Complex networks and public funding: the case of the 2007-2013 Italian program

Properties of vertices evaluated in the present analysis include degree centrality, betweenness centrality, closeness centrality, eccentricity, eigenvector centrality, radiality centrality and PageRank centrality, based on the Google PageRank algorithm [23].

The highest values of all centralities is found in correspondence to public research institutions, like universities and specific research centres. In particular, the Italian National Research Centre ( CNR ) shows the best values for all the indicators. It is worth saying that it is a peculiar vertex of the network, being composed of 104 institutes spread over geographically distributed sites (in all the biggest cities in Italy), and covering a large spectrum of activities in many fields, from pure research, to applied disciplines. Probably, it would be better to split such vertex and consider each site, or department, separately, but the dataset does not contain such details. On the contrary, dividing the CNR in many entities would in a certain sense spoil its central nature in the Italian panorama. Resolving this controversy is interesting, but is over the purposes of the present paper, and is left for a future work, when more detailed Open Data will be available. Apart from cases like the one described, the central role of public research institution for the network structure is clear from all the centralities.

Degree centralities are discussed in detail in the next Section 4.2, since global properties of the network can be inferred from the distribution of such quantities, despite being them local in nature.

Betweenness [24] measures the importance of a node for traffic of information across the network. Large betweenness centrality of a vertex indicates that many shortest paths between couples of other vertices pass through that node. The relevance of this quantity for program evaluation stands in the possibility of assessing the role of institutions/enterprises for the eventual aggregation of ‘far’ nodes. For example, a policymaker interested in promoting a program aimed at aggregating and consolidating the productive system of a region should pay attention not to spoil the edge betweenness of the network of relations between the actors involved in the program.

Closeness centrality indicates whether a node is at a short average distance from every other reachable vertex, with higher closeness meaning shorter distance. A variant is radiality centrality, which gives higher weight to the neighbourhood of the node. From the social/economical point of view these quantities give indication about how easily an institution/enterprise can connect to all the other members of the network (and, so, of the productive system). For example, an enterprise with high closeness centrality could be the right promoter for initiatives like the creation of technological districts, associations or lobbies. Exploiting the information contained in this quantity, a policymaker could more easily head the productive system in the desired direction with focused regulatory interventions.

Eccentricity is the maximum value of the distances between a node and any other node in the network. It gives an idea of how central a vertex is within the network, with smallest values corresponding to more central nodes.

High eigenvector centrality is assigned to vertices that are connected to many other well-connected vertices. It can be used to identify the best way to spread a trend within the productive system represented by the network. A variant of eigenvector centrality is PageRank centrality, which is a way of measuring the importance of a node within a graph. The original algorithm was created by Larry Page and Sergey Brin in 1986 at Stanford University [25‐27] and is widely used by Google to measure the importance of website pages. The algorithm used here is given by the solutions of: where r is the vector of the PageRank centralities for each node, \(\mathcal{ A}\) is the adjacency matrix of the graph, \(\mathcal{ H}\) is the diagonal matrix consisting of \(1/\max\{1,d_{i}\}\), \(d_{i}\) being the degree of the ith vertex, and α is the damping factor, an empirical parameter⁵ usually set to \(\alpha=0.85\) [23]. This is an example of how fruitful the application of social network analysis can be to the evaluation of the effects of public funding, also by means of well known and successful algorithms, as the PageRank one.

In Figure 5 the largest values of all the centralities of the (giant component of the) PON R&C network are reported. As stated before, the CNR has the highest values for all the centralities, probably due to its being scattered along the whole country. Nevertheless, it is important to observe that public research institutions, mainly universities, occupy the top positions for every centrality, while in the lowest positions we find private enterprises, no matter whether they are large or small (except for the Polytechnic of Bari, which is a public university and has a low value of PageRank centrality). The central role of public research institutions for the network of relations underlying (at least part of) the Italian productive system is clear from Figure 5. Eccentricity is not reported in the figure, since a high number of vertices share the same value of this centrality, meaning that the network is somewhat ‘equally spaced’. For degree and betweenness centralities only the largest values are reported, since the smallest ones are trivial.

