  Cycle is the simplest structure that brings redundant paths in network
connectivity and feedback effects in network dynamics. Focusing on cycle
structure, this paper defines a new matrix, named cycle number matrix, to
represent cycle information of a network, and an index, named cycle ratio, to
quantify the node importance. Experiments on real networks suggest that cycle
ratio contains rich information in addition to well-known benchmark indices,
for example, the node rankings by cycle ratio are largely different from
rankings by degree, H-index, coreness, betweenness and articulation ranking,
while the rankings by degree, H-index, coreness are very similar to each other.
Extensive experiments on identifying vital nodes that maintain network
connectivity, facilitate network synchronization and maximize the early reach
of spreading show that cycle ratio is competitive to betweenness and overall
better than other benchmarks. We believe the in-depth analyses on cycle
structure may yield novel insights, metrics, models and algorithms for network
science.
