Abstract We study phase allocation in networks with an assumed cyclic timing frame. Each node chooses where in a cycle of length $ P $ to update; directed edges reward useful phase offsets, and a crowding term penalizes excessive use of the same phase. The model therefore does not explain the origin of the cycle itself. It asks how graph structure shapes the stable allocation of nodes across phases once such a frame is available. We show that the resulting finite timing game converges under asynchronous local improvement moves. Its equilibria are not a single kind of clock: they include diffuse timing, phase clustering, hub- or core-anchored timing, and distributed or defected scaffolds. We analyze these outcomes through circular phase modes, normalized phase entropy, and degree-weighted timing fields. In simulations across synthetic graph families, high-degree momentum terms provide compact diagnostics of the converged phase-entropy organization and improve within-ensemble prediction relative to mean degree alone. Within the controlled graph ensembles studied here, these quantities diagnose phase-regime organization; they should not be read as universal graph-family-independent predictors. Degree-weighted order parameters, interpreted against frozen-random baselines, distinguish learned phase organization from structural concentration artifacts in highly heterogeneous graphs. Additional local-bandit experiments show that sampled decentralized learners can recover similar regime-level outcomes without evaluating all phases at every update.
Abstract Understanding systemic risk in financial systems is of critical importance for both academic research and regulatory policy. A central unresolved issue is whether systemic risk is driven primarily by the financial soundness of individual banks or by the structure of interbank networks. To investigate this question, we develop an enhanced agent-based model that incorporates median absolute deviation and bank credit ratings to characterize heterogeneity in banks’ risk profiles. By combining these features with a bidirectional matching mechanism (BiMM), the model generates a realistic interbank network that reflects key structural characteristics of the Chinese banking system. Systemic risk contagion is then simulated using a weighted DebtRank (W-DebtRank) framework, in which bank category specific weights capture differences in systemic importance. Simulation experiments based on a sample of 400 Chinese banks yield two main findings. First, banks occupying more central positions in the network contribute disproportionately to the amplification of systemic risk. Second, network structural characteristics have a stronger influence on systemic risk propagation than conventional financial soundness indicators, such as capital adequacy and leverage ratios. Overall, this study provides a novel modeling framework for constructing realistic financial networks and offers empirical evidence that network structure plays a dominant role in systemic risk, with important implications for macroprudential regulation.
Abstract The analysis of graph data has gained increasing attention in recent years. This is justified by the numerous applications in which it appears. Several methods exist to predict graphs, but far fewer to quantify the uncertainty of the prediction. The present work proposes an uncertainty quantification methodology for graph data, based on conformal prediction. The method works for both graphs with the same set of nodes (labelled graphs) and graphs with no clear correspondence between the set of nodes across the observed graphs (unlabelled graphs). The unlabelled case deals with the creation of prediction regions embedded in a discrete quotient space. The proposed method does not rely on distributional assumptions, it achieves finite-sample validity, and it identifies interpretable prediction regions. To explore the properties of this novel uncertainty quantification technique, we perform two simulation studies to show the methodology in both the labelled and the unlabelled cases. We showcase the applicability of the method in analysing the performance of different teams during the 2018 FIFA Football World Cup via their player passing networks.
Community detection in temporal networks must reconcile two competing goals: reacting to genuine mesoscale changes while avoiding spurious label fluctuations caused by transient activity. This paper proposes an adaptive spectral framework that places memory in the modularity objective and smoothness in the labels. At time $ \boldsymbol{t} $, an adaptive modularity operator is formed by exponentially aggregating past modularity residuals with a decay rate $ \boldsymbol{\lambda} $ and history weight $ \boldsymbol{\beta} $, so that persistent community evidence accumulates relative to the configuration-model baseline. The partition is initialized by the leading eigenvector of the adaptive operator and refined by recursive bipartitioning with a Kernighan-Lin-style refinement step that directly optimizes a regularized modularity objective, where label persistence is rewarded only on the shared node set $ \boldsymbol{V_{t}\cap V_{t-1}} $. A matrix-free implementation with history truncation enables scalable eigensolver iterations without materializing dense modularity matrices. Experiments on dynamic stochastic block model benchmarks yield robust accuracy across evolution regimes (e.g. TNMI $ \boldsymbol{=0.92,0.87,0.76} $ for transition probabilities $ \boldsymbol{0.1,0.2,0.5} $), and real temporal networks (MIT Reality Mining and Drosophila gene co-expression) show improved temporal consistency while preserving modular structure. Overall, objective-level memory on modularity residuals, coupled with label-level regularization, provides an interpretable and effective tool for tracking evolving mesoscale organization in temporal networks.
