Graph Connectivity Research Articles

Low-weight high-distance error-correcting fermionic encodings

We perform an extended numerical search for practical fermion-to-qubit encodings with error-correcting properties. Ideally, encodings should strike a balance between a number of the seemingly incompatible attributes, such as having a high minimum distance, low-weight fermionic logical operators, a small qubit to fermionic mode ratio and a simple qubit connectivity graph including ancilla qubits for the measurement of stabilizers. Our strategy consists of a three-step procedure in which we: First generate encodings with code distances up to d≤4 by a brute-force enumeration technique; subsequently, we use these encodings as starting points and apply Clifford deformations to them which allows us to identify higher-distance codes with d≤7; finally, we optimize the hardware connectivity graphs of resulting encodings in terms of the graph thickness and the number of connections per qubit. We report multiple promising high-distance encodings which significantly improve the weights of stabilizers and logical operators compared to previously reported alternatives. Published by the American Physical Society 2024

How Can Graph Theory Inform the Dual-stream Model of Speech Processing? A Resting-state Functional Magnetic Resonance Imaging Study of Stroke and Aphasia Symptomology

Abstract The dual-stream model of speech processing describes a cortical network involved in speech processing. However, it is not yet known if the dual-stream model represents actual intrinsic functional brain networks. Furthermore, it is unclear how disruptions after a stroke to the functional connectivity of the dual-stream model's regions are related to speech production and comprehension impairments seen in aphasia. To address these questions, in the present study, we examined two independent resting-state fMRI data sets: (1) 28 neurotypical matched controls and (2) 28 chronic left-hemisphere stroke survivors collected at another site. We successfully identified an intrinsic functional network among the dual-stream model's regions in the control group using functional connectivity. We then used both standard functional connectivity analyses and graph theory approaches to determine how this connectivity may predict performance on clinical aphasia assessments. Our findings provide evidence that the dual-stream model of speech processing is an intrinsic network as measured via resting-state MRI and that functional connectivity of the hub nodes of the dual-stream network defined by graph theory methods, but not overall average network connectivity, is weaker in the stroke group than in the control participants. In addition, the functional connectivity of the hub nodes predicted linguistic impairments on clinical assessments. In particular, the relative strength of connectivity of the right hemisphere's homologues of the left dorsal stream hubs to the left dorsal hubs, versus to the right ventral stream hubs, is a particularly strong predictor of poststroke aphasia severity and symptomology.

On Connectivity of Multidimensional Fuzzy Graphs and its Applications

Connectivity, average connectivity, and its related concepts are prominent notions in fuzzy graph theory that have applications in various areas, including neural networks, transportation problems, cluster analysis, and telecommunication. This work introduces the concepts of connectivity index, average connectivity index, type 2 connectivity index, etc. to a more generalized variant of a fuzzy graph, namely the multidimensional fuzzy graph. Basic results such as the relation of these parameters to vertices and edges, their bounds in some particular multidimensional fuzzy graphs, etc. are studied with proper illustrations. Theorems on the connectivity index of multidimensional fuzzy graphs are obtained after performing operations like direct product, tensor product, composition, join, and so on. Finally, a decision-making problem that can be effectively handled using the connectivity index is demonstrated and compared with others and similar models.

Bounding the beta invariant of 3-connected graphs

The beta invariant is closely related to the Chromatic and Tutte Polynomials and has been extensively studied, see Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22], Crapo [A higher invariant for matroids, J. Combin. Theory 2 (1967) 406–417], Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11], Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277], and others. Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11] established that a [Formula: see text]-connected graph with [Formula: see text] vertices possesses a beta invariant of at least [Formula: see text], reaching equality only when the graph is a wheel or the Prism. Additionally, Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277] provided characterizations for matroids with beta invariant values of two, three, and four. This paper extends the findings of Lee and Wu by offering a comprehensive characterization of all [Formula: see text]-connected graphs with [Formula: see text] vertices that have a beta invariant of either [Formula: see text] or [Formula: see text]. As a consequence, we prove that any [Formula: see text]-connected graph with at least [Formula: see text] vertices other than a Wheel has a beta invariant of at least [Formula: see text].

