Introduction Public search behavior provides a high-frequency external attention signal for understanding changes in market expectations in the digital economy. During periods of macroeconomic adjustment and investment uncertainty, search attention may capture shifts in public concern, information demand, and expectation formation. Methods Using Douyin search data on the theme of “investment” in Shandong Province from 4 June 2022 to 1 August 2025, this study constructs a Public Investment Search Network based on the Visibility Graph algorithm. The analysis examines temporal fluctuation, phase-based evolution, network topology, community differentiation, topological indicators, degree distribution, and robustness under alternative network constructions. Results The results show clear phase-based aggregation and divergence in public investment attention. The network exhibits a heavy-tailed degree distribution and small-world-like characteristics. Attention evolves through a cyclical process of concentration, dispersion, and rebalancing under the combined influence of policy stimuli, market fluctuations, and information diffusion. Changes in clustering coefficient, modularity, volatility, and Shannon entropy further reveal the self-organizing features of public investment search behavior. Discussion The findings suggest that the public investment search network provides a structural representation of collective attention and offers supplementary information for monitoring market signals and changes in public expectations. The study describes the structure of public investment-related attention rather than directly testing firm-level investment responses. Future research may combine search-network indicators with firm-level investment, innovation, or financial data to further examine how external attention signals are incorporated into corporate decisions.

Social media platforms are important spaces for social and sexual networking among men who have sex with men (MSM). However, limited research has examined structural changes in MSM digital social networks or the contribution of different user roles in maintaining network connectivity. This study aims to investigate structural changes and user dynamics in an online MSM network to inform tailored digital public health interventions. We constructed directed social networks among MSM in Zhuhai, China, using large-scale Blued platform data collected in 2021 and 2024. Users were represented as nodes, and follow relationships were represented as directed edges. Key network metrics, including degree distribution, reciprocity, assortativity, community fragmentation and attribute-based homophily, were analysed to identify weak links and structurally important users. The analysis included 9409 valid users in 2021 and 8890 in 2024. The network was significantly sparser and more fragmented, with edge density halving and 70.91% of detected communities containing only two users in 2024. Average degree dropped from 19.52 to 8.00, whereas degree assortativity decreased from -0.0580 to -0.1482, indicating increasing disassortative mixing. Despite reduced platform engagement, younger, versatile, and highly active boundary and common users remained important for sustaining inter-community connectivity. From 2021 to 2024, the MSM network on Blued became more fragmented, disassortative and heterogeneous. These changes may constrain health-message diffusion. Network-informed strategies targeting central and bridging users, while improving access for peripheral users, may enhance equitable information dissemination among MSM communities.

Graph neural networks (GNNs) have excelled in predicting graph properties in various applications ranging from identifying trends in social networks to drug discovery and malware detection. With the abundance of new architectures and increased complexity, GNNs are becoming highly specialized when tested on a few well-known datasets. However, how the performance of GNNs depends on the topological and features properties of graphs is still an open question. In this work, we introduce a comprehensive benchmarking framework for graph machine learning, called HypBench, focusing on the performance of GNNs across varied network structures. Utilizing the geometric soft configuration model in hyperbolic space, we generate synthetic networks with realistic topological properties and node feature vectors. This approach enables us to assess the impact of network properties, such as topology-feature correlation, degree distributions, local density of triangles, and homophily, on the effectiveness of different GNN architectures. Our results highlight the dependency of model performance on the interplay between network structure and node features, providing insights for model selection in various scenarios. This study contributes to the field by offering a versatile tool for evaluating GNNs, thereby assisting in developing and selecting suitable models based on specific data characteristics.

