Graph Connectivity Research Articles

Finite-Time Distributed Algorithms for Verifying and Ensuring Strong Connectivity of Directed Networks

The strong connectivity of a directed graph associated with the communication network topology is crucial in ensuring the convergence of many distributed estimation/control/optimization algorithms. However, the assumption on the network's strong connectivity may not always be satisfied in practice. In addition, information on the overall network topology is often not available, e.g., due to privacy concerns or geographical constraints which calls for a distributed algorithm. This paper aims to fill a crucial gap in the literature due to the absence of a fully distributed algorithm to verify and ensure in finite-time the strong connectivity of a directed network. Specifically, inspired by the maximum consensus algorithm we propose distributed algorithms that enable individual node in a networked system to verify the strong connectivity of a directed graph and further, if necessary, augment a minimum number of new links to ensure the directed graph's strong connectivity. The proposed distributed algorithms are implemented without requiring information of the overall network topology and are scalable as they only require finite storage and converge in finite number of steps. Furthermore, the algorithms also preserve the privacy in terms of the overall network's topology. Finally, the proposed distributed algorithms are demonstrated and evaluated via numerical results.

On computing Banzhaf power index for k-edge connectivity in graphs

On computing Banzhaf power index for k-edge connectivity in graphs

Artist Similarity for Everyone: A Graph Neural Network Approach

Artist similarity plays an important role in organizing, understanding, and subsequently, facilitating discovery in large collections of music. In this paper, we present a hybrid approach to computing similarity between artists using graph neural networks trained with triplet loss. The novelty of using a graph neural network architecture is to combine the topology of a graph of artist connections with content features to embed artists into a vector space that encodes similarity. Additionally, we propose a simple and effective regularization method—<em>connection dropout</em>—which aims at improving results for long-tail artists, for which few existing connections are known. To evaluate the proposed method, we use two datasets: the open OLGA dataset, which contains artist similarities from AllMusic, together with content features from AcousticBrainz, and a larger, proprietary dataset. We find that using graph neural networks yields superior overall results compared to state-of-the-art methods. Beyond the overall evaluation, we investigate the effectiveness of the proposed model for long-tail artists. Such artists may benefit less from graph-based methods, since they typically have few known connections. We show that the proposed regularization approach clearly improves the performance for long-tail artists, without negatively affecting results for well-connected ones; it computes high-quality embeddings and good similarity scores for everyone.

On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. The graph $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become the most common technique in proving prism-hamiltonicity. Špacapan [S. Špacapan. A counterexample to prism-hamiltonicity of 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we answer his question in the negative, by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. In addition, we prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.

Developing an efficient functional connectivity-based geometric deep network for automatic EEG-based visual decoding

Neural decoding is of great importance in computational neuroscience to automatically interpret brain activities in order to address the challenging problem of mind-reading. Analyzing the vision-related EEG records is of great importance to discern the relation between visual perception and brain activity. Considering the recent advances and achievements in the field of deep neural networks, several architectures have been implemented to decode brain activities. In this paper, functional connectivity-based geometric deep network (FC-GDN) is proposed to leverage the spatio-temporal distributed information in EEG recordings evoked by images to directly extract hidden states of high-resolution time samples considering the functional connectivity between EEG channels. To this end, a topological connectivity graph is constructed based on the functional connectivity between EEG channels and time samples of each EEG channel are considered as a graph signal on top of corresponding graph node. Furthermore, a novel graph neural network architecture based on this efficient graph representation of EEG signals is proposed, in which visually provoked EEG recordings are used as training data in order to decode visual perception state of the participants in terms of extracted EEG patterns related to different image categories. The performance of the proposed FC-GDN is evaluated on the EEG-ImageNet dataset, consisting of 40 image categories and each category includes 50 sample images, shown to 6 participants while their EEG signals were recorded. The average accuracy of 98.4% is obtained for FC-GDN, showing an average improvement of 1.1% compared to the best state-of-the-art method.

