Degree Distribution Research Articles

Clustering and degree correlations are ubiquitous in real-world complex networks. Yet, understanding their role in critical phenomena remains a challenge for theoretical studies. Here, we provide the exact solution of site percolation in a model for strongly clustered random graphs, with many overlapping loops and heterogeneous degree distribution. We systematically compare the exact solution with heterogeneous mean-field predictions obtained from a treelike random rewiring of the network, which preserves only the degree sequence. Our results demonstrate a non-trivial interplay between degree heterogeneity, correlations and network topology, which can significantly alter both the percolation threshold and the critical exponents predicted by the heterogeneous mean field. These findings reveal limitations of heterogeneous mean-field theory, demonstrating that the degree distribution alone is insufficient to determine universality classes in complex networks with realistic structural features.

Recurrence, transience and degree distribution for the Tree Builder Random Walk

Introduction Real-world networks possess complex, higher-order structures that are not captured by traditional pairwise analysis methods. Q-analysis provides a powerful mathematical framework based on simplicial complexes to uncover and quantify these multi-node interactions. However, its adoption has been limited by a lack of accessible software tools. Methods We introduce a comprehensive Python package that implements the core methodology of Q-analysis. The package enables the construction of simplicial complexes from graphs or simplex lists and computes a suite of descriptive metrics, including structure vectors (FSV, SSV, TSV) and topological entropy. It features high-performance routines, integration with scikit-learn for machine learning workflows, and tools for statistical inference, such as permutation tests. Results We demonstrate the package’s capabilities through a simulation study, revealing distinct higher-order topological signatures in scale-free versus configurational networks despite identical degree distributions. Application to the DBLP co-authorship dataset uncovered the evolution of collaborative structures over three decades, showing increased collaboration scale and shifts in higher-order connectivity patterns. Finally, in a network physiology application, the package identified significant disruptions in the higher-order organization of fMRI-derived brain networks in Major Depressive Disorder (MDD), characterized by a loss of high-dimensional functional components and increased fragmentation. Discussion The developed package makes Q-analysis accessible to a broad research audience, facilitating the exploration of higher-order interactions in complex systems. The presented applications validate its utility across diverse domains, from social networks to neuroscience. By providing an open-source tool, this work bridges a gap in network science, enabling quantitative analysis of the intricate, multi-node structures that define real-world networks.

Counting and listing triangles in graphs is a fundamental task in network analysis, supporting applications such as community detection, clustering coefficient computation, k-truss decomposition, and triangle centrality. We introduce the cover-edge set, a novel concept that eliminates unnecessary edges during triangle enumeration, thereby improving efficiency. This compact cover-edge set is rapidly constructed using a breadth-first search (BFS) strategy. Using this concept, we develop both sequential and parallel triangle-counting algorithms and conduct comprehensive comparisons with state-of-the-art methods. We also design a benchmarking framework to evaluate our sequential and parallel algorithms in a systematic and reproducible manner. Extensive experiments on the latest Intel Xeon 8480+ processor reveal clear performance differences among algorithms, demonstrate the benefits of various optimization strategies, and show how graph characteristics, such as diameter and degree distribution, affect algorithm performance. Our source code is available on GitHub.

Abstract We describe the asymptotic behaviour of large degrees in random hyperbolic graphs for all values of the curvature parameter $\alpha$ . We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the sparse regime $\alpha>\tfrac{1}{2}$ , the rank at which these two rankings cease to coincide is $n^{1/(1+8\alpha)+o(1)}$ . We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $\alpha = \tfrac{1}{2}$ , which corresponds to the connectivity threshold of the model. For $\alpha < \tfrac{1}{2}$ , the maximum degree is of order $n - O(n^{\alpha + 1/2})$ , whereas for $\alpha \geq \tfrac{1}{2}$ , the maximum degree is of order $n^{1/(2\alpha)}$ . In the $\alpha < \tfrac{1}{2}$ and $\alpha > \tfrac{1}{2}$ cases, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $\alpha < \tfrac{1}{2}$ and Fréchet for $\alpha > \tfrac{1}{2}$ ). This refines previous estimates on the maximum degree for $\alpha > \tfrac{1}{2}$ and extends the study of large degrees to the dense regime $\alpha \leq \tfrac{1}{2}$ .

