Jul 2026· Chaos, Solitons & Fractals· 0 citations· 27 references
Physics
TL;DR
This work generalizes three distance-based topological measures, namely closeness centrality, betweenness centrality and node eccentricity, using this new hypergraph distance, and shows that hypergraphs can be divided into three distinct classes, corresponding to the possible dominance of specific orders of interaction over their general metric structure.
Abstract
Higher-order interactions introduce an additional structural dimension to complex networks, requiring consistent generalizations of classical topological measures. In hypergraphs, the definition of distance between nodes is not unique: beyond the conventional measure derived from clique projection, an alternative formulation that explicitly incorporates the sizes of hyperedges, those of their intersection and their weights has been recently proposed. Here, we generalize three distance-based topological measures, namely closeness centrality, betweenness centrality and node eccentricity, using this new hypergraph distance. Trough tractable illustrative examples, we demonstrate that the differences between results obtained with the two distances are systematic and arise from structurally meaningful features of the higher-order networks. Also, analyzing a series of real-world datasets, we show that hypergraphs can be divided into three distinct classes, corresponding to the possible dominance of specific orders of interaction over their general metric structure. This provides practical guidance on the possibility of limiting the analysis to only some specific interaction orders, reducing its complexity while maintaining the full information of the system.
Hypergraphs describe higher-order interactions that involve more than a pair of nodes. A characteristic feature of hypergraphs is that their robustness can be strongly affected by the different roles of the nodes. Indeed, some nodes might be essential for a hyperedge's function, while others might not be. The loss of a single essential node completely destroys the hyperedge it belongs to, while the loss of a non-essential node has a buffering effect, inducing the hyperedge to simply reduce its size. In order to capture this phenomenology, we formulate a comprehensive theoretical framework for $(k,n)$-core percolation models on hypergraphs, where each node of a hyperedge is an anchor with probability $\theta$, and a hyperedge fails if an anchor node fails. Hypergraph $(k,n)$-core percolation problems can be classified as first-neighbor and second-neighbor problems, indicating that in the pruning process the connectivity is ensured only by the state of the first neighbors or the second neighbors, respectively. We derive self-consistency equations for first-neighbor and second-neighbor (node- and hyperedge-based) pruning processes, and obtain the size of the giant $(k,n)$-core. We obtain the phase diagram, including continuous and discontinuous transitions, and confirm our theory on random hypergraphs using numerical simulations. The results show how the heterogeneity of the nodes'functional roles and the extended range of the interactions affect the robustness of higher-order networks.
Under stochastic block models, it is proved that stronger topological disruptions incur asymptotically larger distances almost surely in both inter-community and intra-community split settings.
S. Bhattacharyya, Huiyan Sang, Bani Mallick· 0 citations
In the framework of complex network analysis, hypergraphs provide a natural generalization for modeling higher-order interactions. In this work, we investigate the extremal structural properties of uniform hypertrees with respect to the Sombor index and the Sombor spectral radius, two degree-based measures that capture nonlinear connectivity patterns. For hypertrees of fixed size, we identify the structures that maximize and second-maximize these descriptors. We show that both the Sombor index and the Sombor spectral radius increase strictly under an edge-releasing operation applied to non-pendant hyperedges, revealing a monotonic structural transformation principle. This result enables us to characterize the hyperstar configuration as the unique maximizer of both measures. Furthermore, by systematically employing edge-moving and edge-releasing operations, we determine the hypertree structure that attains the second-highest values of these indices. Our findings contribute to the understanding of how local structural modifications influence global spectral and topological descriptors in higher-order networks, offering insights relevant to the study of nonlinear and complex systems.
Shashwath S. Shetty, K. Bhat· Match-communications in Math...· 0 citations
This study introduces a new ranking framework that integrates a quasi-Laplacian structural measure with a gravity-inspired aggregation process and demonstrates that the proposed framework consistently outperforms existing techniques in terms of accuracy, resolution, and computational simplicity.
Centrality measures are defined for simple graphs -- directed, undirected, weighted or unweighted. Attributed graphs have to be reduced to simple graphs for computing centrality measures. However, when applications with multiple types of relationships are modeled using multilayer networks (MLNs), simple graph algorithms cannot be directly used. Existing approaches typically analyze MLNs by aggregating layers of an MLN into a single graph, which results in the loss of structural and semantic information. The semantic information loss can be more pronounced particularly, in weighted networks. This work focuses on computing degree centrality in weighted homogeneous multilayer networks (HoMLNs) using a decoupling-based framework. The framework performs independent layer-wise analysis on MLNs without reducing them to simple graphs. The decoupling approach allows use of exiting algorithms for each layer and uses minimal information from individual layers for computing degree centrality of HoMLNs. We propose heuristic-based algorithms that strike a balance between accuracy and efficiency. The proposed methods are evaluated against ground truth (GT) results obtained using Boolean OR aggregation and naive baselines. Experimental results on both synthetic and real-world HoMLN datasets demonstrate that the heuristics achieve accuracy comparable to the ground truth while significantly improving computational efficiency, thereby establishing the scalability and effectiveness of the HoMLN algorithms developed using the decoupling approach.
A. Ayowole-Obi, Abhishek Santra, Sharma Chakravarthy· 0 citations
Understanding the geometry of complex networks is critical for effective modeling and analysis across domains. While discrete notions of Ricci curvature have emerged as powerful tools for characterizing both local and global network structure, existing formulations are largely confined to undirected networks with real-valued weights. This limits the use of curvature-based analysis of directional and complex-weighted relations that arise naturally in many applications, from social and biological systems to quantum and signal-processing networks. In this work, we introduce a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case. We establish fundamental theoretical properties of this new notion, including relations to the magnetic Laplacian and combinatorial upper and lower bounds that relate curvature to cycle structure in local neighborhoods. We further develop computational methods for curvature estimation and demonstrate their utility in community detection on directed networks.
Yu Tian, Eleanor P. Wiesler, Melanie Weber· 0 citations