Table 4 Expression and interpretation of innovation network indicators.

Overall network indicators

Network density

\(D = \frac{2r}{{n(n - 1)}}\)

n is the number of members in the network (the same below); r is the actual number of relationships included in the city contact network

Average path length

\(L = \frac{2}{n(n - 1)}\sum {d_{ij} }\)

D_ij is the shortest distance between city i and city j

Mean clustering coefficient

\(C = \frac{1}{n}\sum {C_{i} = } \frac{1}{n}\sum {\frac{{2E_{i} }}{{k_{i} (k_{i} - 1)}}}\)

C_i is the ratio of the actual number and the theoretical maximum number between all adjacent cities; E_i is the actual number of edges exists in the LAN formed with adjacent cities

Individual network indicators

Degree centrality

\(C_{D} = \frac{l}{n - 1}\)

The l is the number of other cities in the network directly related to a city;

Betweenness Centrality

\(C_{B} = \frac{{2\sum\limits_{j}^{n} {\sum\limits_{k}^{n} {b_{jk} (i)} } }}{{n^{{2}} - 3n + 2}}\)

b_jk is the probability that the city i is on the shortcut between jk

Closeness Centrality

\(C_{c} = \sum\limits_{j = 1}^{n} {d_{ij} }\)

D_ij is the shortcut distance between city i and city j

Network condensed subgroups

E-I index number

\(\begin{aligned} E - I = & \frac{{\rho_{EL} - \rho_{IL} }}{{\rho_{EL} + \rho_{IL} }} \\ \rho_{EL} = & {{eRC_{i} } \mathord{\left/ {\vphantom {{eRC_{i} } {\left( {(n - k)(n - k - 1)/2} \right)}}} \right. \kern-0pt} {\left( {(n - k)(n - k - 1)/2} \right)}} \\ \rho_{IL} = & {{iRC_{i} } \mathord{\left/ {\vphantom {{iRC_{i} } {\left( {k(k - 1)/2} \right)}}} \right. \kern-0pt} {\left( {k(k - 1)/2} \right)}} \\ \end{aligned}\)

eRC_i is total amount of regional external contact; iRC_i is the total connection amount within the region;ρ_EL is out-of-region network density; ρ_IL is the network density outside the region; k is the number of city nodes in the region