Abstract In complex networks, link prediction is used to predict future relationships or missing edges. Due to applications in multiple domains, link prediction has attracted the attention of researchers from the domain. Plenty of algorithms have been proposed for link prediction in the past, but predicting missing links in networks correctly and efficiently is still a challenging problem. The proposed algorithm is suitable for both unipartite and bipartite networks and would perform well on transportation, financial and biological networks. Also, the algorithm has been extended for weighted networks. In this work, we present a novel Similarity-based method for link prediction based on the Current-flow Centrality, Shortest distance, and Clustering coefficient. We performed extensive experiments and the performance of the algorithm has been evaluated using four metrics, namely AUC, Precision, Prediction power and Prediction@K. The Proposed algorithm has been tested on 16 datasets and compared with 14 baseline algorithms. The experimental results show that the novel algorithm outperforms the baseline algorithms based on the given 4 metrics.
We propose an interpretable, network-based measure of tourism resilience that maps destinations on a two-dimensional plane combining pre-shock market diversity and shock-period hierarchisation. Using monthly inbound international arrivals of non-resident foreigners to Colombian cities, we compute (i) pre-shock Shannon entropy of origins (2018-2019) and (ii) the maximum absolute residual from a monthly log-log Katz-size scaling during the COVID-19 shock (2020-2021). Applied to Colombia, the resilience plane identifies a core-centric system: most international arrivals concentrate in a few diversified gateways that nonetheless experienced large hierarchy spikes under stress. A smaller set of "resilient hubs" combine high diversity with low hierarchisation but account for a minor share of volume. Results are robust to thresholding with interquartile cutoffs and to an alternative city-city projection (cosine similarity). The findings suggest that, for major gateways, market diversification alone is insufficient if access remains structurally compressed into a small set of dominant channels; for more fragile destinations, priorities include broadening source portfolios and improving connectivity to regional hubs. The approach is replicable with open data and standard network tools, and is portable to other countries to benchmark destination systems on a common, interpretable resilience scale.
Controlling complex networks remains a central challenge in modern network science and engineering, with broad implications across disciplines such as electronic circuits, gene regulation, metabolic systems, and ecological networks. Structural controllability theory provides a mathematical foundation for identifying a minimal set of driver nodes through which external inputs can steer the entire system. Traditional approaches based on maximum matching, such as Path Finding and Signal Sharing, often yield redundant or non-optimal driver-node selections, particularly in networks lacking a root strongly connected component. In this study, a refined framework is developed by introducing new concepts in matching theory and formulating an Augmenting Trail method for determining maximum matchings in directed networks. The unmatched nodes resulting from this process represent the optimal set of driver nodes required for full structural controllability. Comprehensive experiments on 90 empirical networks and multiple synthetic models demonstrate that the Augmenting Trail method consistently identifies a smaller and structurally more diverse set of driver nodes compared with existing algorithms. Analyses of node-level indices-including degree, coreness, and clustering coefficient-reveal that Augmenting Trail tends to select nodes of moderate connectivity located at the periphery of strongly connected components, thereby ensuring efficient and distributed control. Jaccard similarity assessments confirm that the Augmenting Trail method introduces new, non-redundant driver nodes while maintaining partial overlap with established methods. Furthermore, a strong negative correlation (rho = -0.62, p < 0.001) between degree heterogeneity and the fraction of driver nodes underscores that networks with higher topological variability require fewer control inputs, a relationship consistently observed across Erdos-Renyi, small-world, and clustered network types. Overall, the Augmenting Trail framework achieves a superior balance between control efficiency and structural diversity, bridging theoretical rigor with computational scalability. It provides a generalized foundation for analyzing controllability in large-scale, heterogeneous, and dynamically evolving networked systems.
This paper develops a spectral-topological foundation for long memory in network autoregressions. We consider VAR(1) dynamics driven by the node Laplacian of a sequence of weighted graphs with weakly dependent innovations. Long memory emerges when the near-zero spectrum of the Laplacian is thickened by either of two structural mechanisms: (i) bottlenecks, which trap flows across network cuts; and (ii) long cycles with regularly varying distributions, which yield near-harmonic modes. These imply long-memory bounds for the autocovariances of linear observables and, along with harmonic components, they allow coexistence of random-walk and stationary long-memory behaviors. The framework could be useful to the design of new applications of network models in economics.