Dynamic graph spatial-temporal dependence information extraction for remaining useful life prediction of rolling bearings

As a powerful tool for learning high-dimensional data representation, graph neural networks (GNN) have been applied to predict the remaining useful life (RUL) of rolling bearings. Existing GNN-based RUL prediction methods predominantly rely on constant pre-constructed graphs. However, the degradation of bearings is a dynamic process, and the dependence information between features may change at different moments of degradation. This article introduces a method for RUL prediction based on dynamic graph spatial-temporal dependence information extraction. The raw signal is segmented into multiple periods, and multiple features of each period data are extracted. Then, the correlation coefficient analysis is conducted, and the feature connection graph of each period is constructed based on different analytical results, thereby dynamically mapping the degradation process. The graph data is fed into graph convolutional networks (GCN) to extract spatial dependence between the graph node features in different periods. To make up for the shortcomings of GCN in temporal dependence extraction, the TimesNet module is introduced. TimesNet considers the two-dimensional changes of time series data and can extract the temporal dependence of graph data within and between different time cycles. Experimental results based on the PHM2012 dataset show that the average RUL prediction error of the proposed method is 17.4%, outperforming other comparative methods.

Inductive Subgraph Embedding for Link Prediction

AbstractLink prediction, which aims to infer missing edges or predict future edges based on currently observed graph connections, has emerged as a powerful technique for diverse applications such as recommendation, relation completion, etc. While there is rich literature on link prediction based on node representation learning, direct link embedding is relatively less studied and less understood. One common practice in previous work characterizes a link by manipulate the embeddings of its incident node pairs, which is not capable of capturing effective link features. Moreover, common link prediction methods such as random walks and graph auto-encoder usually rely on full-graph training, suffering from poor scalability and high resource consumption on large-scale graphs. In this paper, we propose Inductive Subgraph Embedding for Link Prediciton (SE4LP) — an end-to-end scalable representation learning framework for link prediction, which utilizes the strong correlation between central links and their neighborhood subgraphs to characterize links. We sample the “link-centric induced subgraphs” as input, with a subgraph-level contrastive discrimination as pretext task, to learn the intrinsic and structural link features via subgraph classification. Extensive experiments on five datasets demonstrate that SE4LP has significant superiority in link prediction in terms of performance and scalability, when compared with state-of-the-art methods. Moreover, further analysis demonstrate that introducing self-supervision in link prediction can significantly reduce the dependence on training data and improve the generalization and scalability of model.

Decentralized Privacy Preservation for Critical Connections in Graphs

Decentralized Privacy Preservation for Critical Connections in Graphs

Adaptive connectivity-preserving formation control with a dynamic event-triggering mechanism

Connectivity preservation for real-life multi-agent systems is key to designing an effective and reliable control algorithm to achieve desired objectives. But the coexistence of system nonlinearities and uncertainties makes it difficult for the algorithm design, and especially in the event-triggered setting, such difficulty becomes severer due to the demand for integration of multiple ingredients. This paper focuses on proposing an adaptive connectivity-preserving formation control strategy that incorporates a dynamic event-triggering mechanism, in the scenario of the unknown control directions and intrinsic nonlinearities coupling with parameter uncertainties. First, a cluster of potential functions serving as control barrier functions are given to maintain the prescribed connectivity of communication graph. Second, two types of dynamic gains (one type is based on a Nussbaum function) are introduced for agents to cope with the system nonlinearities, uncertainties and the adverse effect of the execution error. As such, an adaptive event-triggered formation protocol is constructed such that the desired formation with connectivity preservation is achieved while a positive minimum inter-execution interval is ensured for each agent to avoid Zeno behavior. Particularly, the threshold in the developed event-triggering mechanism is dynamically adjusted online, rather than static, which better improves resource efficiency. A simulation example is given to demonstrate the effectiveness and advantage of the proposed strategy.

Structure connectivity of a class of Cayley graph

As a generalization of connectivity, structure connectivity was proposed. For connected graphs G and T, the T-structure connectivity κ ( G ; T ) (resp. T-substructure connectivity κ s ( G ; T ) ) of G is the minimum cardinality of a set of subgraphs F of G that each is isomorphic to T (resp. a connected subgraph of T) such that G − F is disconnected or the singleton. n-dimensional graph Q n + is a class of Cayley graph, which obtained by adding edges to n-dimensional hypercubes. In this paper, we study star, path and cycle structure connectivity and substructure connectivity of graph Q n + for n ≥ 5 . For star ( K 1 , m ) structure, K 1 , m -structure (substructure) connectivity of Q n + is ⌈ n + 1 2 ⌉ for 2 ≤ m ≤ n − 1 . For path ( P k ) and cycle ( C k ) structures, we show that if 3 ≤ k ≤ 2 n , then P k -structure (substructure) connectivity is ⌈ 2 ( n + 1 ) k + 1 ⌉ and ⌈ 2 ( n + 1 ) k ⌉ for odd k and even k, respectively. We also obtain that P k -structure connectivity is equal to C k -substructure connectivity. Further, we calculate κ ( Q n + ; C 2 k ) = ⌈ n + 1 k ⌉ for 3 ≤ k ≤ n .

