The characteristics of cycle-nodes-ratio and its application to network classification

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 Erdos--Renyi (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. (c) 2021 Elsevier B.V. All rights reserved.