Understanding the dynamical patterns of public attention towards traditional cultural practices such as Tai Chi is crucial for cultural heritage preservation and promotion strategies. This study reveals the chaotic dynamics underlying Tai Chi public attention through an integrated analytical framework combining Horizontal Visibility Graphs (HVG), Autoencoder neural networks, and Sparse Identification of Nonlinear Dynamics (SINDy). Using daily Baidu Index data (2014-2024) from four representative Chinese provinces (Beijing, Shanghai, Guangdong, Henan), we construct HVGs and quantify chaos through degree distribution exponents (λ). Results reveal significant regional heterogeneity: HVG λ ranges from 0.291 (Beijing PC, strong chaos) to 0.437 (Shanghai PC, quasi-periodic), with corresponding maximum Lyapunov exponents of 0.025-0.036bits/day (Rosenstein method) and 0.016-0.217bits/day (Wolf method); the consistent positivity across both estimation methods provides cross-validated confirmation of chaotic dynamics. Autoencoder-based dimensionality reduction achieves reconstruction correlations of 0.896-0.918, enabling discrete SINDy to identify sparse governing equations (37-38 active terms from 48 candidates) with normalized root-mean-square errors of 11.7-12.8%. The dynamical conclusions are triangulated across three analytically independent characterizations-HVG topological classification, Lyapunov-based phase space analysis, and SINDy equation structure-which converge on consistent regional rankings, with Beijing exhibiting the strongest chaotic signatures and Henan the most structured quasi-periodic behavior. Phase space analysis confirms diverse attractor geometries: Beijing exhibits space-filling chaotic trajectories while Shanghai displays quasi-periodic structures. Comparative evaluation against ARIMA, VAR, and LSTM baselines within the same latent space confirms that SINDy achieves superior predictive fidelity (NRMSE 17.6-22.4%) while uniquely providing interpretable governing equations and chaos diagnostics inaccessible to alternative approaches. This interdisciplinary framework bridges complex network theory, deep learning, and dynamical systems analysis, offering a rigorous quantitative paradigm for studying the temporal evolution of public attention in cultural phenomena.

Empirical growing networks vs minimal models: Evidence and challenges from Software Heritage and APS citation datasets

Complex network time-series analysis by the Visibility Graph (VG) method is applied to an experimental set of drain current signals from nano-transistor devices (nano-MOSFETs). Electric current in nano-MOSFETs has noisy fluctuations produced by different physical mechanisms, including thermal, electron-hole recombination, and the effect of ion traps present at the gate region. The combination of these mechanisms results in a complex power spectrum, which may contain "corners" at one or more critical frequencies, switching from the well-known 1/f "flicker" noise to 1/f 2 "Brownian" tails, or contain flat regions at low frequencies. More recent studies have shown that current fluctuations in a fully depleted nano-MOSFET may contain low-dimensional chaotic dynamics with critical intermittent behavior. Consequently, noisy current signals in nano-MOSFETs constitute an excellent testbed to assess the ability of the VG method in capturing and discerning subtle features of dynamics under complex scenarios. Using graph metrics, such as clustering coefficient, assortativity, and the rich-club coefficient, we show that the VG structure consistently discerns differences in the nature of noise between fresh and stressed (faulty) transistors and between stochastic and low-dimensional chaotic dynamics. Using three types of surrogate time series, we show that these differences are statistically significant. Moreover, we show that graph metrics other than the degree distribution are crucial in capturing features of complex system dynamics that are mixtures of various types of noise and possibly deterministic chaos. In the case of noisy signals from nano-devices, this has a direct application to device classification and fault detection.

Multi-host pathogens vary in how they utilise different hosts, yet the traits determining which species occupy central positions in transmission networks remain poorly understood. We tested whether the wild bird-avian influenza virus (AIV) network exhibits a scale-free structure, implying that hub hosts disproportionately contribute to transmission and whether ecological and evolutionary traits jointly predict hub status. Using global infection records, we constructed a bipartite network linking 247 bird species to 105 AIV subtypes. Degree distributions followed a power-law pattern, confirming substantial heterogeneity in host importance. We identified 23 hub species, exclusively from Anseriformes and Charadriiformes. Interpretable machine learning revealed that hub species share strong flightability, prolonged water-surface foraging, greater longevity and higher diversification rates, indicating that both ecological exposure and evolutionary history influence hub status. These findings provide insight into host-pathogen network dynamics and highlight priority species for targeted AIV surveillance and control.

Topology and fragility of European high-voltage networks: A cross-country comparative analysis

Sleep deprivation (SD) changes brain-wide dynamics, but the circuit-level perturbation that can generate this systems-level shift remains unclear. We scanned 26 participants at seven time points across 36 h of continuous wakefulness and assessed criticality from resting-state functional Magnetic Resonance Imaging (rs-fMRI) blood-oxygen-level-dependent (BOLD) signals using neuronal avalanche metrics (branching ratio and mean avalanche size). The branching ratio increased from 0.98 at baseline to 1.08 after 36 h, indicating a progressive shift from near-critical to supercritical propagation. Interestingly, the shift was heterogeneous. Visual and sensorimotor networks showed the largest deviations, whereas the limbic network remained close to criticality. Criticality changes tracked accumulated subjective sleep pressure but were largely dissociated from psychomotor vigilance lapses. SD also reshaped functional network organization, with the functional connectivity (FC) degree distribution shifting toward more high-degree nodes. In a recurrent excitatory-inhibitory network model, gamma-band power provided an interpretable proxy for effective gain and inhibitory control. Using this proxy, selectively reducing inhibitory efficacy was sufficient to capture the direction of the near-critical-to-supercritical drift and a limbic-like resilience pattern, supporting inhibitory decay as a plausible candidate circuit-level mechanism linking SD to large-scale propagation instability.