In the mining field, hydraulic fracturing of coal - seam boreholes generates a large number of weak microseismic signals. The accurate identification of these signals is crucial for subsequent positioning and inversion. However, when dealing with such signals, traditional automatic microseismic waveform identification algorithms have difficulty in accurately identifying weak waveforms and are prone to misjudging background noise. This study innovatively introduces the deep - learning convolutional neural network (CNN), integrating the concepts and methods of computational communication to analyze microseismic signals. 8,341 pieces of background noise data and 5,860 pieces of microseismic data are carefully selected from the data of coal - seam borehole hydraulic fracturing. After adding noise at 12 levels and performing translation with 10 different degrees of displacement, 101,123 pieces of background noise and 102,546 effective waveforms are obtained. Subsequently, by applying the information - propagation dynamics model of computational communication, microseismic signals are regarded as information carriers. A signal - propagation network is constructed, and features such as network degree distribution are extracted. These features, combined with traditional time - domain and frequency - domain features, are converted into time - domain and Fourier images and then input into a two - dimensional CNN model. Experiments show that the time - domain CNN model achieves a precision rate of 100% and a recall rate of 68% in microseismic event identification, significantly outperforming traditional methods such as AIC, STA/LTA, and the Fourier CNN model. Furthermore, the time-frequency fusion CNN model—integrating time-domain waveforms, Fourier frequency-domain features, and time-frequency characteristics (e.g., short-time Fourier transform)—achieves an identical precision rate of 100% and a higher recall rate of 72%, outperforming the single-domain time-domain CNN model. The integration of computational communication concepts (e.g., signal propagation network topological features) and multi-domain features enables the model to capture comprehensive spatiotemporal and dynamic signal characteristics, further validating its superiority in identifying weak microseismic signals with low signal-to-noise ratios (SNR).This indicates that the combination of time - domain images and computational - communication technology is more suitable as the input data for the CNN model. It can effectively distinguish microseismic waveforms from background noise, opening up a new path for the identification of mine microseismic signals and demonstrating the application potential of computational communication in this field.

Abstract Career choices are shaped by students’ experiences, knowledge, and skill sets across time, reflecting not only disciplinary interests but also exposure to evolving fields such as data science (DSC). Despite a surge in interest and enrollment in data science degrees, the United States faces a growing demand for data literacy across multiple sectors. Online learning environments have become entry points for students’ initial engagement with DSC, offering accessibility and supporting workforce needs. Nevertheless, the interdisciplinary essence of DSC means that clear career paths remain ambiguous, especially for those applying DSC knowledge within various disciplines. While national data sources provide valuable overviews of degree distributions, more granular analysis at the course level is warranted to understand nuanced student trajectories. Project-based online learning, though proven valuable in in-person settings, remains underexplored in online DSC education. This study employs curriculum analytics and Sankey diagram visualizations to investigate course enrollment patterns and career trajectories among students after enrolling in an introductory online project-based DSC course. We built a longitudinal dataset by following 35 students between Fall 2022 and Spring 2024, tracking their subsequent course enrollments over time. Demographic and academic data were sourced from institutional enrollment records, allowing subgroup analysis based on major, gender, race, first-generation status, and achievement. Our exploratory analysis reveals patterns indicating that continued DSC course enrollment appears prevalent among nonwhite, male, STEM-major, and academically proficient students, whereas first-generation students exhibit no persistence. We illustrate how Sankey diagrams, though not establishing causality, provide actionable insights for program and curriculum development in DSC education.

Dynamical phase transitions (DPTs) characterize critical changes in system behavior occurring at finite times, providing a lens to study nonequilibrium phenomena beyond conventional equilibrium physics. While extensively studied in quantum systems, DPTs have remained largely unexplored in classical settings. Recent experiments on complex systems, from social networks to financial markets, have revealed abrupt dynamical changes analogous to quantum DPTs, motivating the search for a theoretical understanding. Here, we present a minimal model for nonequilibrium networks, demonstrating that nonlinear interactions among network edges naturally give rise to DPTs. Specifically, we show that network degree diverges at a finite critical time, following a universal hyperbolic scaling, consistent with empirical observations. Our analytical results predict that key network properties, including degree distributions and clustering coefficients, exhibit critical scaling as criticality approaches. These findings establish a theoretical foundation for understanding emergent nonequilibrium criticality across diverse complex systems.