Cross-view graph matching for incomplete multi-view clustering

Multi-view clustering (MVC) focuses on adaptively partitioning data from diverse sources into the respective groups and has been widely studied under the assumption of complete data. However, real-world applications often encounter a more realistic incomplete multi-view clustering (IMVC) problem, where data samples are missing in certain views. There are two challenges in IMVC: 1) how to reduce the impact of the missing instances; 2) how to effectively extract the consistent information to cluster the multi-view data. To address the challenges, we propose an adaptive graph learning framework for IMVC, which optimizes the missing information to fit the intrinsic structure of each view and clusters the multi-view data by cross-view graph matching. The proposed method mainly consists of three steps. Firstly, owing to the outstanding performance of the intrinsic structure of data, we adapt it to complete the missing data of each view. Secondly, the connection graph of each view from a projection space is adaptively constructed wherein the data points are connected if and only if they belong to the same cluster. Thirdly, we further introduce a cross-view graph matching strategy to appropriately utilize complementary multi-views information and preserve view-specific semantic information. We develop an iterative algorithm for solving the proposed model. Numerical experiments on several standard datasets demonstrate the effectiveness of the proposed method.

Hamiltonian in 5-Connected Graph

One of the branches of mathematics that studies the properties of graphs is graph theory. The purpose of this study is to find out how to prove the Hamiltonian on a 5-connected graph. Through the stages, including modeling a complete graph, as well as modeling a 5-connected graph using vertices and cut edges, it was found that is a graph that satisfies the characteristics of a 5-connected graph. Analysis of Hamiltonian on 5-connected graph that 5-connected graph is Hamilton's invention, but it is not uniquely Hamiltonian because it has more than one Hamiltonian circuit.Keywords: k-connected graph, Hamiltonian, vertex cuts

Distribution of subtree sums

Motivated by the conjectures of Bondy and Malkevitch on (almost-)pancyclicity of 4-connected planar graphs, we study the distribution of subtree weights in trees with positive integer vertex weights. Let T be a tree on N1 vertices and let c:V(T)→N be vertex weights such that ∑v∈V(T)c(v)=N2. Let TS(T;c) denote the set of weights of subtrees of T, and [N1,K]N≔{k∈N:N1≤k≤K}. We show that, for every integer N1≤K≤N2, if N2<2N1−1, we have |TS(T;c)∩[N1,K]N|≥K−N1+12. As a corollary, we show that, for every planar hamiltonian graph G with minimum degree at least 4 and for every integer K with N≤K≤|V(G)|, where N≔⌈|V(G)|2⌉+3, there exist K−N+12 distinct cycle lengths in [N,K]N.

A linear algorithm for semi-external cutnode computation

In the literature, many algorithms have been proposed for finding cutnodes on undirected graphs, since cutnodes are crucial to graph connectivity. Here, a cutnode of an undirected graph G is a node of G, whose deletion will cause a reachable pair of the other nodes in G to be unreachable. Currently, the difficulty of maintaining the entire G in the main memory makes researchers pay attention to compute cutnodes on semi-external memory model. This paper shows that traditional semi-external algorithms are limited by their unbounded time and I/O consumption, making them impractical when G is relatively large or complex. Thus, we propose a linear semi-external cutnode computation algorithm, named SECN. Assuming that G has n nodes and m edges, and B is the disk block size. SECN is the first that can find all the cutnodes of G in O(m+n) time and with O(mB) I/O cost on the semi-external memory model, as far as we know. SECN also has a smaller minimum memory space requirement than traditional algorithms. Our experimental evaluation conducted on both synthetic and real graphs confirms that SECN significantly outperforms existing algorithms (up to 103 times faster).

Analyzing connectivity reliability and critical units for highway networks in high-intensity seismic region using Bayesian network

It is important to evaluate the connectivity reliability of highway networks for the emergency response and rehabilitation of transportation systems in high-intensity seismic regions. Given the complexity and uncertainty of seismic damages of highway networks in high-intensity seismic region, this paper describes a Bayesian network (BN) model for evaluating the network connectivity reliability and identifying critical units. The empirical prediction method is employed to compute the connectivity probability of highway units based on the structural damage of units under earthquakes. A success tree is used to construct the network connectivity graph. Then, the network connectivity graph is converted into the BN model by BN method with the connectivity probability of highway units as prior probability. Sensitivity analysis and Bayesian updating are performed in BN to identify critical units and dynamically assess the connectivity reliability of highway network. The proposed model is applied to a highway network composed of G213 and S9 in the Wenchuan Earthquake. The results show that the BN model integrates the structural damage of units with the functional performance of the highway network in high-intensity seismic region. Bayesian updating allows the posterior probability of segments and origin–destination pairs to be computed, providing an online evaluation of the functional performance of the highway network. The identification of critical units at each stage enables seismic reinforcement priority, thus contributing to the rehabilitation on connectivity reliability of network system.

