Node Degree Research Articles

Detecting planted partition in sparse multilayer networks

Abstract Multilayer networks are used to represent the interdependence between the relational data of individuals interacting with each other via different types of relationships. To study the information-theoretic phase transitions in detecting the presence of planted partition among the nodes of a multilayer network with additional nodewise covariate information and diverging average degree, Ma and Nandy (2023, IEEE Trans. Inf. Theory, 69, 3203–3239) introduced Multi-Layer Contextual Stochastic Block Model. In this paper, we consider the problem of detecting planted partitions in the Multi-Layer Contextual Stochastic Block Model, when the average node degrees for each network are greater than $1$. We establish the sharp phase transition threshold for detecting the planted bi-partition. Above the phase-transition threshold testing the presence of a bi-partition is possible, whereas below the threshold no procedure to identify the planted bi-partition can perform better than random guessing. We further establish that the derived detection threshold coincides with the threshold for weak recovery of the partition and provides a quasi-polynomial time algorithm to estimate it.

A Grid-Wide Comprehensive Evaluation Method of Power Quality Based on Complex Network Theory

To achieve a hierarchical and quantitative evaluation of grid-wide power quality in the distribution network, reflecting the overall power quality level of the distribution network, a comprehensive evaluation method for power quality in a grid-wide system based on complex network theory is proposed. Firstly, based on the propagation characteristics of power quality disturbances, a power quality evaluation index system is constructed. Secondly, to reflect the constraint effect of the local power quality level of nodes on the overall power quality level of the distribution system, corresponding indices such as improved node degree, improved node electrical betweenness, and node self-healing capability are proposed based on complex network theory, and the power quality influence degree of nodes is calculated. Then, the GRA-ANP (Grey Relational Analysis–Analytic Network Process) subjective weight calculation method is improved by introducing grey relational analysis to address the impact of differences in different decision-making results. Based on power quality monitoring data, the entropy weight method is used for objective weighting. To avoid the partiality of a single weight evaluation result, the game equilibrium algorithm is employed to calculate the comprehensive weight of each power quality index. Subsequently, considering the correlation and dependency among indices, the VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) method is used to obtain the power quality grade of each node. Combining this with the calculation of the power quality influence degree of nodes, the overall power quality grade of the distribution network is determined, achieving a hierarchical and quantitative evaluation of power quality in the entire distribution system. Finally, through a case study analysis of an improved 13-node distribution network, it is verified that the proposed method can fully extract data information and produce comprehensive and accurate power quality assessment results by comparing it with other methods. This provides strong support for the safe and stable operation of the distribution system and the subsequent optimization and management of power quality.

Networks and their degree distribution, leading to a new concept of small worlds

The degree distribution, referred to as the delta-sequence of a network is studied. Using the non-normalized Lorenz curve, we apply a generalized form of the classical majorization partial order.Next, we introduce a new class of small worlds, namely those based on the degrees of nodes in a network. Similar to a previous study, small worlds are defined as sequences of networks with certain limiting properties. We distinguish between three types of small worlds: those based on the highest degree, those based on the average degree, and those based on the median degree. We show that these new classes of small worlds are different from those introduced previously based on the diameter of the network or the average and median distance between nodes. However, there exist sequences of networks that qualify as small worlds in both senses of the word, with stars being an example. Our approach enables the comparison of two networks with an equal number of nodes in terms of their “small-worldliness”.Finally, we introduced neighboring arrays based on the degrees of the zeroth and first-order neighbors.

Spatiotemporal Variations in Co-Occurrence Patterns of Planktonic Prokaryotic Microorganisms along the Yangtze River.