The surface of asteroid (25143) Itokawa shows both fresh and mature terrains, despite its short space-weathering timescale of approximately $10^3$ years, as inferred from recent studies. Seismic shaking triggered by the impact that formed the 8-meter crater Kamoi has been proposed as a possible explanation for the diversity. This study aims to examine whether the seismic shaking induced by the impact might account for the observed spatial variations in space weathering and might further constrain the internal structure of Itokawa. Assuming that the Kamoi crater was formed by a recent impact, we conducted three-dimensional seismic wave propagation simulations and applied a simplified landslide model to estimate surface accelerations and boulder displacements. Our results show that even a low-energy case (1% of the nominal seismic energy) produces surface accelerations sufficient to destabilize the surface materials. The simulated boulder displacements are consistent with the observed distribution of space-weathering degrees even on the opposite hemisphere. We estimate the seismic diffusivity to be 1,000-2,000 mathrm and the seismic efficiency to be in the range of $5.0 $ to $5.0 $, implying that the interior of Itokawa contains blocks tens of meters across and acts as a strongly scattering medium. Our findings provide unique dynamical evidence, based on seismic wave propagation modeling, that supports the hypothesis that the interior of Itokawa truly is a rubble pile.

This study aims to address the issue of identifying diverse intercity travel modes in high population density areas and analyzing their network structure characteristics over different time periods. Consequently, a management model combining complex network theory (CNT) with a random forest classification (RFC) algorithm—referred to as the CNT-RFC model—is proposed. The study utilizes publicly available migration data from the National Bureau of Statistics and transportation departments from January 2021 to December 2023. Network structure features are extracted through node degree distributions, edge connections, centrality metrics, and community detection algorithms. Integrating RFC enables precise identification of travel modes and uncovers the spatio-temporal heterogeneity of intercity travel patterns. Experimental results demonstrate that the optimized model outperforms comparative methods on key metrics including accuracy, precision, recall, and F1 score. Specifically, for leisure travel identification, the CNT-RFC model achieves an accuracy of 0.947, precision of 0.928, and F1 score of 0.947, surpassing advanced models such as the Funnel Tabular Transformer and Graph Transformer for Node and Edge Representation Learning. Paired sample t-tests confirm that these improvements are statistically significant (p < 0.05) with a very large effect size (Cohen’s d > 3.5). Network structure analysis reveals a decline in the small-world coefficient to 0.58 during holiday periods, an increase in average travel distance to 25 km, and a rise in average adaptation cost to 0.30, indicating significant structural reconfiguration. Sensitivity analysis related to the pandemic further validates the model’s robustness, showing only a slight accuracy decline of 0.0101 in 2022, while the centrality of the high-speed rail network decreased by 0.0720, confirming the pandemic’s asymmetric impact across transportation network layers. This study provides robust scientific evidence and effective solutions for regional transportation planning and policy formulation, thereby effectively addressing traffic congestion, optimizing travel routes, and improving intercity transportation efficiency.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-19392-x.

ABSTRACTComplex networks are prevalent in various real‐world domains, including social networks, biological networks, communication networks, and information networks. Identifying the significance of nodes within these networks provides numerous advantages, such as detecting influential individuals, identifying structurally similar nodes, and facilitating community analysis. However, due to the absence of a universally accepted definition of node importance, multiple centrality measures have been developed to quantify it. Each centrality measure evaluates node significance from a distinct perspective. By utilizing multiple measures in conjunction or integrating them, a more comprehensive understanding of node importance can be achieved. Combining multiple centrality measures into a linear combination involves determining a coefficient for each sub‐centrality measure, which can be computationally intensive. Additionally, developing separate combined measures for different graphs requires repeating this process for each case. This study proposes the use of the Arithmetic Optimization Algorithm (AOA) to determine these coefficients. It also suggests that generating a combined centrality measure from a synthetic Representative Graph, created by averaging the degree distributions of similar graphs, may produce effective results for the original graphs. Experimental results indicate that the AOA algorithm can quickly determine coefficients for combined centrality measures, and the Representative Graph method performs effectively.

Context. Complex systems are characterised by many highly interconnected dynamical units that exhibit non-linear properties. The type of pulsating stars known as δ Sct stars have intrinsic brightness variations that require non-linear models to be described comprehensively. These stars span a broad range of properties, from low to high amplitudes, as well as a broad range of complex features. We applied the complex network approach to δ Sct stars as non-linear complex systems. Aims. Differences among the constructed networks, which might appear in the network metrics of low-amplitude and high-amplitude δ Sct stars, can indicate intrinsic asteroseismic differences that are essential for classification of pulsating stars. Additionally, the relations between the asteroseismic parameters and network metrics (such as degree or clustering distributions) can lead us to a better understanding of pulsating stars dynamics. Methods. By using the horizontal visibility algorithm, we mapped the TESS light curves of 69 δ Sct stars to undirected horizontal visibility graphs (HVGs), where the graph nodes represent the light curve points. This allowed us to measure the morphological characteristics of HVGs, such as the distribution of links between nodes (degree distribution) and the average fraction of triangles around each node (average clustering coefficient). Results. The average clustering coefficients for HADS and LADS display two different linear correlations with the peak-to-peak amplitude of the TESS light curves that naturally separates them into two groups. This novel approach enabled us to obtain this result without having to use an ad hoc criterion, for the first time. Exponential fits on HVG degree distributions for both HADS and LADS stars offer indices that suggest correlated stochastic generating processes for δ Sct light curves. By applying the theoretical expression for the HVGs degree distribution of random time series, we have the ability to distinguish significant pulsations from the background noise, which could become a practical tool in frequency analyses of stars going forward.

