A cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics. An in-depth understanding of which cycles are important and what role they play on network structure and dynamics, however, is still lacking. In this paper, we define the cycle number matrix, a matrix enclosing the information about cycles in a network, and the cycle ratio, an index that quantifies node importance. Experiments on real networks suggest that cycle ratio contains rich information in addition to well-known benchmark indices. For example, node rankings by cycle ratio are largely different from rankings by degree, H-index, and coreness, which are very similar indices. Numerical experiments on identifying vital nodes for network connectivity and synchronization and maximizing the early reach of spreading show that the cycle ratio performs overall better than other benchmarks. Finally, we highlight a significant difference between the distribution of shorter cycles in real and model networks. We believe our in-depth analyses on cycle structure may yield insights, metrics, models, and algorithms for network science.

Diverse higher-order structures, foundational for supporting a network’s “meta-functions”, play a vital role in structure, functionality, and the emergence of complex dynamics. Nevertheless, the problem of dismantling them has been consistently overlooked. In this paper, we introduce the concept of dismantling higher-order structures, with the objective of disrupting not only network connectivity but also eradicating all higher-order structures in each branch, thereby ensuring thorough functional paralysis. Given the diversity and unknown specifics of higher-order structures, identifying and targeting them individually is not practical or even feasible. Fortunately, their close association with k-cores arises from their internal high connectivity. Thus, we transform higher-order structure measurement into measurements on k-cores with corresponding orders. Furthermore, we propose the Belief Propagation-guided Higher-order Dismantling (BPHD) algorithm, minimizing dismantling costs while achieving maximal disruption to connectivity and higher-order structures, ultimately converting the network into a forest. BPHD exhibits the explosive vulnerability of network higher-order structures, counterintuitively showcasing decreasing dismantling costs with increasing structural complexity. Our findings offer a novel approach for dismantling malignant networks, emphasizing the substantial challenges inherent in safeguarding against such malicious attacks.

Identifying influential nodes to maximize the spread of information within networks is a vital combinatorial optimization problem with extensive practical applications. Unlike proposing a specific node ranking method to identify vital nodes, this study introduces CycRank, a universal framework to optimize the strategies for selecting vital nodes in existing methods by leveraging cycle structures. The experimental results demonstrate that, compared to directly selecting top-k nodes from centrality rankings and state-of-the-art optimization frameworks, the influencers identified by CycRank increase the average dissemination range by up to 17%. Additionally, regardless of the centrality measures or network types, these influencers exhibit lower degree and greater average distances, effectively striking a delicate trade-off between their influence, dispersion, and hub properties. Our study not only paves the way for novel strategies in vital nodes identification but also underscores the unique potential of underappreciated cycle structures.