In complex networks, identifying key nodes has attracted significant attention from researchers in areas such as information diffusion, network attacks, and epidemic spreading. In recent years, most studies have focused on identifying influential nodes in unweighted and undirected networks. However, many existing methods are not directly applicable to weighted and directed complex networks, as they typically consider only the immediate neighbors of a node without accounting for the interactions and structural information among them. Furthermore, traditional centrality measures for node ranking are often limited to single-feature factors and suffer from issues such as poor scalability, unequal neighbor contributions, and sensitivity to network topology. Incorporating multidimensional factors and edge weights into node importance evaluation can significantly enhance the identification of key nodes in large-scale and heterogeneous networks. To address these limitations, this article introduces a Multidimensional Entropy-based Weighted Semi-Local (MEWSL) centrality for identifying key nodes in weighted complex networks. To better capture semi-local neighbor influence and improve scalability, MEWSL constructs weighted semi-local subgraphs for each node in a distributed manner. It integrates multiple factors-such as the degree of neighboring nodes, the number of shortest paths, and the average shortest path-to quantify node importance within weighted semi-local subgraphs. Information entropy is employed to assign weights to different contributing factors, and MEWSL computes the final node ranking by aggregating these factors through a weighted sum. The proposed metric considers not only the relationship between a node and its neighbors but also the interactions among neighboring nodes by incorporating edge weights. MEWSL is evaluated on real-world networks using the susceptible-infected-removed (SIR) model and Kendall's correlation, demonstrating more effective node ranking with lower computational complexity than existing methods.
In this paper, we apply a role classification method to international trade networks derived from the OECD-ICIO dataset. Each node in the network corresponds to a country in the dataset and is associated with a feature vector based on network centrality indicators. After reducing redundant information via principal component analysis, we group countries using fuzzy clustering. This defines distinct roles, to which each country belongs with an appropriate membership level. Our analysis then focuses on how each countrys role in the international production and trade network has evolved between 1995 and 2022 and the related economic implications. The results show that, while many countries (such as the USA and Germany) have essentially maintained their position over time, many others have changed dramatically. For instance, although at different levels, China and Vietnam have transitioned to higher-level roles, whereas the United Kingdom, Japan and Italy, among others, have deteriorated from leader to second-tier country status. Interestingly, a sharp downgrading of role can occur even when economic and network indicators undergo only modest changes, if competing countries increase their performance disproportionately. The work concludes with a separate analysis of some of the most important goods and services sectors. Overall, the results reveal growing polarization, with a small number of leading countries progressively moving away from the rest of the world in terms of economic dominance.
We consider an exact algorithm for generating connected Erd & odblac;s-R & eacute;nyi random graphs $ \boldsymbol{G(n, p)} $. The method is based on coupling the graph exploration process to an inhomogeneous Poisson random walk, which yields an exact sampler that runs in $ \boldsymbol{O(n)} $ time in the sparse regime $ \boldsymbol{p\,=\,c/n} $. We also show how the method extends to the $ \boldsymbol{G(n, M)} $ model via an additional acceptance-rejection step.
In this paper, we propose a class of asymmetric networks with two different coupling styles to highlight the impact of network parameters and leader position on network coherence. Firstly, the exact analytical expressions for the leaderless coherence of two networks are derived. Secondly, we select several distinct schemes for assigning leaders to the networks and derive the corresponding expressions between leader-follower coherence and network parameters. Thirdly, the coherence and robustness properties of the networks are further analyzed based on the previously derived theoretical results. Finally, the Kirchhoff index, global mean first-passage time, and Laplacian energy of the networks are analyzed. Based on the analysis results of this paper, we can conclude that leader position and network parameters have a profound impact on coherence. Specifically, the coupled networks with no chain connections in the middle exhibit better coherence performance. The analytical results obtained above provide a new perspective and theoretical foundation for the optimal design of coordination mechanisms in distributed systems.
This work introduces a novel measure of centrality for simplicial complexes of arbitrary dimension $ k $, called $ k $-Laplacian centrality, which extends the concept of Laplacian centrality from graph theory. Through experimentation on real-world data and comparison with existing centrality measures, our findings reveal that $ k $-Laplacian centrality assigns higher centrality to simplices that form the core of dense, cohesive communities.
Urban scaling laws reveal how cities evolve as their populations grow, yet the precise quantitative relationship between urban size and spatial proximity influence remains underexplored. We analyze 5252 Brazilian cities to establish a scaling law linking average closeness centrality, $ \langle c_{C} angle -a measure of the influence of spatial proximity on the dissemination of information in street networks-to city population size $ N $. Our results demonstrate that $ \langle c_{C} angle $ decays sublinearly as $ N{-\sigma} $, with a characteristic exponent $ \sigma\approx 0.41 $. We show that this specific scaling behavior arises from the fractal interplay between infrastructure and population, characterized by an effective network dimension $ d\approx 2.20 $, which exceeds that of a regular 2D grid. The slower decline in closeness centrality $ \sigma\,\lt\,0.5 $ highlights a mitigating effect: while urban growth naturally reduces average spatial proximity, the network's capacity to form topological shortcuts partially preserves navigability and enhances connectivity. By integrating the Molinero & Thurner model [1]-which offers a mechanistic explanation for the observed exponent-with network centrality metrics, our work provides a framework to reconcile infrastructure efficiency with the influence of equitable spatial proximity in growing cities.