Humans organize semantic knowledge into complex networks that encode relations between concepts. The structure of those networks has broad implications for human cognitive processes, and for theories of semantic development. Evidence from large lexical networks such as those derived from word associations suggest that semantic networks are characterized by high sparsity and clustering while maintaining short average paths between concepts, a phenomenon known as a “small‐world” network. It has also been argued that those networks are “scale‐free,” meaning that the number of connections (or degree) between concepts follows a power‐law distribution, whereby most concepts have few connections, while a few have many. However, the scale‐free property is still debated, and the extent to which the lexical evidence reflects the naturally occurring semantic regularities of the environment has not been investigated systematically. To address this, we collected and analyzed semantic descriptors, human evaluations, and similarity judgments from four large datasets of naturalistic stimuli across three modalities (visual, auditory, and audio‐visual) comprising 7916 stimuli and 610,841 human responses. By connecting concepts that co‐occur as descriptors of the same stimuli, we construct “multimodal” semantic networks. We show that these networks exhibit a clear small‐world structure with a degree distribution that is best captured by a truncated power law (i.e., the most‐connected concepts are less common than predicted by a perfect power law). We further show that these networks are predictive of human sensory judgments on these domains, as well as reaction times in an independent lexical decision task. Finally, we show that multimodal networks also share overlapping themes with previously analyzed lexical networks, which upon a more rigorous reanalysis are revealed to be truncated too. Our findings shed new light on the origins of the structure of semantic networks by tying it to the semantic regularities of the environment.

Exploring the evolution of landscape patterns along scenic byways and future development trends is beneficial for coordinating ecological security protection and scientifically guiding tourism development. With the Grass Skyline of Bashang Grassland, Hebei Province as an example, we constructed a landscape pattern indicator system for scenic byways, and used Fragstats, optimal parameters-based geographical detector, and ARIMA-MOP-PLUS models to compare the evolutionary differences in landscape patterns before (2000-2010) and after the construction (2010-2020). We further examined the driving factors of post-construction evolution, and predicted landscape pattern changes under four scenarios for 2035: tourism-economic-priority development, inertial development, ecological-priority development, and ecotourism development. Results showed that cultivated land, forest, and grassland were the main landscape types within the 5 km buffer zone along the Grass Skyline scenic byway. Compared to the pre-construction period, the overall landscape connectivity in the corridor area decreased, shape complexity and distribution evenness increased, fragmentation intensified, and disturbance degree increased by 4.6% during the post-construction period. Tourism development and road transportation had the strongest impacts on construction land expansion, with a contribution rate of 60.0%. Topographic-soil and socio-economic factors were the main driving forces for spatial differentiation of landscape disturbance degree, with a q-value of 39.9%. The interaction between slope and other factors was the strongest. By 2035, ecological-priority development and ecotourism development scenarios showed lower disturbance degrees than other scenarios, at 0.52 and 0.53, respectively. The ecotourism development scenario reduced the disturbance degree distribution range by 3.21 km2 by balancing the proportion of ecological land and construction land. Our results could provide decision support for optimizing tourism resource allocation along the Grass Skyline and mitigating the negative impacts of tourism development on the environment of the Bashang Plateau.

The article shows practical application of rising graphs for building models of knowledge localization net in cities of different scale. Results of modeling based on three types of rising graphs were compared: they are casual, preferred joining and mixed type that uses different combinations of casualty and preference in arc shaping. The author carried out mathematic description of the model for the latter type of rising graph. Methodology of assessing adequacy and efficiency of the model was put forward, which covers not only identifying the form of dependence between real and theoretical models of distribution of degrees, dynamics of node degree and average local clusterization factor but also analysis of the inner structure of net. As a result the author substantiated impossibility to describe the real net of knowledge localization by a uniform model. Models of rising casual graph and of mixed type are considered the most adequate. Dynamics of clusterization factor can be modeled mainly by graph of preferable joining but with condition of disparity of the indicator rate. The obtained conclusions make the objective of developing models for small nets more acute.