Bacteria and archaea are foundational life forms on Earth and play crucial roles in the development of our planet's biological hierarchy. Their interactions influence various aspects of life, including eukaryotic cell biology, molecular biology, and ecological dynamics. However, the coexistence network patterns of these microorganisms within natural river ecosystems, vital for nutrient cycling and environmental health, are not well understood. To address this knowledge gap, we systematically explored the non-random coexistence patterns of planktonic bacteria and archaea in the 6000-km stretch of the Yangtze River by using high-throughput sequencing technology. By analyzing the O/R ratio, representing the divergence between observed (O%) and random (R%) co-existence incidences, and the module composition, we found a preference of both bacteria and archaea for intradomain associations over interdomain associations. Seasons notably influenced the co-existence of bacteria and archaea, and archaea played a more crucial role in spring as evidenced by their predominant presence of interphyla co-existence and more species as keystone ones. The autumn network was characterized by a higher node or edge number, greater graph density, node degree, degree centralization, and nearest neighbor degree, indicating a more complex and interconnected structure. Landforms markedly affected microbial associations, with more complex networks and more core species found in plain and non-source areas. Distance-decay analysis suggested the importance of geographical distance in shaping bacteria and archaea co-existence patterns (more pronounced in spring). Natural, nutrient, and metal factors, including water temperature, NH4+-N, Fe, Al, and Ni were identified as crucial determinants shaping the co-occurrence patterns. Overall, these findings revealed the dynamics of prokaryotic taxa coexistence patterns in response to varying environmental conditions and further contributed to a broader understanding of microbial ecology in freshwater biogeochemical cycling.

Nodal degree centrality in the default mode-like network of the TgF344-AD Alzheimer’s disease rat model as a measure of early network alterations

This study investigates brain network alterations in the default mode-like network (DMLN) at early stages of disease progression in a rat model of Alzheimer’s disease (AD) with application in the development of early diagnostic biomarkers of AD in translational studies. Thirteen male TgF344-AD (TG) rats, and eleven male wild-types (WT) littermates underwent longitudinal resting-state fMRI at the age of 4 and 6 months (pre and early-plaque stages of AD). Alterations in connectivity within DMLN were characterized by calculating the nodal degree (ND), a graph theoretical measure of centrality. The ND values of the left CA2 subregion of the hippocampus was found to be significantly lower in the 4-month-old TG cohort compared to the age-matched WT littermates. Moreover, a lower ND value (hypo-connectivity) was observed in the right prelimbic cortex (prL) and basal forebrain in the 6-month-old TG cohort, compared to the same age WT cohort. Indeed, the ND pattern in the DMLN in both TG and WT cohorts showed significant differences across the two time points that represent pre-plaque and early plaque stages of disease progression. Our findings indicate that lower nodal degree (hypo-connectivity) in the left CA2 in the pre-plaque stage of AD and hypo-connectivity between the basal forebrain and the DMLN regions in the early-plaque stage demonstrated differences in comparison to healthy controls. These results suggest that a graph-theoretical measure such as the nodal degree, can characterize brain networks and improve our insights into the mechanisms underlying Alzheimer’s disease.

Least angle regression, relaxed lasso, and elastic net for algebraic multigrid of systems of elliptic partial differential equations