Analysis of EEG brain connectivity of children with ADHD using graph theory and directional information transfer.

Research shows that Attention Deficit Hyperactivity Disorder (ADHD) is related to a disorder in brain networks. The purpose of this study is to use an effective connectivity measure and graph theory to examine the impairments of brain connectivity in ADHD. Weighted directed graphs based on electroencephalography (EEG) signals of 61 children with ADHD and 60 healthy children were constructed. The edges between two nodes (electrodes) were calculated by Phase Transfer Entropy (PTE). PTE is calculated for five frequency bands: delta, theta, alpha, beta, and gamma. The graph theory measures were divided into two categories: global and local. Statistical analysis with global measures indicates that in children with ADHD, the segregation of brain connectivity increases while the integration of the brain connectivity decreases compared tohealthy children. These brain network differences were identified in the delta and theta frequency bands. The classification accuracy of 89.4% is obtained for both in-degree and strength measures in the theta band. Our result indicated local graph measures classified ADHD and healthy subjects with accuracy of 91.2 and 90% in theta anddelta bands, respectively. Our analysis may provide a new understanding of the differences in the EEG brain network of children with ADHD and healthy children.

Quantum variational learning for quantum error-correcting codes

Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., ((n,2n&amp;#x2212;6,3))2 for n from 7 to 14. We also found new ((6,2,3))2 and ((7,2,3))2 codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a ((7,3,3))2 code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.

The maximum size of a nonhamiltonian graph with given order and connectivity

Motivated by work of Erdős, Ota determined the maximum size g(n,k) of a k-connected nonhamiltonian graph of order n in 1995. But for some pairs n,k, the maximum size is not attained by a graph of connectivity k. For example, g(15,3)=77 is attained by a unique graph of connectivity 7, not 3. In this paper we obtain more precise information by determining the maximum size of a nonhamiltonian graph of order n and connectivity k, and determining the extremal graphs. Consequently we solve the corresponding problem for nontraceable graphs.

Complete family reduction and spanning connectivity in line graphs

Complete families of connected graphs, introduced by Catlin in the 1980s, have been known useful in the study of certain graphical properties that are closed under taking contractions. We show that given any complete family C of connected graphs such that C contains graphs with sufficiently many edge-disjoint spanning trees, for any real number a and b with 0<a<1, there exists a finite obstacle family F=F(a,b,C) such that for any simple graph G on n vertices satisfying the Ore-type degree conditionmin⁡{dG(u)+dG(v):u,v∈V(G) and uv∉E(G)}≥an+b, either G∈C or G can be contracted to a member in F. This result is applied to the study of spanning connectivity of line graphs. The spanning connectivity is the largest integer s such that for any k with 0≤k≤s and for any u,v∈V(G) with u≠v, G has a spanning subgraph H consisting of k internally disjoint (u,v)-paths. Z. Ryjáček and P. Vrána in [J. Graph Theory, 66 (2011) 152-173] prove that a fascinating conjecture of Thomassen on hamiltonian line graphs is equivalent to that every essentially 4-edge-connected graph has a 2-spanning-connected line graph. We prove that for any essentially 3-edge-connected graph G and any positive integer s, if G satisfies an Ore degree condition lower bounded by an arbitrary linear function in the number of vertices, then L(G) is s-spanning-connected with only finitely many contraction obstacles. When s=3, we determine a finite graph family J′(n) such that for every simple graph G on n≥156 vertices with κ(L(G))≥3 and satisfying d(u)+d(v)≥2(n−6)5 for any pair of nonadjacent vertices u and v, we have either κ⁎(L(G))≥3 or G is contractible to a member in J′(n).

Sharp upper bounds on the Q-index of (minimally) 2-connected graphs with given size

A 2-connected graph G is minimally 2-connected if deleting any arbitrary chosen edge of G always leaves a graph which is not 2-connected. In this paper, we give sharp upper bounds on the Q -index of (minimally) 2-connected graphs with given size, and characterize the corresponding extremal graphs completely.

GNGraph: Self-Organizing Maps for Autonomous Aerial Vehicle Planning

The present letter tackles the problem of planning a collision-free path in a known environment from a general point of view. We address the problem by using an unsupervised learning algorithm to generate a sparse graph representing the topological structure of the environment and use it for planning paths in 3D spaces. We propose GNGraph, an integrated solution combining the Growing Neural Gas algorithm to generate the sparse graph, a stop criterion to guarantee the graph's connectivity and a collision check to assess the edges and nodes validity. The proposed solution has been tested on simulated and real environment maps, and compared against a state-of-the-art graph planning algorithm among other global planning methods.

