uses a popularity-aware embedding generation to construct multi-grained popularity features, and uses the idea of joint learning to capture the signals within and between different granularities of popularity features that are relevant for modeling user preferences. Additionally, we propose a multistage stacked training strategy to speed up model convergence. We conduct extensive experiments on three public datasets to show the state-of-the-art performance of JMP-GCF.

Planar and Toroidal Morphs Made Easier

We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater’s asymmetric extension of Tutte’s classical spring-embedding theorem. First, we give a very simple algorithm to construct piecewise-linear morphs between planar straight-line graphs. Our algorithm entirely avoids the classical edge-collapsing strategy dating back to Cairns; instead, in each morphing step, we interpolate the pair of weights associated with a single edge. Specifically, given equivalent straight-line drawings $\Gamma_0$ and $\Gamma_1$ of the same 3-connected planar graph $G$, with the same convex outer face, we construct a morph from $\Gamma_0$ to $\Gamma_1$ that consists of $O(n)$ unidirectional morphing steps, in $O(n^{1+\omega/2})$ time, where $n$ is the number of vertices and $\omega$ is the matrix multiplication constant. Second, we describe a natural extension of barycentric interpolation to geodesic graphs on the flat torus. Barycentric interpolation cannot be applied directly in this setting, because the linear systems defining intermediate vertex positions are not necessarily solvable. We describe a simple scaling strategy that circumvents this issue. Computing the appropriate scaling requires $O(n^{\omega/2})$ time, after which we can compute the drawing at any point in the morph in $O(n^{\omega/2})$ time. Our algorithm is considerably simpler than the recent algorithm of Chambers, Erickson, Lin, and Parsa, and produces more natural morphs. Our techniques also yield a simple proof of a conjecture of Connelly, Henderson, Ho, and Starbird for geodesic torus triangulations.

Quantum density peak clustering

Clustering algorithms are of fundamental importance when dealing with large unstructured datasets and discovering new patterns and correlations therein, with applications ranging from scientific research to medical imaging and marketing analysis. In this work, we introduce a quantum version of the density peak clustering algorithm, built upon a quantum routine for minimum finding. We prove a quantum speedup for a decision version of density peak clustering depending on the structure of the dataset. Specifically, the speedup is dependent on the heights of the trees of the induced graph of nearest-highers, i.e. the graph of connections to the nearest elements with higher density. We discuss this condition, showing that our algorithm is particularly suitable for high-dimensional datasets. Finally, we benchmark our proposal with a toy problem on a real quantum device.

A General Relationship Between Optimality Criteria and Connectivity Indices for Active Graph-SLAM

Quantifying uncertainty is a key stage in active simultaneous localization and mapping (SLAM), as it allows to identify the most informative actions to execute. However, dealing with full covariance or even Fisher information matrices (FIMs) is computationally heavy and easily becomes intractable for online systems. In this letter, we study the paradigm of active graph-SLAM formulated over the special Euclidean group <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SE(n)</i> , and propose a general relationship between the FIM of the system and the Laplacian matrix of the underlying pose-graph. This link makes possible to use graph connectivity indices as utility functions with optimality guarantees, since they approximate the well-known optimality criteria that stem from optimal design theory. Experimental validation demonstrates that the proposed method leads to equivalent decisions for active SLAM in a fraction of the time.

On smallest 3-polytopes of given graph radius

The 3-polytopes are planar, 3-connected graphs. A classical question is, given r≥3, is the 2(r−1)-gonal prism K2×C2(r−1) the unique 3-polytope of graph radius r and smallest size? Under some extra assumptions, we answer this question in the positive.

Spanning trees with at most [formula omitted] leaves in 2-connected [formula omitted]-free graphs

A vertex with degree one and a vertex with degree at least three are called a leaf and a branch vertex in a tree, respectively. In this paper, we obtain that every 2-connected K1,r-free graph G contains a spanning tree with at most k leaves if α(G)≤k+⌈k+1r−3⌉−⌊1|r−k−3|+1⌋, where k≥2 and r≥4. The upper bound is best possible. Furthermore, we prove that if a connected K1,4-free graph G satisfies that α(G)≤2k+5, then G contains either a spanning tree with at most k branch vertices or a block B with α(B)≤2. A related conjecture for 2-connected claw-free graphs is also posed.

Proper (Strong) Rainbow Connection and Proper (Strong) Rainbow Vertex Connection of Some Special Graphs

The proper rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is rainbow vertex connected. The proper strong rainbow vertex connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors needed to properly color the vertices of [Formula: see text] so that [Formula: see text] is strong rainbow vertex connected. These two concepts are inspired by the concept of proper (strong) rainbow connection number of graphs. In this paper, we first determine the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], pencil graphs, circular ladders or Möbius ladders. Secondly, we obtain the values of [Formula: see text] and [Formula: see text] for some special graphs, such as all cubic graphs of order [Formula: see text], paths, cycles, wheels, complete multipartite graphs, pencil graphs, circular ladders and Möbius ladders. Finally, we characterize all the connected graphs [Formula: see text] with [Formula: see text] and [Formula: see text].

The Threshold of Symmetry in Random Graphs with Specified Degree Sequences

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterization captures the phase transition of the symmetry of the random graphs. This phase transition coincides with that of the graph connectivity. We also show that there are unbounded degree sequences where these two thresholds do not coincide.

Component (edge) connectivity of pancake graphs

The l-component (edge) connectivity of a graph G, denoted by cκl(G) (cλl(G)), is the minimum number of vertices (edges) whose removal from G results in a disconnected graph with at least l components. The pancake graph Pn is a popular underlying topology for distributed systems. In the paper, we determine the cκl(Pn) and cλl(Pn) for 3 ≤ l ≤ 5.