AbstractIn a sequence of papers, the author examined several statistical affinity measures for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (Cnodes) in algebraic multigrid (AMG) for systems of elliptic partial differential equations (PDEs). These measures were applied to a set of relaxed vectors that exposes the problematic error components. Once the CDOFs are determined using any one of these measures, the interpolation operator is constructed in a bootstrap AMG (BAMG) procedure. However, in a recent paper of Kahl and Rottmann, the statistical least angle regression (LARS) method was utilized in the coarsening procedure and shown to be promising in the CDOF selection. This method is generally used in the statistics community to select the most relevant variables in constructing a parsimonious model for a very complicated and high‐dimensional model or data set (i.e., variable selection for a “reduced” model). As pointed out by Kahl and Rottmann, the LARS procedure has the ability to detect group relations between variables, which can be more useful than binary relations that are derived from strength‐of‐connection, or affinity measures, between pairs of variables. Moreover, by using an updated Cholesky factorization approach in the regression computation, the LARS procedure can be performed efficiently even when the original set of variables is large; and due to the LARS formulation itself (i.e., its ‐norm constraint), sparse interpolation operators can be generated. In this article, we extend the LARS coarsening approach to systems of PDEs. Furthermore, we incorporate some modifications to the LARS approach based on the so‐called elastic net and relaxed lasso methods, which are well known and thoroughly analyzed in the statistics community for ameliorating several major issues with LARS as a variable selection procedure. We note that the original LARS coarsening approach may have addressed some of these issues in similar or other ways but due to the limited details provided there, it is difficult to determine the extent of their similarities. Incorporating these modifications (or effecting them in similar ways) leads to improved robustness in the LARS coarsening procedure, and numerical experiments indicate that the changes lead to faster convergence in the multigrid method. Moreover, the relaxed lasso modification permits an indirect BAMG (iBAMG) extension to the interpolation operator. This iBAMG extension applied in an intra‐ or inter‐variable interpolation setting (i.e., nodal‐based coarsening), as well as in variable‐based coarsening, which will not preserve the nodal structure of a finest‐level discretization on the lower levels of the multilevel hierarchy, will be examined. For the variable‐based coarsening, because of the parsimonious feature of LARS, the performance is reasonably good when applied to systems of PDEs albeit at a substantial additional cost over a nodal‐based procedure.

Complex Network View of the Sun’s Magnetic Patches. I. Identification

Solar and stellar magnetic patches (i.e., magnetic fluxes that reach the surface from the interior) are believed to be the primary sources of a star’s atmospheric conditions. Here, we apply the complex network approach and investigate its efficacy in the identification of these features. For this purpose, we use the line-of-sight magnetograms provided by the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. We construct the magnetic network following a specific visibility graph condition between pairs of pixels with opposite polarities and search for possible links between these regions. The complex network facilitates the construction of node degrees and PageRank images, and applying the downhill algorithm to node-degree images allows for the grouping of pixels into features corresponding to one-to-one matches with magnetogram patches. This approach promisingly serves to identify the nontrivial morphological structure of the magnetic patches for small and large sizes. We observe that the changes in the features of the node-degree images effectively correspond to the cospatial magnetic patches over time. Through visual assessment, we estimate an average false-negative error rate of approximately 1% in identifying small-scale features (one or two pixels in size).

Node2vec2rank: Large Scale and Stable Graph Differential Analysis via Multi-Layer Node Embeddings and Ranking.

Computational methods in biology can infer large molecular interaction networks from multiple data sources and at different resolutions, creating unprecedented opportunities to explore the mechanisms driving complex biological phenomena. Networks can be built to represent distinct conditions and compared to uncover graph-level differences-such as when comparing patterns of gene-gene interactions that change between biological states. Given the importance of the graph comparison problem, there is a clear and growing need for robust and scalable methods that can identify meaningful differences. We introduce node2vec2rank (n2v2r), a method for graph differential analysis that ranks nodes according to the disparities of their representations in joint latent embedding spaces. Improving upon previous bag-of-features approaches, we take advantage of recent advances in machine learning and statistics to compare graphs in higher-order structures and in a data-driven manner. Formulated as a multi-layer spectral embedding algorithm, n2v2r is computationally efficient, incorporates stability as a key feature, and can provably identify the correct ranking of differences between graphs in an overall procedure that adheres to veridical data science principles. By better adapting to the data, node2vec2rank clearly outperformed the commonly used node degree in finding complex differences in simulated data. In the real-world applications of breast cancer subtype characterization, analysis of cell cycle in single-cell data, and searching for sex differences in lung adenocarcinoma, node2vec2rank found meaningful biological differences enabling the hypothesis generation for therapeutic candidates. Software and analysis pipelines implementing n2v2r and used for the analyses presented here are publicly available.