Generally, networks are classified into two sides of inequality and equality with respect to the number of links at nodes by the types of degree distributions. One side includes many social, technological, and biological networks which consist of a few nodes with many links, and many nodes with a few links, whereas the other side consists of all nodes with an equal number of links. In comprehensive investigations between them, we have found that, as a more equal network, the tolerance of whole connectivity is stronger without fragmentation against the malfunction of nodes in a wide class of randomized networks. However, we newly find that all networks which include typical well-known network structure between them become extremely vulnerable, if a strong modular (or community) structure is added with commonalities of areas, interests, religions, purpose, and so on. These results will encourage avoiding too dense unions by connecting nodes and taking into account the balanced resource allocation between intra- and inter-links of weak communities. We must reconsider not only efficiency but also tolerance against attacks or disasters, unless no community that is really impossible.

Analyzing 9000 urban areas' street networks, we identify properties, including extreme betweenness centrality heterogeneity, that typical spatial network models fail to explain. Accordingly we propose a universal, parsimonious, generative model based on a two-step mechanism that begins with a spanning tree as a backbone then iteratively adds edges to match empirical degree distributions. Controlled by a single parameter representing lattice-equivalent node density, it accurately reproduces key universal properties to bridge the gap between empirical observations and generative models.

Abstract The entanglement properties of quantum states associated with directed graphs are investigated. Using a measure derived from the Fubini–Study metric, multipartite entanglement is quantitatively related to the local connectivity of the graph. In Entanglement in Directed Graph States (2025), arXiv:2505.10716, it is demonstrated that the vertex degree distribution fully determines this entanglement measure and remains invariant under vertex relabeling, highlighting its topological character. As a consequence, the measure depends only on the total degree of each vertex, making it independent of the distinction between incoming and outgoing edges. This framework is applied to several specific graph structures, including hierarchical networks, neural network–inspired graphs, full binary tree and linear bridged cycle graphs, demonstrating how their combinatorial properties influence entanglement distribution. These results provide a geometric perspective on quantum correlations in complex systems, offering potential applications in the design and analysis of quantum networks.

In drug-safety monitoring systems, adverse events (AEs) associated with the use of medical products often consist of complex patterns of clinical events. Network analysis (NA) was used for pattern recognition and characterizing the Vaccine Adverse Event Reporting System (VAERS), but limited applications of NA to the US Food and Drug Administration (FDA) Adverse Event Reporting System (FAERS) left its network description incomplete. In this analysis, the network properties of FAERS were characterized and leveraged to facilitate pattern discovery. Reported AE information in FAERS is represented using preferred terms (PTs) in Medical Dictionary for Regulatory Activities terminology. The FAERS subsets were analyzed with drugs and PTs as nodes and interconnections as edges. Global characteristics, like the scale-free nature of the distribution, were examined to explore theoretical and structural considerations. Metrics that assess connectivity and edge weighting algorithms based on report co-occurrence or clustering were applied. Serious AE reports from 2016 to 2023 (2,062,099) were represented as a network of 20,965 nodes (16,847 PTs and 4116 drugs) with more than four million interconnections. Characteristics of FAERS subnetworks were determined with heavy-tailed degree distributions, high local clustering, and low diameters. Complexities related to structural and evolutionary characteristics were revealed as the log-normal model fits the degree distribution better than the power law. Network-based techniques identified clinically relevant patterns and clustering patterns representative of known adverse drug reactions. Comparisons to VAERS reveal similarities in networks of AE reporting systems. This initial systematic application of NA to FAERS describes the overall network characteristics of the FAERS database and provides insight into the use of network applications in drug safety research.

Due to the tidal locking, the far side of the Moon is permanently turned away from the Earth. Its polarization characteristics are still poorly understood, limiting our knowledge of material composition and evolution. Previous studies have indicated a correlation between the distributions of degree of polarization (DOP) and the iron oxide (FeO) abundance on the Moon, suggesting a new approach to infer the polarization characteristics of the lunar far side from FeO abundance distribution. Three critical issues have been analyzed: (1) A linear regression model between DOP and FeO abundance is proposed based on control points from ground-based near side polarization images. (2) The DOP distribution of the lunar far side is estimated, based on the established model, revealing significant hemispheric differences in polarization characteristics. (3) The relationship between DOP and lunar phase angle is examined, with the fitted values demonstrating strong agreement with the observations in both magnitude and variation trend. These insights offer valuable guidance for comprehensive polarimetric studies of the Moon.