Deep reinforcement learning based topology optimization of triangular meshes

Abstract The statistical method of visibility graph (VG) has been becoming widely employed for analyzing the topological properties of signals in various scientific fields. The VG method is based on transforming time series into graphs or networks, whose nodes are the series values linked between each other by edges drawn on the basis of specific ‘visibility’ criteria. The number of the edges departing from each node is the degree of that particular node. In this paper, the VG is utilized to analyze the topological properties of Moderate Resolution Imaging Spectroradiometer (MODIS) satellite evapotranspiration time series of pixels covering olive orchards in various areas of southern Italy, some of which are affected by Xylella fastidiosa infection. Xylella fastidiosa is a very dangerous phytobacterium that causes desiccation and then death of olive trees. By converting the investigated MODIS time series in networks by using the VG, we focused on evaluating the discrimination capability between infected and uninfected sites by analyzing two informational quantities: the fisher information measure (FIM) and the Shannon entropy of the connection degree distribution. The results of the receiver operating characteristic analysis indicate that the Shannon entropy of the degree distribution demonstrates strong discrimination capability between infected and healthy pixels, while the FIM show a much less discrimination power. These findings suggest that the VG method combined with information theory is highly effective in identifying satellite pixels covering infected vegetated areas and holds significant potential as a valuable tool for infection detection of large-scale areas.

The assortative behavior of a network is the tendency of similar (or dissimilar) nodes to connect to each other. This tendency can have an influence on various properties of the network, such as its robustness or the dynamics of spreading processes. In this paper, we study degree assortativity both in real-world networks and in several generative models for networks with heavy-tailed degree distribution based on latent spaces. In particular, we study Chung-Lu Graphs and Geometric Inhomogeneous Random Graphs (GIRGs). Previous research on assortativity has primarily focused on measuring the degree assortativity in real-world networks using the Pearson assortativity coefficient, despite reservations against this coefficient. We rigorously confirm these reservations by mathematically proving that the Pearson assortativity coefficient does not measure assortativity in any network with sufficiently heavy-tailed degree distributions, which is typical for real-world networks. Moreover, we find that other single-valued assortativity coefficients also do not sufficiently capture the wiring preferences of nodes, which often vary greatly by node degree. We therefore take a more fine-grained approach, analyzing a wide range of conditional and joint weight and degree distributions of connected nodes, both numerically in real-world networks and mathematically in the generative graph models. We provide several methods of visualizing the results. We show that the generative models are assortativity-neutral, while many real-world networks are not. Therefore, we also propose an extension of the GIRG model which retains the manifold desirable properties induced by the degree distribution and the latent space, but also exhibits tunable assortativity. We analyze the resulting model mathematically, and give a fine-grained quantification of its assortativity.

Accurately identifying urban functional zones and revealing their spatial interaction characteristics is crucial for understanding urban operational mechanisms and optimizing spatial layouts. Addressing the limitations of traditional research in simultaneously capturing static functional attributes and dynamic resident travel behaviors, this study takes the central urban area of Lhasa as the research object, integrating ride-hailing trajectory data with Point of Interest (POI) data to conduct research on urban functional zone identification and spatial interaction characteristics. First, Thiessen polygons were used to quantify the spatial influence range of POIs, and an address matching algorithm was employed to associate ride-hailing origins and destinations (ODs) with POIs. A weighted land use intensity index was constructed, and functional zones were precisely identified using information entropy and K-Means clustering. Secondly, with basic research units as nodes and OD flows as edges, a directed weighted spatial interaction network was constructed. Complex-network indicators and the Infomap community detection algorithm were utilized to analyze network characteristics, node importance, and community interaction patterns. The results show that: (1) The functional mixing degree in the study area exhibits a pattern of “highly composite core, relatively differentiated periphery.” Eight functional zone types, including commercial–residential mixed, science–education–culture, and transportation service zones, were ultimately identified. Residential areas form the base, while the core area features multi-functional agglomeration. (2) The spatial interaction network exhibits typical small-world effects, while its degree distribution is better characterized by a lognormal distribution rather than a power law. Node importance is dominated by betweenness centrality, with Lhasa Station, the Potala Palace, and core commercial areas constituting key hubs. (3) The network can be divided into four functionally coupled communities: the core multi-functional area, the western industry–residence integrated area, the eastern science–education-dominated area, and the southern transportation hub area, forming a “core leading, two wings supporting” center–subcenter spatial organization pattern. This study verifies the effectiveness of integrating trajectory and POI data for identifying urban functional zones and provides a new perspective for understanding the spatial structure and planning of plateau cities.