Application of Deep Learning in Social Network Public Opinion Sentiment

With the rapid development of social network platform, in order to purify the network environment, prevent the abuse of public opinion information, and control public opinion in a very short time, this paper proposes the application research of deep learning in social network public opinion sentiment recognition and analysis. Through the construction of social network model and social network knowledge map and the analysis of key technology, a network public opinion algorithm based on deep learning is proposed, and a competitive public opinion information communication model in online social networks is constructed, and then a simulation experiment is conducted on the improved model of the constructed social networks. The result shows that in public opinion management and control, for users with larger node degree, this technology can effectively understand and obtain public opinion information in a very short time with faster and wider information propagation speed, so as to realize the effective control of public opinion information and the processing of massive information.

A comprehensive generalization of the Friendship Paradox to weights and attributes

The Friendship Paradox is a simple and powerful statement about node degrees in a graph. However, it only applies to undirected graphs with no edge weights, and the only node characteristic it concerns is degree. Since many social networks are more complex than that, it is useful to generalize this phenomenon, if possible, and a number of papers have proposed different generalizations. Here, we unify these generalizations in a common framework, retaining the focus on undirected graphs and allowing for weighted edges and for numeric node attributes other than degree to be considered, since this extension allows for a clean characterization and links to the original concepts most naturally. While the original Friendship Paradox and the Weighted Friendship Paradox hold for all graphs, considering non-degree attributes actually makes the extensions fail around 50% of the time, given random attribute assignment. We provide simple correlation-based rules to see whether an attribute-based version of the paradox holds. In addition to theory, our simulation and data results show how all the concepts can be applied to synthetic and real networks. Where applicable, we draw connections to prior work to make this an accessible and comprehensive paper that lets one understand the math behind the Friendship Paradox and its basic extensions.

Age-related changes of node degree in the multiple-demand network predict fluid intelligence

Fluid intelligence is an individual's innate ability to cope with complex situations and is gradually reduced across adults aging. The realization of fluid intelligence requires the simultaneous activity of multiple brain regions and depends on the structural connection of distributed brain regions. Uncovering the structural features of brain connections associated with fluid intelligence decline will provide reference for the development of intervention and treatment programs for cognitive decline. Using structural magnetic resonance imaging data of 454 healthy participants (18–87 years) from the Cam-CAN dataset, we constructed structural similarity network for each participant and calculated the node degree. Spearman correlation analysis showed that age was positively correlated with degree centrality in the cingulate cortex, left insula and subcortical regions, while negatively correlated with that in the orbito-frontal cortex, right middle temporal and precentral regions. Partial least squares (PLS) regression showed that the first PLS components explained 32 % (second PLS component: 20 %, pperm < 0.001) of the variance in fluid intelligence. Additionally, the degree centralities of anterior insula, supplementary motor area, prefrontal, orbito-frontal and anterior cingulate cortices, which are critical nodes of the multiple-demand network (MDN), were linked to fluid intelligence. Increased degree centrality in anterior cingulate cortex and left insula partially mediated age-related decline in fluid intelligence. Collectively, these findings suggest that the structural stability of MDN might contribute to the maintenance of fluid intelligence.

DTPBFT:A dynamic and highly trusted blockchain consensus algorithm for UAV swarm

With the continuous development of UAV technology, the application of UAV swarm in the military field is gradually becoming the focus of research around the world. Although it can bring a series of benefits in autonomous cooperation, the traditional UAV management technology is prone to hacker attacks due to many security issues such as a single point of failure brought by centralized management. Because of the advantages of distributed, tamper-proof, and traceability, blockchain is applied to UAV swarm to solve some of the security problems caused by centralized management. However, due to the limitations of its consensus algorithm Practical Byzantine Fault Tolerance (PBFT), its communication complexity will increase rapidly with the increase of the number of nodes, which also leads to the poor scalability of the algorithm and can only be applied to small-scale networks. To use the PBFT algorithm in large-scale networks such as UAV swarm, a dynamic and highly trusted PBFT (DTPBFT) algorithm is proposed in this paper. Firstly, a new consensus algorithm model including consensus layer and verification layer is designed. Then, a consensus node election scheme based on trust mechanism is proposed under this model. The trust degree of nodes in the blockchain network is comprehensively evaluated by selecting several representative indicators, and the weight factor of each indicator is calculated by entropy weight method. It not only reduces the communication complexity, but also ensures the reliability and dynamic update of consensus nodes. Experiments show that when the number of UAVs is 200, the consensus time of DTPBFT is 0.24 s, which indicates that this algorithm can support large-scale UAV swarm without causing communication congestion, so it has good scalability. In addition, experiments also show that DTPBFT can tolerate more than 13 malicious nodes, which improves the fault tolerance rate of PBFT.