The last centrality considered here is edge betweenness, which is a property of the edges of the network (rather than vertices), and measures how central a link is for the connections between nodes. It is measured by counting the number of shortest paths the edge belongs to, and gives a quantitative idea of how much the relation between two institutions/enterprises is important for the ‘communication’ between all the actors composing the network. As the number of edges is much larger than the number of vertices, and since it is necessary to evaluate the shortest path between any couple of nodes, the calculation of such centrality is a resource intensive process. The largest values of the edge betweenness for the (giant component of the) PON R&C network are reported in Figure 6. Also in this case, the most important relationships (edges) between the nodes of the network are the ones between public research institutions, while small enterprises give small to no contribution to the geodesics.

The first property analysed here is the degree distribution of the vertices, i.e. the frequencies of the degree centralities described in the previous Section 4.1. The importance of such distribution stands in the possibility of inferring from it information about the topology of the graph, and in particular to understand if the network is scale-free [28]. The property of being scale-free is shared by many real networks, showing power law-shaped degree distributions \(P(k)=A k^{-\gamma}\), with exponents usually varying in the range \(2<\gamma<3\), which have the same form at all scales.

This is of particular interest since power laws are commonly associated with second-order phase transitions in dynamical systems. Phase transitions in complex networks represent an interesting research field [29, 30], but the graph considered here is static, so no considerations can be made in this respect. Anyway, this is an interesting perspective for a future work, in which dynamics can be taken into account.

The degree distribution of the PON R&C network is shown in Figure 7. The tail (starting from the upper bound of the median interval, \(m=7\)) is fitted very well to a power-law function of the form \(P(k)=A k^{-\gamma}\) with \(A=4.156\pm0.375\) and \(\gamma=1.998\pm0.040\). To obtain the fits, a nonlinear regression based on Newton method [31] has been used. A comparison with another fit, to an exponential distribution \(P(k)=A e^{-\gamma k}\) with \(A=0.241\pm0.011\) and \(\gamma=0.164\pm0.005\), shows that the former fits the distribution slightly better than the latter, with \(R_{\mathrm{pow}}^{2}=0.935\) and \(R_{\mathrm{exp}}^{2}=0.929\). Such a small difference between the values of \(R^{2}\) is not a strong indication of the fact that a power-law fits the distribution better than an exponential law, but together with the fact (shown below) that higher moments grow, it is sufficient to assess the power-law nature of the distribution. In fact, for a power-law distribution with tail of \(\mathcal{O} (x^{-\nu} )\) the moments of order \(n\geqslant(\nu-1)\) diverge, and in general higher order moments are larger in size with respect to the lower order ones (this is not true for exponential distributions). Standing the known difficulties in evaluating the nature of the degree distribution, due to noise coming from the finiteness of the sample (especially from boundary values), the present result is satisfactory in assessing the property of the PON R&C network of being scale-free. More refined methods could be used to evaluate the parameter γ with higher precision like e.g. the Kolmogorov-Smirnov test [32], but this is outside the purposes of the present work.

The distribution has an expected value \(\langle k\rangle=12.661\), mode \(M=6\), median \(6< m<7\), standard deviation \(\sigma=23.781\), skewness \(s=8.876\) and kurtosis \(\kappa=113.882\) (all the moments are shown in Table 2).

