Research productivity and the quality of interregional knowledge networks

Abstract

This paper estimates the impact of interregional knowledge flows on the productivity of research at the regional level. We develop the novel index of ‘ego network quality’ in order to measure the contribution of knowledge accessed from the interregional network to the production of new knowledge inside the region. Quality of interregional knowledge networks is related to the level of knowledge accumulated by the partners (‘knowledge potential’), the extent of collaboration among partners (‘local connectivity’) and the position of partners in the entire knowledge network (‘global embeddedness’). Ego network quality impact on the productivity of research in scientific publications and patenting at the regional level is tested with co-patenting and EU Framework Program collaboration data for 189 European NUTS 2 regions.

Appendix

1.1 An alternative formula for ENQ

The formula of ENQ can be converted into a comprehensive form if we define the sequence of node-generated subgraphs $S_d^i$, where $S_d^i$ stands for a subgarph containing nodes, which are at most distance $d$ from node $i$, and the edges between all such nodes. Let us use the following notation:

$$\begin{aligned} C_d^i =\frac{\sum _{j:r_{ij} \le \,d} {\sum _{l:r_{il} \le \,d} {a_{jl}}}}{2} \end{aligned}$$

(11)

In other words, $C_d^i$ gives the weighted number of edges counted in the subgraph containing nodes at distance $d$ or less from node $i$, that is, the weighted number of edges in $S_d^i$. It is easy to see that we can now substitute the previously given measure for $\mathrm{LC}_d^i$ by $\left( {C_d^i -C_{d-1}^i } \right)$. Using $C_0^i =0$, by definition (the node at distance 0 from node $i$ is itself; therefore, we have a trivial graph where the number of edges is zero by definition.), the expression in Eq. (6) can be reformulated in the following form:

$$\begin{aligned} \mathrm{ENQ}^{i}=\mathrm{KP}_1^i \frac{C_1^i }{N_1^i }+W_2 \mathrm{KP}_2^i \frac{C_z^i -C_1^i }{N_z^i }+\cdots +W_{M-1} \mathrm{KP}_{M-1}^i \frac{C_{M-1}^i -C_{M-z}^i }{N_{M-1}^i }\qquad \quad \end{aligned}$$

(12)

Again, using $C_0^i =0$ and $W_1 =1$, this formula can be written more comprehensively:

$$\begin{aligned} \mathrm{ENQ}^{i}=\sum \limits _{d=1}^{M-1} {W_d \mathrm{KP}_d^i \frac{C_d^i -C_{d-1}^i }{N_d^i }} \end{aligned}$$

(13)

It can be demonstrated that if we are not using distance weights (i.e., the distance to all partners is taken to be 1) and knowledge levels are identical across partners normalized to 1, ENQ counts the weighted number of links in the whole network. On the other hand, if knowledge weights are not considered but distance weights are in effect, ENQ shows common features with the distance-weighted sum of degrees in the network

1.2 ENQ without distance and knowledge weights

Consider the situation in which distance weights are not considered, that is, $W_d =1$ for all $d$, and knowledge levels are identical across nodes, that is, we can use $k_i =1$ for all $i$, therefore $\mathrm{KP}_d^i =\sum _{j:r_{ij} =d} {k_j =N_d^i }$. This way the formula in (A3) collapses to

$$\begin{aligned} \mathrm{ENQ}^{i}=\sum \limits _{d=1}^{S-1} {C_d^i -C_{d-1}^i =C_{M-1}^i} \end{aligned}$$

(14)

As $S_{M-1}^i$ is the largest subgraph possible, containing all nodes, in this very special case $\mathrm{ENQ}_i$ is simply the weighted number of links in the network, irrespective of the node in question.

1.3 ENQ without knowledge weights

If knowledge weights are not used but distance weights are in effect, we have the following formula for ENQ:

$$\begin{aligned} \mathrm{ENQ}^{i}=\sum \limits _{d=1}^{S-1} {W_d } \left( {\sum \limits _{j:r_{ij} =d-1} {\sum \limits _{k:r_{ik} =d} {a_{jk} } } +\frac{\sum \nolimits _{j:r_{ij} =d} {\sum _{k:r_{ik} =d} {a_{jk} } } }{2}} \right) \end{aligned}$$

(15)

The first term in the parenthesis counts the number of links connecting distance with distance $d-1$, whereas the second term gives the number of links between nodes at distance $d$. We should add the expression $\sum _{j,g_{ij} =d} {\sum _{k,g_{ik} =d+1} {a_{jk}}}$ to those in the parenthesis, and multiply the second term by 2. This expression simply counts the weighted number of links between nodes at distance $d$ and $d+1$, whereas the multiplication double-counts the links among nodes at distance $d$. It is easy to see that after this modification, the number in the parenthesis gives the number of links connecting nodes at distance $d$ with nodes at distances $d-1$ and $d+1$ and twice the links among nodes at distance $d$. In other words, this number is the sum of (weighted) degrees of nodes at distance $d$. This expansion, however, results in no substantial change in the ENQ measure. The original measure counts every link once, whereas the modified version counts every link twice. If the adjacency matrix is symmetric, which we assume, this modification, then it means a simple multiplication by 2 on the level of the overall index. Using all this, we can write

$$\begin{aligned} \mathrm{DIST}^{i}=\sum \limits _{d=1}^{S-1} {W_d } \left( {\sum \limits _{j:r_{ij} =d} {\sum \limits _k {a_{jk} } } } \right)=\sum \limits _{d=1}^{S-1} {W_d } \sum \limits _{j:r_{ij} =d} {\mathrm{DEG}_j} \end{aligned}$$

(16)

This last measure is nothing but the distance-weighted sum of degrees in the network, which, as shown above, is twice the ENQ measure:

$$\begin{aligned} \mathrm{DIST}^{i}=2\mathrm{ENQ}^{i} \end{aligned}$$

(17)

On the other hand, this last expression bears a close resemblance to the eigenvector centrality (Bonacich 1972, 2007), which also reflects a distance-weighted sum of degrees in a network, although it uses a recursive definition with exponential weights leading to an eigenvector problem. This means that our ENQ index, when knowledge levels are homogenous, reflects similar properties to eigenvector centrality, which is a comprehensive measure of network position taking into account the whole structure around a given node from its immediate neighborhood to farther parts of the network.