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including clustering, graph classification, and graph neural networks. Defining persistent homology for graphs is relatively straightforward, as graphs possess distinct intrinsic distances and a simplicial complex structure. However, hypergraphs present a challenge in preserving topological information since they may not have a simplicial complex structure. In this paper, we define two persistent homology filtrations for hypergraphs using their barycentric subdivision in defining persistent homology to extract different topological features within hypergraphs. To showcase the effectiveness, we employ these features in the hypergraph classification problem on four different real-world hypergraphs. We also compare their performance to the widely used simplicial complex closure filtration and also graph neural network models. Experimental results demonstrate that our persistent homology filtrations extract meaningful topological features that are effective in classifying hypergraphs and outperform the baseline models.
The availability of network datasets advances research in network science, machine learning and related fields by enabling empirical analyses and their reproducibility, algorithm development, model validation and benchmarking. Existing repositories, such as SNAP and Netzschleuder, have made traditional network datasets widely accessible with metadata, metrics, and basic visualizations. However, they primarily focus on pairwise interactions, limiting data access to systems with many-body interactions. To address this gap, we created hypergraphx-data, a repository of real-world hypergraph datasets for higher-order network analysis, spanning different domains from social networks to biology and finance, and supporting configurations such as weighted, directed, temporal, and multiplex hypergraphs. Each dataset includes relational information and metadata, provided in an open JSON format and a binarized format for Hypergraphx. We provide a user-friendly interface to facilitate browsing, filtering, and accessing the datasets, while also ensuring integrity and reproducibility through hash-based verification and data versioning. The repository is available at https://hgx-team.github.io/hypergraphx-data
Graph Neural Networks (GNNs) have become the dominant deep learning model for learning on graph-structured data, enabling breakthroughs in fields ranging from bioinformatics to social network analysis. Yet, as any deep learning model they suffer from 'black box' syndrome. Their decisions making process remains largely unknown. In this work, we want to advance our understanding of GNN explainability. We evaluate the performance and stability of GNNExplainer, a widely used posthoc interpretability method, on the simple task of random graph classification. Using three very different GNN architectures, Graph Convolutional Networks, Graph Attention Networks, and Graph Isomorphism Networks, we examine the explainability of models trained to distinguish between Erdos-Renyi and Barabasi-Albert random graphs, as well as between dk-randomized variants of four real-world networks. Our results show that despite the models achieving perfect classification accuracy, feature importance values identified by GNNExplainer exhibit substantial variability across architectures, hyperparameters, and random seed values. Moreover, the extracted explanations often fail to align with theoretical expectations based on established graph properties, such as degree distributions and degree correlations. These findings indicate that explanations produced by GNNExplainer are highly model-, configuration-, and seed value-dependent, challenging its reliability for deriving general insights into GNN decision mechanisms. Our work highlights fundamental limitations in the current generation of GNN explanations using GNNExplainer and suggests the need for more stable, theoretically grounded approaches to explainability in graph-based learning.
Addressing the overlooked stochastic, language-specific interactions on heterogeneous networks, we put forward a multilingual stochastic IE2SR rumor model with variable contact rates. We first develop a deterministic IE2SR model and prove the existence and uniqueness of its equilibrium. Subsequently, we extend it to a stochastic version, deriving explicit conditions for rumor existence and extinction using the strong law of large numbers. Numerical simulations confirm our theoretical findings and reveal that both contact rates and stochastic noise significantly impact rumor thresholds, offering practical insights for rumor control in multilingual environments.
The Laplacian spectrum reflects the intrinsic characteristics and properties of a network, which can be used to quantify and evaluate performance metrics, such as connectivity, robustness, information propagation efficiency, consistency, and so on. Firstly, we systematically analyze the topological characteristics of a class of polygonal networks. Secondly, the Laplacian spectrum is calculated through matrix decomposition and an analysis of the characteristic polynomial. Finally, as an application of the obtained results, we derive expressions for the Kirchhoff index, the number of spanning trees, the mean first-passage time, and the network coherence.
We revisit the classical friendship paradox which states that on an average one's friends have at least as many friends as oneself and generalize it to a variety of network centrality indices. For a broad class of spectral centralities on connected undirected graphs-degree, eigenvector centrality, walk counts, Katz centrality and PageRank, we show that the average centrality of a node's neighbours always exceeds the global average centrality. We further prove an analogous result for PageRank on strongly connected directed graphs. For degree, this recovers the classical friendship paradox, while for the other centralities it yields new instances of what we call the centrality paradox. We also compare our neighbour-averaged formulation with edge-sampled versions studied previously in the literature.
