Towards the cycle structures in complex network: A new perspective

TL;DR

This paper introduces cycle-based node characteristics to better understand network importance and influence, highlighting the significance of cycles in network structure and proposing a new hypernetwork representation.

Contribution

It defines cycle number and cycle ratio as novel metrics, explores their relationship to node importance, and introduces a cycle number matrix for analyzing complex networks.

Findings

Nodes with higher cycle ratio are more crucial for connectivity

Cycle number better quantifies influence in cycle-based spreading

Converting networks into hypernetworks via cycles offers new structural insights

Abstract

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.