Edge computing task scheduling method based on user’s social relations: a construction and solution for Smart City Library

In the realm of the development of a Smart City Library, the integration of robust edge computing is vital. The research suggests a novel task-scheduling model for edge computing, leveraging user’s social relationships. Analyzing these connections involves constructing a user’s social relationship graph by implementing mathematical convolution and the Jaccard similarity ratio. This precise quantification of social ties ensures secure and reliable task scheduling. An equipment connection graph of a user equipment service is also crafted based on Euclidean distance, aligning task scheduling with device-to-device (D2D) communication conditions. Combining a user’s social relationship graph and a user’s device-service device connection graph creates a task-device bipartite graph. On the other hand, the calculation of a task execution cost and edge weight determination finalize a scheduling model. Implementing the proposed method for constructing a model for edge computing task scheduling based on utilizing the Kuhn–Munkres (KM) algorithm demonstrates positive impacts, which are few delays and less energy consumption, on edge computing task scheduling. For instance, when the social threshold score changes from 02. To 0.6, the total task execution delay time increases from 23 to 32, which is the best when compared with other algorithms. The approach strengthens security and reliability while decreasing task execution delays and energy consumption. This research advances edge computing for Smart City Libraries, promising transformative implications.

Improved rate-distance trade-offs for quantum codes with restricted connectivity

Abstract For quantum error-correcting codes to be realizable, it is important that the qubits subject to the code constraints exhibit some form of limited connectivity. The works of Bravyi and Terhal (2009 New J. Phys. 11 043029) (BT) and Bravyi et al (2010 Phys. Rev. Lett. 104 050503) (BPT) established that geometric locality constrains code properties—for instance [ [ n , k , d ] ] quantum codes defined by local checks on the D-dimensional lattice must obey k d 2 / ( D − 1 ) ⩽ O ( n ) . Baspin and Krishna (2022 Quantum 6 711) studied the more general question of how the connectivity graph associated with a quantum code constrains the code parameters. These trade-offs apply to a richer class of codes compared to the BPT and BT bounds, which only capture geometrically-local codes. We extend and improve this work, establishing a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph. We also obtain a distance bound that covers all stabilizer codes with a particular separation profile, rather than only LDPC codes.

Enhancing non-invasive pre-surgical evaluation through functional connectivity and graph theory in drug-resistant focal epilepsy

BackgroundEpilepsy, characterized as a network disorder, involves widely distributed areas following seizure propagation from a limited onset zone. Accurate delineation of the epileptogenic zone (EZ) is crucial for successful surgery in drug-resistant focal epilepsy. While visual analysis of scalp electroencephalogram (EEG) primarily elucidates seizure spreading patterns, we employed brain connectivity techniques and graph theory principles during the pre-ictal to ictal transition to define the epileptogenic network. MethodCortical sources were reconstructed from 40-channel scalp EEG in five patients during pre-surgical evaluation for focal drug-resistant epilepsy. Temporal Granger connectivity was estimated ten seconds before seizure and at seizure onset. Results have been analyzed using some centrality indices taken from Graph theory (Outdegree, Hubness). A new lateralization index is proposed by taking into account the sum of the most relevant hubness values across left and right regions of interest. ResultsIn three patients with positive surgical outcomes, analysis of the most relevant Hubness regions closely aligned with clinical hypotheses, demonstrating consistency in EZ lateralization and location. In one patient, the method provides unreliable results due to the abundant movement artifacts preceding the seizure. In a fifth patient with poor surgical outcome, the proposed method suggests a wider epileptic network compared with the clinically suspected EZ, providing intriguing new indications beyond those obtained with traditional electro-clinical analysis. ConclusionsThe proposed method could serve as an additional tool during pre-surgical non-invasive evaluation, complementing data obtained from EEG visual inspection. It represents a first step toward a more sophisticated analysis of seizure onset based on connectivity imbalances, electrical propagation, and graph theory principles.

