Towards the cycle structures in complex network: A new perspective

Stars and cycles are basic structures in network construction. The former has been well studied in network analysis, while the latter attracted rare attention. A node together with its neighbors constitute a neighborhood star-structure where the basic assumption is two nodes interact through their direct connection. A cycle is a closed loop with many nodes who can influence each other even without direct connection. Here we show their difference and relationship in understanding network structure and function. We define two cycle-based node characteristics, namely cycle number and cycle ratio, which can be used to measure a node’s importance. Numerical analyses on six disparate real networks suggest that the nodes with higher cycle ratio are more important to network connectivity, while cycle number can better quantify a node influence of cycle-based spreading than the common star-based node centralities. We also find that an ordinary network can be converted into a hypernetwork by considering its basic cycles as hyperedges, meanwhile, a new matrix called the cycle number matrix is captured. We hope that this paper can open a new direction of understanding both local and global structures of network and its function.

💡 Research Summary

This paper shifts the focus of network analysis from the traditionally dominant star‑based view to the often‑overlooked cycle structures. While star structures assume interactions only between directly connected nodes, cycles form closed loops that enable indirect influence among multiple nodes. The authors introduce two cycle‑centric node importance metrics: (1) Cycle Number, the count of basic (non‑redundant) cycles that a node participates in, and (2) Cycle Ratio, the sum of the proportions of the node’s presence in the basic cycles of all nodes that share its cycles.

To evaluate these metrics, the authors analyze six diverse real‑world networks (social, biological, infrastructural, etc.). They compare the new metrics against classic star‑based centralities such as degree, k‑core coreness, and H‑index. In robustness experiments where nodes are removed in descending order of a given metric, networks disintegrate most rapidly when removal follows the Cycle Ratio ranking, indicating that nodes with high Cycle Ratio are crucial for overall connectivity. In spreading simulations that mimic information diffusion along cycles, seeding the process with nodes of high Cycle Number yields the largest infected fraction, showing that Cycle Number better captures a node’s potential to drive cycle‑based propagation.

A further contribution is the formal conversion of an ordinary graph into a hypergraph by treating each basic cycle as a hyperedge. Using the hypergraph’s incidence matrix, the authors derive three new matrix representations:

• Cycle Number Matrix (B) – entry (i,j) counts the number of cycles shared by nodes i and j; diagonal entries equal each node’s Cycle Number.
• Cycle Ratio Matrix – obtained by normalizing each column of B by its diagonal element; column sums equal the corresponding node’s Cycle Ratio.
• Spread Matrix – a binary version of B (diagonal set to zero) that encodes whether two nodes share at least one cycle; it serves as a compact operator for cycle‑based diffusion dynamics.

These matrices play a role analogous to the adjacency matrix in star‑centric analysis, providing tools to study both structural properties (e.g., clustering of cycles) and dynamical processes (e.g., diffusion, synchronization) from a cycle‑centric perspective.

The paper’s key contributions are: (1) defining and empirically validating cycle‑based importance measures, (2) establishing a mathematically sound bridge between ordinary graphs and hypergraphs via cycle‑derived hyperedges, and (3) introducing matrix‑based frameworks (Cycle Number, Cycle Ratio, Spread matrices) that enable simultaneous structural and dynamical analyses of networks. The authors argue that these insights are especially relevant for systems where multi‑party interactions dominate, such as group communications on social platforms, brain networks where cliques and cavities (higher‑order cycles) underlie cognition, and engineered systems adopting cycle‑based coordination. Overall, the work opens a new avenue for understanding and exploiting the mesoscopic cycle architecture that underlies many complex networks.