Learning Topological Operations on Meshes with Application to Block Decomposition of Polygons

We present a learning based framework for mesh quality improvement on unstructured triangular and quadrilateral meshes. Our model learns to improve mesh quality according to a prescribed objective function purely via self-play reinforcement learning with no prior heuristics. The actions performed on the mesh are standard local and global element operations. The goal is to minimize the deviation of the node degrees from their ideal values, which in the case of interior vertices leads to a minimization of irregular nodes.

Quantitative study on the acupoint effects based on complex network analysis: with the examples of the textbook networks formed by Chinese Acupuncture and Moxibustion Therapy and Practical Acupuncture and Moxibustion

By extracting the acupoint names and their main indications from cases in Chinese Acupuncture and Moxibustion Therapy and Practical Acupuncture and Moxibustion, the acupoints and their main indications are represented in a reduced dimension, establishing an "acupoint-indication" linkage. Using complex network detection results (node degree values), the specificity of acupoints was assessed. The small-world characteristics of the "acupoint-indication" network are utilized to analyze the consistency of acupoint selection in acupuncture prescriptions and strategies to avoid redundant acupoints. The results show that the "acupoint-indication" network formed by both texts exhibited an approximate "long-tail" distribution, with a large number of node degree values concentrated between 0 and 4 000, while a few nodes have degree values exceeding 10 000. There are significant differences in the number and distribution of nodes with degree values> 10 000 between the two texts. Chinese Acupuncture and Moxibustion Therapy includes 11 acupoints with multiple edges across the body, whereas Practical Acupuncture and Moxibustion contains only 2 such acupoints, located in the lower limbs. Clinically, some acupoints have a broad therapeutic effect and appear in numerous prescriptions. The division of acupoints based on node degree values can coarsely evaluate the body region specificity of acupoints' regulatory effects. The "acupoint-indication" network of Chinese Acupuncture and Moxibustion Therapy has a higher number of edges than that of Practical Acupuncture and Moxibustion, which might be related to the different historical contexts of the two texts. In the future, diagnostic and therapeutic patterns with historical continuity can be utilized to optimize acupuncture prescriptions.

Node Load and Location-Based Clustering Protocol for Underwater Acoustic Sensor Networks

Clustering protocols for underwater acoustic sensor networks (UASNs) have gained widespread attention due to their importance in reducing network complexity. Congestion occurs when the intra-cluster load is greater than the upper limit of the intra-cluster information transmission capacity, which leads to a dramatic deterioration of network performance despite the reduction of network complexity. To avoid congestion, we propose a node load and location-based clustering protocol for UASNs (LLCP). First, a node load and location-based optimization mechanism is proposed. The number of cluster members is optimized based on node load and location to maximize the number of cluster members while avoiding congestion. Then, a node degree and location-based cluster member selection mechanism is proposed to select the optimal cluster members. Finally, a priority-based clustering mechanism is proposed. The node clustering order is adjusted based on the clustering priority to maximize the reduction of network complexity by increasing the average number of cluster members. Simulation results show that our proposed LLCP minimizes the network complexity while avoiding congestion.

Vascular Risk Factors and Brain Health in Aging: Insights from a Community-Based Cohort Study.

