  Cycles, which can be found in many different kinds of networks, make the
problems more intractable, especially when dealing with dynamical processes on
networks. On the contrary, tree networks in which no cycle exists, are
simplifications and usually allow for analyticity. There lacks a quantity,
however, to tell the ratio of cycles which determines the extent of network
being close to tree networks. Therefore we introduce the term Cycle Nodes Ratio
(CNR) to describe the ratio of number of nodes belonging to cycles to the
number of total nodes, and provide an algorithm to calculate CNR. CNR is
studied in both network models and real networks. The CNR remains unchanged in
different sized Erd\"os R\'enyi (ER) networks with the same average degree, and
increases with the average degree, which yields a critical turning point. The
approximate analytical solutions of CNR in ER networks are given, which fits
the simulations well. Furthermore, the difference between CNR and two-core
ratio (TCR) is analyzed. The critical phenomenon is explored by analysing the
giant component of networks. We compare the CNR in network models and real
networks, and find the latter is generally smaller. Combining the
coarse-graining method can distinguish the CNR structure of networks with high
average degree. The CNR is also applied to four different kinds of
transportation networks and fungal networks, which give rise to different zones
of effect. It is interesting to see that CNR is very useful in network
recognition of machine learning.