Brain Connectivity Based Graph Convolutional Networks and Its Application to Infant Age Prediction.

Infancy is a critical period for the human brain development, and brain age is one of the indices for the brain development status associated with neuroimaging data. The difference between the predicted age based on neuroimaging and the chronological age can provide an important early indicator of deviation from the normal developmental trajectory. In this study, we utilize the Graph Convolutional Network (GCN) to predict the infant brain age based on resting-state fMRI data. The brain connectivity obtained from rs-fMRI can be represented as a graph with brain regions as nodes and functional connections as edges. However, since the brain connectivity is a fully connected graph with features on edges, current GCN cannot be directly used for it is a node-based method for sparse graphs. Hence, we propose an edge-based Graph Path Convolution (GPC) method, which aggregates the information from different paths and can be naturally applied on dense graphs. We refer the whole model as Brain Connectivity Graph Convolutional Networks (BC-GCN). Further, two upgraded network structures are proposed by including the residual and attention modules, referred as BC-GCN-Res and BC-GCN-SE to emphasize the information of the original data and enhance influential channels. Moreover, we design a two-stage coarse-to-fine framework, which determines the age group first and then predicts the age using group-specific BC-GCN-SE models. To avoid accumulated errors from the first stage, a cross-group training strategy is adopted for the second stage regression models. We conduct experiments on infant fMRI scans from 6 to 811 days of age. The coarse-to-fine framework shows significant improvements when being applied to several models (reducing error over 10 days). Comparing with state-of-the-art methods, our proposed model BC-GCN-SE with coarse-to-fine framework reduces the mean absolute error of the prediction from >70 days to 49.9 days. The code is now available at https://github.com/SCUT-Xinlab/BC-GCN.

Dynamic Graph-Based Feature Learning With Few Edges Considering Noisy Samples for Rotating Machinery Fault Diagnosis

Due to its ability to learn the relationship among nodes from graph data, the graph convolution network (GCN) has received extensive attention. In the machine fault diagnosis field, it needs to construct input graphs reflecting features and relationships of the monitoring signals. Thus, the quality of the input graph affects the diagnostic performance. But it still has two limitations: 1) the constructed input graph usually has redundant edges, consuming excessive computational costs; 2) the constructed input graph cannot reflect the relationship between the noisy signals well. In order to overcome them, a dynamic graph-based feature learning with few edges considering noisy samples is proposed for rotating machinery fault diagnosis in this article. Noisy vibration signals are converted into one spectrum feature-based static graph, where redundant edges are simplified by the distance metric function. Edge connections of the input static graph are updated according to the relationship among high-level features extracted by the GCN. Based on this, dynamic input graphs are reconstructed as new graph representations for noisy samples. To verify the effectiveness of the proposed method, validation experiments were conducted on practical platforms, and results show that the dynamic input graph with few edges can effectively improve the diagnostic performance under different SNRs.

On Connectivity of Conditional Configuration Graphs under Destruction

Configuration graphs with node degrees being independent identically distributed random variables following the power-law distribution with a shift are studied. We consider conditional graphs that consist of only one connected component. Computer simulations are used to study how the connectivity of such graphs changes under the two types of destruction processes: the “target attack” and the “random breakdown”. We propose models for the dependence of the probability of graph connectivity destruction on the graph size, the node degree distribution parameter, and the number of deleted nodes.

The proper 2-connection number and size of graphs

An edge-coloured graph G is called properly k -connected if any two vertices are connected by at least k internally vertex-disjoint paths whose edges are properly coloured. The proper k -connection number of a k -connected graph G , denoted by p c k ( G ) , is the minimum number of colours that are needed in order to make it properly k -connected. Let f k ( n , r ) be the minimum integer such that every k -connected graph of order n and size at least f k ( n , r ) has proper k -connection number at most r . Our main results are as follows: (1) Let G be a 2-connected graph of order n ≥ 19 . If | E ( G ) | ≥ n − 3 2 + 7 , then p c 2 ( G ) = 2 . (2) Let G be a hamiltonian graph G of order n ≥ 50 and size | E ( G ) | > n 2 4 . Then p c 2 ( G ) = 2 . We also determine lower and upper bounds for f 2 ( n , r ) .