Graph neural networks in multi-stained pathological imaging: extended comparative analysis of Radiomic features.

This study investigates the application of Radiomic features within graph neural networks (GNNs) for the classification of multiple-epitope-ligand cartography (MELC) pathology samples. It aims to enhance the diagnosis of often misdiagnosed skin diseases such as eczema, lymphoma, and melanoma. The novel contribution lies in integrating Radiomic features with GNNs and comparing their efficacy against traditional multi-stain profiles. We utilized GNNs to process multiple pathological slides as cell-level graphs, comparing their performance with XGBoost and Random Forest classifiers. The analysis included two feature types: multi-stain profiles and Radiomic features. Dimensionality reduction techniques such as UMAP and t-SNE were applied to optimize the feature space, and graph connectivity was based on spatial and feature closeness. Integrating Radiomic features into spatially connected graphs significantly improved classification accuracy over traditional models. The application of UMAP further enhanced the performance of GNNs, particularly in classifying diseases with similar pathological features. The GNN model outperformed baseline methods, demonstrating its robustness in handling complex histopathological data. Radiomic features processed through GNNs show significant promise for multi-disease classification, improving diagnostic accuracy. This study's findings suggest that integrating advanced imaging analysis with graph-based modeling can lead to better diagnostic tools. Future research should expand these methods to a wider range of diseases to validate their generalizability and effectiveness.

Transparent sparse graph pathway network for analyzing the internal relationship of lung cancer.

While it is important to find the key biomarkers and improve the accuracy of disease models, it is equally important to understand their interaction relationships. In this study, a transparent sparse graph pathway network (TSGPN) is proposed based on the structure of graph neural networks. This network simulates the action of genes in vivo, adds to prior knowledge, and improves the model's accuracy. First, the graph connection was constructed according to protein-protein interaction networks and competing endogenous RNA (ceRNA) networks, from which some noise or unimportant connections were spontaneously removed based on the graph attention mechanism and hard concrete estimation. This realized the reconstruction of the ceRNA network representing the influence of other genes in the disease on mRNA. Next, the gene-based interpretation was transformed into a pathway-based interpretation based on the pathway database, and the hidden layer was added to realize the high-dimensional analysis of the pathway. Finally, the experimental results showed that the proposed TSGPN method is superior to other comparison methods in F1 score and AUC, and more importantly, it can effectively display the role of genes. Through data analysis applied to lung cancer prognosis, ten pathways related to LUSC prognosis were found, as well as the key biomarkers closely related to these pathways, such as HOXA10, hsa-mir-182, and LINC02544. The relationship between them was also reconstructed to better explain the internal mechanism of the disease.

Coding Dynamics of the Striatal Networks During Learning.

The rat dorsomedial (DMS) and dorsolateral striatum (DLS), equivalent to caudate nucleus and putamen in primates, are required for goal-directed and habit behaviour, respectively. However, it is still unclear whether and how this functional dichotomy emerges in the course of learning. In this study, we investigated this issue by recording DMS and DLS single neuron activity in rats performing a continuous spatial alternation task, from the acquisition to optimized performance. We first applied a classical analytical approach to identify task-related activity based on the modifications of single neuron firing rate in relation to specific task events or maze trajectories. We then used an innovative approach based on Hawkes process to reconstruct a directed connectivity graph of simultaneously recorded neurons, that was used to decode animal behavior. This approach enabled us to better unravel the role of DMSand DLS neural networks across learning stages. We showed that DMS and DLS display different task-related activity throughout learning stages, and the proportion of coding neurons over time decreases in the DMS and increases in the DLS. Despite these major differences, the decoding power of both networks increases during learning. These results suggest that DMS and DLS neural networks gradually reorganize in different ways in order to progressively increase their control over the behavioral performance.

Spectral Bipartition via Gap Cut on DNA Sequences

Deoxyribonucleic Acid (DNA) and graph partitioning are two distinct fields of study which can be linked in the structure of biological networks. Graph partitioning has been extensively studied but not its application in the biological field. This research explored on the application of spectral graph partitioning in DNA splicing, aiming to simulate the cleavage of DNA by performing spectral bipartition on DNA sequences in a DNA splicing system. This research incorporates Fiedler theory and algebraic graph theory, which are commonly utilized in network analysis and the analysis of graph connectivity. Some DNA sequences of even length are selected and expressed in graphical representations. The adjacency matrix, Laplacian matrix, and degree matrix are computed from the graphs, as well as the Fiedler value and Fiedler vector associated with the graphs. Gap cut is used as a method of spectral bipartition which produces two partitions of DNA sequence of unequal lengths. The generalizations of gap cut on DNA sequences of even length are provided as lemmas and theorem.