Visibility graphs transform time series into complex networks where nodes represent time indices and edges encode temporal visibility relationships. While these graphs preserve structural properties of time series, the semantic interpretation of derived network characteristics remains challenging, limiting interpretability and raising questions about what information visibility graphs actually encode. This study enhances visibility graph interpretability using Statistical Process Control (SPC) as an external lens. We construct zone-labeled horizontal visibility graphs (HVGs) that preserve ordinal visibility properties while labeling nodes according to SPC zone classifications (A, B, C zones and control limit violations). Following Six Sigma practices, we quantize the vertical axis into control chart zones and establish explicit correspondences between SPC patterns and graph signatures, including degree distributions, edge weights, motifs, and local communities. The zone-labeled HVG framework successfully maps SPC-charted time series to interpretable network representations. Long runs and trends correspond to skewed in/out degrees and elongated paths, alternation patterns show elevated local clustering, while limit excursions manifest as hubs or articulation points partitioning the graph. This integration provides structured subgraph-level explanations for process alarms, demonstrating that visibility graphs inherently carry actionable information aligned with classical SPC rules, enabling explainable anomaly detection in industrial processes and quality control applications.

Gene-sharing networks provide a powerful framework to study the evolution of viruses and mobile genetic elements. These bipartite networks, which link genes to the genomes that contain them, exhibit characteristic degree distributions: a scale-free distribution for genes and an exponential-like decay for genomes. Here, we propose a mechanistic model that explains these patterns through fundamental evolutionary processes including horizontal gene transfer, capture of new genes, emergence of new genomes, and gene loss. Using a mean-field approximation, we derive analytical expressions for the asymptotic gene and genome degree distributions, recapitulating a power-law distribution for genes and an exponential distribution for genomes. Numerical simulations validate these predictions and yield parameter values that closely fit empirical data from dsDNA viruses, RNA viruses, and prokaryotic pangenomes. This simple model with only two parameters provides a generative framework for bipartite gene-sharing networks, offering qualitative and quantitative insights into the main evolutionary forces driving genome plasticity. Setting the gene loss rate to zero, the gene and genome degree distributions of the model closely fit the empirically observed distributions. Thus, evolution of viruses appears to be dominated by gene gain, in agreement with the results of independent reconstructions of viral evolution.

Given a graph G defined in a domain \(\mathcal {G}\) , we investigate locally differentially private mechanisms to release a degree sequence on \(\mathcal {G}\) that accurately approximates the actual degree distribution. Existing solutions for this problem mostly use graph projection techniques based on edge deletion process, using a threshold parameter \(\theta\) to bound node degrees. However, this approach presents a fundamental trade-off in threshold parameter selection. While large \(\theta\) values introduce substantial noise in the released degree sequence, small \(\theta\) values result in more edges removed than necessary. Furthermore, \(\theta\) selection leads to an excessive communication cost. To remedy existing solutions’ deficiencies, we present CADR-LDP, an efficient framework incorporating encryption techniques and differentially private mechanisms to release the degree sequence. In CADR-LDP, we first use the crypto-assisted Optimal- \(\theta\) -Selection method to select the optimal parameter with a low communication cost. Then, we use the LPEA-LOW method to add some edges for each node with the edge addition process in local projection. LPEA-LOW prioritizes the projection with low-degree nodes, which can retain more edges for such nodes and reduce the projection error. Theoretical analysis shows that CADR-LDP satisfies \(\epsilon\) -node local differential privacy. The experimental results on eight graph datasets show that our solution outperforms existing methods.