Another way of assessing if a network is scale-free consists in evaluating the distribution of local clustering coefficients, i.e. the number of edges connecting the neighbours of each vertex v, divided by the number of edges of a complete graph of the same cardinality of the neighbourhood of v [33]. The PON R&C network represents a special case, in which local clustering coefficients are less important, the majority of them being close to 1 by construction. In fact, since the graph is a union of complete graphs, it is likely that the neighbourhood of a vertex is fully connected, implying the closeness to one of the local clustering coefficient. The global clustering coefficient \(\mathcal{C}\), i.e. the fraction of paths of length two that are closed (over all paths of length two), is much more significant instead, and it takes a small value \(\mathcal{C}=0.215\) for the giant component, meaning that the network is not strongly clustered. From the political and sociological point of view, this is an interesting point, since the network is made by ‘scattered’ relationships, despite being composed of ‘closed’ TSAs.

Other important features that can guide the policymaker in evaluating the effects of the program or planning future ones are vertex connectivity \(V_{c}\) and edge connectivity \(E_{c}\), i.e. the smallest number of vertices or edges to be removed in order to disconnect the graph, respectively. For the case under examination such quantities take value \(V_{c}=1\) and \(E_{c}=1\), meaning that the removal of a single node or edge can be catastrophic for network connectivity. Identifying and monitoring such nodes/edges can be very important in case of low values of such parameters, in order to keep the network of relations tightly connected.

Another important property of scale-free networks is that they are small world networks [34]. This means that relatively short paths exist between any two nodes (with respect to the large size of the graph), with an average shortest path length⁶ \(L\sim \mathcal{ O}(\log N)\), N being the total number of vertices. This is due to the existence of links between vertices belonging to farther parts of the graph, having the role of connecting them and reducing distances to few hops. Usually, in scale-free networks such vertices are the hubs and the small-world property is enhanced when \(2<\gamma<3\) where \(L\sim\mathcal { O}(\log\log N)\) (while it is \(L\sim\mathcal{ O}(\log N)\) when \(\gamma >3\)) [35]. For the PON R&C network, \(\gamma=1.998\pm0.040\), \(L=2.532\), \(\log N=6.645\) and \(\log\log N=1.889\), so the small world property is enhanced, as expected when the vertex distribution follows a power-law with \(2<\gamma<3\). Again, this cannot be considered a smoking gun proving that the network is scale-free, but just another indication in addition to the ones mentioned above.

Other global features of the network are the radius \(\mathcal{ R}\) and diameter \(\mathcal{ D}\) of the graph, defined as the minimum and the maximum eccentricity of all vertices, respectively, the eccentricity being the longest shortest path from a source node to every other vertex in the graph. For the PON R&C network \(\mathcal{ D}=5\) and \(\mathcal{ R}=3\), meaning that no vertex is more than 5 hops far from any other node, and that the farthest destination is never closer than 3 hops from any source. From the point of view of program evaluation, this means that PON R&C has been successful in creating (or intersecting) a network of close relationships between the funded actors. Being interested in promoting such a relationship network while defining the program, these could be good ex post indicators of the goodness of the obtained results.

