Table 2 Common characters of complex networks

Scale-free

It is a most notable characteristic in complex systems. It was used to desibe the finding that most nodes in a network have few neighbors while few nodes have large amount of neighbors. In most cases, the connectivity distribution asymptotically follows a power law [43]. It can be expressed in $P\left(k\right)~k^{-y}$, where P(k) is the number of nodes with k degrees, k is connectivity/degrees and γ is a constant.

Small-world

It is a terminology in network analyses to depict the average distance between nodes in a network is short, usually logarithmically with the total number of nodes [44]. It means the network nodes are always closely related with each other.

Modularity

It was used to demonstrate a network which could be naturally divided into communities or modules [45]. Each module in gene regulation networks is considered as a functional unit which consisted of several elementary genes and performed an identifiable task [23, 46]. A modularity value can be calculated by Newman’s method [45] which was used to measure how well a network is able to be separated into modules. The value is between 0 to 1.

Hierarchy

It was used to depict the networks which could be arranged into a hierarchy of groups representing in a tree structure. Several studies demonstrated that metabolic networks are usually accompanied by a hierarchical modularity [37, 44]. It was potentially consistent with the notion that the accumulation of many local changes affects the small highly integrated modules more than the larger, less integrated modules [37]. One of the most important signatures for hierarchical modular organizations is that the scaling of clustering coefficient follows C(k) ~ k^−γ (scaling law), in which k is connectivity and γ is a constant [47].