This work introduces a non-parametric partition of a network derived from its persistent homology, defined using the concept of a persistence surface, and demonstrates that persistent homology can reveal distinctive structural features that are not detected by conventional methods.
Abstract
Determining network structures that are neither local nor global is an area of research that has received considerable attention. The study of these intermediate structures has been primarily concerned with the detection of network communities but also includes the examination of network roles, core and peripheral structure, etc. In an increasingly relevant line of research, persistent homology has also been used to analyze the shape of a network in terms of the network’s cycle structure and its higher-dimensional analogues. In this work, we bring these two perspectives together by introducing a non-parametric partition of a network derived from its persistent homology. This partition, which we call the network’s persistence partition, is defined using the concept of a
persistence surface
, assigning to each node a measure of its individual persistence relative to its position in the network. We examine the extent to which this partition aligns with standard notions of network roles defined via combinatorial equivalence. We then compare how persistence partitions relate to communities and to the core–periphery structure of a network. Our analysis draws on both real and synthetic networks and demonstrates that persistent homology can reveal distinctive structural features that are not detected by conventional methods.
The proposed percolation process displays a universal phase transition, characterized by the abrupt and simultaneous disappearance of both the giant connected component and the largest 2-core, across networks with markedly different degree distributions, indicating that the physics of network dismantling is insensitive to a broad range of topological properties.
L. Cirigliano, Claudio Castellano, Minsuk Kim et al.· 0 citations
The results show that smaller clusters are generally more vulnerable to attacks on central nodes, whereas larger and less centralized clusters retain more topological efficiency.
In complex network analysis, the identification of influential nodes is a fundamental issue, which is closely related to the structural robustness of the network and the dynamics of propagation processes. Current research primarily focuses on mesoscale features based on the smallest cycles or local features derived from star-shaped structures. However, the role of neighboring nodes that are connected to a given node but do not participate in its smallest cycles remains underexplored in network analysis. To address this issue, this paper proposes a hybrid centrality measure that integrates information from both smallest-cycle structures and non-smallest-cycle structures associated with each target node. The smallest-cycle structures considered in this method are identified only within the imposed local search range and do not necessarily correspond to the true smallest cycles in the full graph. Specifically, the extent of a node’s involvement in mesoscale structures is characterized by the number of the smallest cycles it participates in, while its local structural heterogeneity is represented by the number of neighboring nodes connected to it that do not belong to any smallest cycles. These two aspects are then unified into a single node importance metric through a weighted integration strategy. This paper evaluates node importance from multiple perspectives, including propagation capability analysis based on the SI model, network robustness testing through node attack simulations, and ranking accuracy assessment using Kendall correlation coefficient. The experimental results demonstrate that the proposed method achieves competitive or superior performance compared with the selected baseline methods under the experimental settings considered in this work. The findings indicate that integrating smallest-cycle and non-smallest-cycle features provides a more comprehensive characterization of a node’s role in complex networks. This study offers a novel perspective on the integration of multi-scale structural information in complex networks and presents an effective new approach for the identification of important nodes.
Fu-Rui Tan, Xiao-long Chen, Ruijie Wang et al.· Entropy· 0 citations
This work introduces a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case and establishes 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.
Yu Tian, Eleanor P. Wiesler, Melanie Weber· 0 citations
The Internet, a living cell, a circle of friends, a billion-dollar construction project: these systems share almost nothing -- yet, drawn as networks, they look astonishingly alike. Each has a few giant hubs among a multitude of sparsely connected nodes, short paths between any two parts, dense local clustering, communities, and many redundant routes. For two decades such patterns have been credited to"preferential attachment,"the rich getting richer -- a rule that, taken literally, asks every newcomer to survey the whole network before it links. This book makes a simpler case, and defends it one mechanism at a time: the global regularities of real networks are not imposed from above but emerge from purely local rules, in which each new node acts only on a node it has reached and that node's immediate neighbours. A surfer following links, a friend introducing a friend, a gene copied with its connections -- none consults the network as a whole, yet each builds, in the aggregate, the full and unmistakable signature of a real complex system. Written for the curious reader as much as the specialist, with the ideas told in plain language and the mathematics set aside in boxes that can be skipped, it shows how citation graphs, the web, social ties, protein interactions, and project schedules all grow themselves from the same handful of local rules -- one local decision at a time.
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.
E. Vasilyeva, L. Tupikina, D. Musatov et al.· Chaos, Solitons & Fracta...· 0 citations
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