The centre of the graph⁷ is shown in Figure 4. It includes public research centres like CNR (which is also the main hub) and ENEA, all the major Universities involved in the program (Bari, Calabria, Catanzaro, Foggia, Naples, Palermo, Salento, Salerno), private research centres like CETMA, and also some large private enterprises like Avio S.p.A., Engeneering S.p.A., IBM, SELEX S.p.A., and EXEURA S.r.l. This is a strong indication that the network of funded projects gravitates around large poles involving research centres (public and private), which turn out to have a key role in aggregating entities. This can also be an explanation for the scale-free property of the graph, since preferential attachment is known to be a generating mechanism for this kind of networks [36‐38], in which nodes prefer to link to vertices with high degree. It is reasonable to imagine that many small actors prefer forming TSAs including large research organisations, which are usually able to get more funds, rather than form TSAs between themselves. From the point of view of social networks and relationships, it is particularly interesting to study such feature side-by-side with assortativity [39], which indicates whether nodes of the graph tend to connect with their connectivity peers (vertices with similar degree) or not. In the first case the network is said to be assortative, while in the second case it is anti-assortative. This feature is quantitatively measured through the assortativity coefficient r, whose range is \(-1\leqslant r\leqslant1\), \(r=1\) (−1) meaning a perfectly (anti-)assortative graph and \(r=0\) indicating no particular preference for the majority of the nodes. In the present network, \(r=-0.173\), meaning that the graph is slightly anti-assortative. This means that the productive system funded by such program has a little tendency not to form lobbies among important actors, but to associate strongly connected hubs to smaller and less connected enterprises/institutions. From the socio-economical point of view, it seems reasonable that small enterprises turn to larger ones or to big research centres to benefit from sharing and collaborations. This is an interesting result, since most social networks show assortative mixing by node degree [40], and it also has some implications on the topology of the network. First, anti-assortative networks are more susceptible to the removal of high-degree nodes (here represented by universities and research centres), which is an indication for the policymaker of the importance that public research has in the productive system, and of the possible disruptive effect of its underestimation. Second, in anti-assortative networks epidemics span to larger portions of the nodes than in similar assortative ones. This means that being anti-assortative is preferable for the spreading of knowledge and know-how in the productive system, making it more efficient. It is worth noting that in a recent work [12] a social network similar to the one studied here, concerning the funding of FP7 (Seventh Framework Programme) European research projects, has been found to be anti-assortative as well, and conclusions close to the ones put foward here are drawn. This could be an indication of some structural feature shared by graphs constructed starting from public funding programs, and we plan to further investigate this point in a future work. Lastly, link efficiency is a measure of traffic capacity within the network, representing how efficiently information can be transmitted along the graph. This parameter takes the very high value \(\xi=0.999\) in the PON R&C network, which is a strong indicator of robustness for the relations between vertices, especially in a graph with small density \(\rho=0.017\) as the one under examination.

The study of global properties of the PON R&C graph is given as an example showing how network analysis provides a concrete way of examining the role of funded actors within a program, supporting its ex post evaluation with the introduction of rather innovative indicators. In particular, it is able to describe the structure of the productive system, highlighting the key nodes for network connectivity, or vertices that have a central role, through quantitative (thus evaluable) indicators, which for the present case are summarised in Table 3.

The community structure of a graph is a global property, but a separate section is dedicated to it, since it has a special role, of particular importance for program evaluation. In the PON R&C network 15 communities are found with the Newman-Girvan algorithm [41], composed of 207, 136, 129, 113, 75, 29, 8, 7, 7, 7, 6, 6, 5, 5, and 4 vertices, respectively. The algorithm consists in recursively removing from the graph the edges with the highest edge betweenness and recalculating the edge betweenness for the new graph obtained at each step. This procedure generates a dendrogram of sets of communities, from which the set with largest modularity is chosen. The community structure of the PON R&C network is shown in Figure 8. The reason why it is so important is that it represents an unbiased way of discovering the existence of groups within a certain network of relationships, and highlighting such groups can be very important for the analysis of a productive system like the one described by the graph under examination. The PON R&C network shows strongly heterogeneous communities, with hugely populated groups and very small ones. An important point, that can be interesting for an ex post program evaluator, is that when communities grow in size, they tend to include important nodes. For example, the biggest community, made of 186 vertices, include the CNR in it, which shows the record values for all the centralities, as stated in Section 4.1.

Moreover, comparing the community structure coming from network analysis with the one expected on the basis of external (economical, political, and social) considerations can enrich the evaluation by introducing a different point of view on the system under examination, not driven by ‘human’ considerations, but purely mathematical in nature. The distribution in percentage of action areas within each community is shown in Figure 9. It can be seen that the communities found with network analysis are not directly linked to action areas, at least the largest ones. Other algorithms can be used to extract the community structure of a graph, like the leading eigenvector [42], the multi-level modularity [43] or the spin-glass [44], and refined methods can be used to which one gives the most significant results, like e.g. a consensus analysis [45]. This kind of study is outside the purposes of the present paper, since it must be policy-driven, rather than research-driven, in order to be of interest for program evaluation. We plan to develop similar analyses in future work.