Adaptive Neighbors Graph Learning for Large-Scale Data Clustering using Vector Quantization and Self-Regularization

In traditional adaptive neighbors graph learning (ANGL)-based clustering, the time complexity is more than O(n2), where n is the number of data points, which is not scalable for large-scale data problems in real applications. Subsequently, ANGL adds a balance regularization to its objective function to avoid the sparse over-fitting problem in the learned similarity graph matrix. Still, the regularization may leads to many weak connections between data points in different clusters. To address these problems, we propose a new fast clustering method, namely, Adaptive Neighbors Graph Learning for Large-Scale Data Clustering using Vector Quantization and Self-Regularization (ANGL-LDC), to perform vector quantization (VQ) on original data and feed the obtained VQ data as the input in the n×n similarity graph matrix learning. Hence, the n×n similarity graph matrix learning problem is simplified to weighted m×m(m≪n) graph learning problem, where m is the number of distinct points and weight is the duplicate times of distinct points in VQ data. Consequently, the time complexity of ANGL-LDC is much lower than that of ANGL. At the same time, we propose a new ANGL objective function with a graph connection self-regularization mechanism, where the ANGL-LDC objective function will get an infinity value if the value of one graph connection is equal to 1. Therefore, ANGL-LDC naturally avoids obtaining the sparse over-fitting problem since we need to minimize the value of ANGL-LDC’s objective function. Experimental results on synthetic and real-world datasets demonstrate the scalability and effectiveness of ANGL-LDC.

D $d$‐connectivity of the random graph with restricted budget

AbstractIn this short note, we consider a graph process recently introduced by Frieze, Krivelevich and Michaeli. In their model, the edges of the complete graph are ordered uniformly at random and are then revealed consecutively to a player called Builder. At every round, Builder must decide if they accept the edge proposed at this round or not. We prove that, for every , Builder can construct a spanning ‐connected graph after rounds by accepting edges with probability converging to 1 as . This settles a conjecture of Frieze, Krivelevich and Michaeli.

Computing the cut locus, Voronoi diagram, and signed distance function of polygons

This paper presents a new method for the computation of the generalized Voronoi diagram of planar polygons. First, we show that the vertices of the cut locus can be computed efficiently. This is achieved by enumerating the tripoints of the polygon, a superset of the cut locus vertices. This is the set of all points that are of equal distance to three distinct topological entities. Then our algorithm identifies and connects the appropriate tripoints to form the cut locus vertex connectivity graph, where edges define linear or parabolic boundary segments between the Voronoi regions, resulting in the generalized Voronoi diagram. Our proposed method is validated on complex polygon soups. We apply the algorithm to represent the exact signed distance function of the polygon by augmenting the Voronoi regions with linear and radial functions, calculating the cut locus both inside and outside.

Multi-Graph Assessment of Temporal and Extratemporal Lobe Epilepsy in Resting-State fMRI

Epilepsy is a common neurological disorder that affects millions of people worldwide, disrupting brain networks and causing recurrent seizures. In this regard, investigating the distinctive characteristics of brain connectivity is crucial to understanding the underlying neural processes of epilepsy. However, the various graph-theory frameworks and different estimation measures may yield significant variability among the results of different studies. On this premise, this study investigates the brain network topological variations between patients with temporal lobe epilepsy (TLE) and extratemporal lobe epilepsy (ETLE) using both directed and undirected network connectivity methods as well as different graph-theory metrics. Our results reveal distinct topological differences in connectivity graphs between the two epilepsy groups, with TLE patients displaying more disassortative graphs at lower density levels compared to ETLE patients. Moreover, we highlight the variations in the hub regions across different network metrics, underscoring the importance of considering various centrality measures for a comprehensive understanding of brain network dynamics in epilepsy. Our findings suggest that the differences in brain network organization between TLE and ETLE patients could be attributed to the unique characteristics of each epilepsy type, offering insights into potential biomarkers for type-specific epilepsy diagnosis and treatment.