Table 1 The relevant parameters of the network and its specific meaning

Number of nodes

Number of metabolites in metabolic network, n

Number of edges

Two nodes participating in the same reaction are connected by edges, \(\user2{e}_{\user2{n}}\) is the number of edges connected to node n. The number of edges of the network is the sum of \(\user2{e}_{\user2{n}} .\)

Average number of neighbors

Number of neighbors \(\user2{k}_{\user2{n}}\) is the number of nodes that react with metabolite n

Network diameter

The largest distance between two nodes, d

Network radius

The minimum among the non-zero eccentricities of the nodes in the network, r

Characteristic path length

The expected distance between two connected nodes, l

Clustering coefficient

The clustering coefficient \(\user2{C}_{\user2{n}}\) of a node n is defined as \(\user2{C}_{\user2{n}} = 2\user2{e}_{\user2{n}} /\left( {\user2{k}_{\user2{n}} \left( {\user2{k}_{\user2{n}} - 1} \right)} \right).\) The network clustering coefficient is the average of the clustering coefficients for all nodes in the network

Network density

The density of interconnected edges between nodes in the network,\(\user2{mean}\left( {\user2{k}_{\user2{n}} } \right)/\left( {\user2{n} - 1} \right)\)

Network heterogeneity

The tendency of a network to contain hub nodes,\(\sqrt {\user2{variance}\left( {\user2{k}_{\user2{n}} } \right)} /\user2{mean}\left( {\user2{k}_{\user2{n}} } \right)\)

Network centralization

The concentration of other nodes connected to a node similar to a stellar,\(\max \left( {\user2{k}_{\user2{n}} } \right)/\user2{n}\)