Abstract
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.
Highlights
A cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics
Denote by Si the set of the shortest cycles associated with node i, and S 1⁄4 ∪ i2V Si the set of all shortesht ciycles of G, we define the so-called cycle number matrix C 1⁄4 cij N N to characterize the cycle structure of G, where N=|V| is the number of nodes in G, and cij is the number of cycles in S that pass through both nodes i and j if i≠j
The basic idea underlying such an index is that if cycles are important in maintaining connectivity and interacting dynamics, a node involved in many cycles should be vital
Summary
A cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics. Recent studies have uncovered the topological properties of cycles, including the distribution of cycles of different sizes in real and artificial networks[12,13,14,15,16], the effect of degree correlations on the loops of scale-free networks[17], as well as the significant roles of the cycles in network functions related to storage[18], synchronizability[19], and controllability[20]. Considering a simple network where direction and weight of a link are ignored and self-loops are not allowed, a cycle is the simplest structure providing redundant paths to all involved node pairs.
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