The non-commuting graph $\Gamma(G)$ of a non-abelian group $G$ has vertex set $G\setminus{Z(G)}$, with edges connecting elements that do not commute. In this paper, we provide a complete structural and spectral analysis of $\Gamma (\text{Dic}_n)$ for the dicyclic groups of order $4n$. Our main result is a decomposition $\Gamma (\text{Dic}_n) \cong\bar{K}_{2n-2}\vee(nK_2)^c$, where $\bar{K}_{2n-2}$ is the empty graph on $2n-2$ vertices and $(nK_2)^c$ is the complement of a perfect matching on $2n$ vertices. Using this decomposition, we determine the degree distribution, showing that the graph is regular only when $n = 2$, and compute fundamental parameters: diameter 2, girth 3 (for $n\ge3$), clique number $n + 1$ and independence number $2n-2$. We analyze the spectrum via equitable partitions, providing a complete description of the adjacency eigenvalues and their multiplicities. This work contributes to the growing literature on algebraically defined graphs by providing one of the few complete spectral characterizations for a non-trivial family of non-abelian groups. The paper concludes with explicit examples and a comparison with dihedral groups. Received: December 16, 2025Revised: January 31, 2026Accepted: February 9, 2026

Abstract In this paper, we address a long-standing challenge in self-organized criticality (SOC) systems by examining the connection between sandpile dynamics on two-dimensional lattices of different sizes and complex networks. Our approach employs a similarity-based transfer function characterized by two parameters, $\mathcal{R}=(r_1, r_2)$. Here, $r_1$ quantifies the similarity of local activities, while $r_2$ governs the filtration process used to convert a weighted network into a binary one. We reveal that the degree centrality distribution $p(k)$ of the resulting network follows a generalized Gamma distribution (GGD), with exponents depending on $\mathcal{R}$. Based on the numerical results, we uncover the fact that there are two regimes: dilute and dense regimes distinguished by the presence of a gap, i.e. the drop amount at the tail of the degree distribution function of nodes to zero. The dense regime is identified by a gapless $p(k)$ (i.e., $\Delta \approx 0$, gap value at the cutoff), while the dilute regime is gapful (i.e., $\Delta > 0$). Furthermore, the Shannon entropy is observed to decrease linearly with increasing $r_2$, whereas its variation with $r_1$ is more gradual. An analytical expression for the Shannon entropy is proposed. To characterize the network structure, we investigate the clustering coefficient ($cc$), eigenvalue centrality ($e$), closeness centrality ($c$), and betweenness centrality ($b$). The distributions of $cc$, $e$, and $c$ exhibit peaked profiles, while $b$ displays a power-law distribution over a finite interval of $k$. Additionally, we explore correlations between the exponents and identify a specific parameter regime of $\mathcal{R}$ and $k$ where the $e-k$, $c-k$, and $b-k$ correlations become negative.

Understanding how local interactions generate social polarization is a central challenge in the study of collective dynamics. Prior work has established that homophily and social balance can trigger a first-order phase transition to polarization in homogeneous small-world networks. However, real-world social networks are structurally heterogeneous, featuring power-law degree distributions dominated by a few highly connected hubs. In this study, we investigate how this structural heterogeneity affects polarization dynamics by introducing the co-evolutionary opinion model on scale-free networks generated via a modified Holme-Kim model. We find that, in contrast to the discontinuous transition in small-world networks, social polarization for scale-free networks grows continuously with increasing connectivity, emerging earlier but reaching a lower magnitude. Individual-level analysis reveals that polarization is a hierarchical process concentrated among large-degree hubs, while most peripheral nodes remain weakly polarized. Counterfactual experiments demonstrate that neutralizing only the top 5% of large-degree nodes is sufficient to suppress system-wide polarization to the random baseline. Our findings reveal that network topology fundamentally alters the nature of polarization transitions and suggest that targeted interventions on influential nodes may be more effective than broad-based approaches for managing social polarization in heterogeneous societies.

We investigate coherent quantum transport on multilayer complex networks composed of treelike scale-free layers. The general principles governing transport efficiency are uncovered under continuous-time quantum walks by tuning the degree distribution exponent γ, which interpolates between starlike and chainlike structures, and the interlink probability q. We show that global transport efficiency improves with increasing γ, increasing the number of layers, and increasing interlayer connectivity. The symmetry breaking and the decrease of spectral degeneracy of Laplacian maximally enhance transport efficiency when exactly one interlink between any pair of adjacent layers is removed. Notably, due to reduced structural symmetries nonidentical layers yield better transport than identical ones.