The aging population and high rates of Alzheimer's disease (AD) create significant medical burdens, prompting a need for early prevention. Targeting modifiable risk factors like vascular risk factors (VRFs), closely linked to AD, may provide a promising strategy for intervention. This study investigates how VRFs influence cognitive performance and brain structures in a community-based cohort. In this cross-sectional study, 4,667 participants over 50 years old, drawn from the Beijing Ageing Brain Rejuvenation Initiative project, were meticulously examined. Cognitive function and VRFs (diabetes mellitus, hypertension, hyperlipidemia, obesity, and smoking), were comprehensively assessed through one-to-one interviews. Additionally, a subset of participants (n = 719) underwent MRI, encompassing T1-weighted and diffusion-weighted scans, to elucidate gray matter volume and white matter structural network organization. The findings unveil diabetes as a potent detriment to memory, manifesting in atrophy within the right supramarginal gyrus and diminished nodal efficiency and degree centrality in the right inferior parietal lobe. Hypertension solely impaired memory without significant structural changes. Intriguingly, individuals with comorbid diabetes and hypertension exhibited the most pronounced deficits in both brain structure and cognitive performance. Remarkably, hyperlipidemia emerged as a factor associated with enhanced cognition, and preservation of brain structure. This study illuminates the intricate associations between VRFs and the varied patterns of cognitive and brain structural damage. Notably, the synergistic effect of diabetes and hypertension emerges as particularly deleterious. These findings underscore the imperative to tailor interventions for patients with distinct VRF comorbidities, especially when addressing cognitive decline and structural brain changes.

Limit Law for Zagreb and Wiener Indices of Random Exponential Recursive Trees

The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes.

Physical networks as network-of-networks

Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a network-of-networks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degree-volume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the network-of-networks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.

Oscillation-Specific Nodal Differences in Parkinson's Disease Patients with Anxiety.

Parkinson's disease (PD) is a common neurodegenerative disorder that is predominantly known for its motor symptoms but is also accompanied by non-motor symptoms, including anxiety. The underlying neurobiological substrates and brain network changes associated with comorbid anxiety in PD require further exploration. An analysis of oscillation-specific nodal properties in patients with and without anxiety was conducted using resting-state functional magnetic resonance imaging (rs-fMRI) and graph theory. We used a band-pass filtering approach to differentiate oscillatory frequency bands for subsequent functional connectivity (FC) and graph analyses. The study included 68 non-anxiety PD (naPD) patients, 62 anxiety PD (aPD) patients, and 64 healthy controls (NC). Analyses of nodal betweenness centrality (BC), degree centrality (DC), and efficiency were conducted across multiple frequency bands. The findings indicated no significant differences in BC among naPD, aPD, and NC within the 0.01-0.08 Hz frequency range. However, we observed a specific reduction in BC at narrower frequency ranges in aPD patients, as well as differing patterns of change in DC and efficiency, which are believed to reflect the neurophysiological bases of anxiety symptoms in PD. Differential oscillation-specific nodal characteristics have been identified in PD patients with anxiety, suggesting potential dysregulations in brain network dynamics. These findings emphasize the complexity of brain network alterations in anxiety-associated PD and identify oscillatory frequencies as potential biomarkers. The study highlights the importance of considering oscillatory frequency bands in the analysis of brain network changes.

Spatial Analysis of Kuwait Street Networks

Objective: Since only little is known about street networks (SNs) in developing countries, the overarching aim of this study is to spatially analyze Kuwait SN griddedness, transportation efficiency, and connectivity, using several indicators (i.e., orientation-order, street circuity, average street length, average node degree, proportion of dead-ends, and proportion of four-way intersections). Methodology: This work utilizes the python package OSMnx which obtains, models, and analyzes the open-sourced data repository OpenStreetMap to analyze the Kuwait built environment. This analysis has been complemented with qualitative information. Results: the key findings are: (I) orientation-order is lowest in some of the coastal areas and high in some modular residential areas; (II) Furthermore, compared to residential areas, non-residential areas have coarse-grained SNs, lower intersections connectivity and higher street circuity. Conclusion: Through acknowledging these features, more sustainable urban planning can be undertaken for expanding the built area in Kuwait. Further, the obtained data was made available for conducting further studies.