Network of European Union–funded collaborative research and development projects

We describe collaboration networks consisting of research projects funded by the European Union (EU) and the organizations involved in those projects. The networks are substantial in terms of size, complexity, and potential impact on research policies and national economies in the EU. In empirical determinations of the network properties, we observe characteristics similar to those of other collaboration networks, including scale-free degree distributions, small diameter, and high clustering. We present some plausible models for the formation and structure of networks with the observed properties.

Figure 1

Distribution of project sizes. The size of a project is defined as the number of organizations taking part in the project. The tails of the distributions are power laws; for FP4, we show a power-law fit to the data with exponent $-4.1$.

Figure 2

Distribution of organization sizes. The size of an organization is defined here as the number of projects in which it takes part. The tails of the distributions are power laws; for FP4, we show a power-law fit to the data with exponent $-1.8$.

Figure 3

Degree distribution of projects projection. The distributiond show a structure formed from the superposition of two power laws; for FP4, we show a power-law fit to the high-degree data with exponent $-2.2$.

Figure 4

Degree distribution of organizations projection. The tails of the distributions are power laws; for FP4, we show a power-law fit to the data with exponent $-2.0$.

Figure 5

Relation between degree and number of triangles in the projects projection. For each degree value, we show the mean number of triangles, conditioned on the vertices with the given degree. For low degree values, a strong linear relationship is observed, but the strength of the relationship weakens with high degrees. Here, we show only the data for FP4 for comprehensibility; similar results hold for the other Framework Programs.

Figure 6

Relation between degree and number of triangles in the organizations projection. For each degree value, we show the mean number of triangles, conditioned on the vertices with the given degree. For low degree values, a strong linear relationship is observed, weakening slightly with higher degrees (compare with Fig. 5). Here, we show only the data for FP4 for comprehensibility; similar results hold for the other Framework Programs.

Figure 7

Distribution of edge multiplicities in the projects projection. An almost perfect power-law distribution is observed for all Framework Programs. The multiplicity is strongly indicative of correlations in the edge formation rules, possibly caused by memory effects due to prior collaborative experiences amongst the participating organizations or by the fact that organizations are active in a wider set of complementary activities.

Figure 8

Distribution of edge multiplicities in the projects projection. An almost perfect power-law distribution is observed for all Framework Programs. As with the P graphs in Fig. 7, the high multiplicities are indicative of correlations in the edge formation rules.

Figure 9

Simulated degree distribution for the O graphs. The empirical FP4 data are the same as in Fig. 3. The PO model takes as input the empirical organization sizes and project sizes, and randomly pairs an organization to a project using the Molloy-Reed algorithm described in Sec. 4B. During that pairing, both the multiplicative and the additive degree-degree correlations produce networks that are very similar to the empirical O graph with respect to the degree.

Figure 10

Simulated degree distribution for the P graphs. The empirical FP4 data are the same as in Fig. 4. With respect to the degree, the Molloy-Reed algorithm with multiplicative degree-degree correlation produces a network that more closely matches the empirical P graph than the modified Molloy-Reed algorithm with additive degree coupling.

Figure 11

Simulated triangle degree dependence for the O graphs. The empirical FP4 data are the same as in Fig. 6. With respect to the mean triangle number (conditioned on the degree), the Molloy-Reed algorithm with multiplicative degree-degree correlation produces a network more similiar to the empirical O graph than the modified Molloy-Reed algorithm with additive degree coupling.

Figure 12

Simulated edge multiplicity for the O graphs. The empirical FP4 data are the same as in Fig. 8. The two Molloy-Reed algorithms are unable to generate networks reproducing the empirically observed edge multiplicity, in terms of either the exponent or the absolute numbers; the empirical case has nonrandom features that a more advanced model needs to imitate. Also shown is a P model network, in which only the empirical project sizes are taken as input for random sets of organizations; the organization sizes automatically form a Poisson-like distribution. The P model has even smaller edge multiplicities. All models show scale-free edge multiplicities.