So far, we have limited our study of the relativity collaboration network to the co-authorship networks, using different criteria for the choice of co-authorship edges, length of the edges and the permanence of nodes in the network. This section will add, one layer at a time, the three other kinds of collaboration edges of our multilayer network, which have been retrieved through the historical analysis of biographical data: collaboration, influence and copresence at institution edges (see “Collaboration edges: definition and criteria” section).
Collaboration + ExtendedCo-author network [all_nodes—8-year length edges]
The first layer we added is that formed by collaboration edges as described in “Collaboration edges: definition and criteria” section. Since we are interested in historically relevant transformations that might be visualized in sudden changes in network parameters, in the following analyses we will consider that all edges last only 8 years and discuss only what the different edge types add to our previous studies in a cumulative fashion. As far as the nodes are concerned, we will not filter out those inactive in the specific year of the year-graphs, for the reasons explained in “Co-authorship extended [all_nodes – 8-year length edges]” section. We call the flattened multilayer network including the extended co-authorship and the collaboration relations with the all-nodes and 8-year rules the Collaboration_AllNodes_8Years.
Introducing a new kind of edge manually retrieved from biographical data has the obvious effect of significantly increasing the number of edges with respect to the Co-authorExtended_AllNodes_8Years network analyzed in “Co-authorship extended [all_nodes – 8-year length edges]” section. Most interestingly, however, adding a new layer provides a different periodization of a sudden shift in the network topology toward the formation of a giant component. The number of nodes and edges of the largest connected component of the new collaboration graph shows a radical transformation occurring a few years before it becomes evident in the co-authorship network (Fig. 12a). The relevance of this shift is clearly visible if one looks at the first derivative of the number of edges of the largest component over time, which shows the first high peak in 1960 (Fig. 12b). Various indicators of the dynamics of the Collaboration_AllNodes_8Years network show, then, that a change occurred between 1959 and 1960, suggesting a different dynamic which does not appear in the co-authorship network. This seems to be confirmed also by the comparison of various parameters of the Collaboration_AllNodes_8Years network to the random-graph and scale-free models (See Fig. 23 in “Appendix 1”).
As was the case in the previous section, the 8-year rule does not lead to the stabilization of the largest component, while the assumption that the edges stay permanently in the network does (Fig. 13). The 8-year rule allows for sudden social changes to become visible, while the assumption of unlimited length for the edge might provide some insight on the intellectual stability of the field.Footnote 26
The networks in Fig. 14 show that the transformation can be interpreted as driven by specific movements of scientists. In 1959 the collaboration network has three large components, similar to the ones identified as separate components in the 1961 co-authorship network in Fig. 8: the UK-based group mostly centered around Bondi at King’s College London; the US-based group mostly connected to John Wheeler at Princeton University; and the US-based group mostly connected with Peter Bergmann at Syracuse University, which in this network is already connected with Jordan’s group by 1959. Contrary to the co-authorship network in Fig. 8, the shift between 1959 and 1960 corresponds to the merging of all these three components.
The degree centrality measures in the Collaboration_AllNodes_8Years network in 1960 reveal at least two junior scholars together with research center leaders Bergmann, Bondi and Wheeler as the most central actors in shaping the network structure. These junior scholars are German physicist Rainer Kurt Sachs and British physicist Felix Pirani. Notice that these scientists are different from those identified as central in the previous co-authorship analyses. The closeness centrality measures give an even greater relative central position to those scholars connecting various groups such as Pirani, Robinson, Sachs and the senior relativity expert John L. Synge, who at the time was a leader of a smaller group at the Dublin Institute for Advanced Studies (see Table 4 in “Appendix 2”). Bondi and Pirani reach a high betweenness centrality in the early 1950s, displacing Einstein as the major figure already by the mid-1950s, together with Bergmann, who, however, is not attached to the largest component until 1960 (Fig. 15).
Influence + Collaboration + ExtendedCo-author network [all_nodes—8-year length edges]
We then added the third layer of “Construction of the multilayer social network” section: influence, which includes the two relations PhD with and influenced by described in “Collaboration edges: definition and criteria” section. We use the same criteria for the permanence of nodes and the length of edges, and call this network the Influence_AllNodes_8Years.
In spite of the fact that adding a new layer necessarily implies more connections than in the previous analysis in “Collaboration+ExtendedCo-author network [all_nodes – 8-year length edges]” section, the general picture concerning the relevant changes in the topology of the network and the formation of a giant component remains essentially unaltered. The number of edges and nodes in the largest component is greater than in the Collaboration_AllNodes_8Years from the early 1950s and these values more than double already by 1957. The radical change in the structure, however, occurs again the same year, between 1959 and 1960 (Fig. 16).
While the general picture is not modified in any substantial way by including relations of influence, this analysis shows more clearly the effect of the war on the topology of the network: the number of nodes and edges in the last years of the 1930s is significantly greater than in the previous analysis. It is possible to identify the increase of the largest component as a social group based on influence relations, more than collaborations, but this group is disrupted by World War II and never materializes into a more connected network. After the disruption one has to wait until 1955 to see the same number of edges in the largest component as before World War II. After 20 years, however, those scholars belonging to the largest connected component have changed significantly.
The influence edges alter the interpretation of the connection between the different groups in the collaboration network before and after the major shift occurred. According to this new picture, in 1959 there are only two, rather than three, major components. The largest one was centered on scientists based at Cambridge University and at King’s College London, but also includes Leopold Infeld’s group in Poland and Synge’s in Dublin. The second group is mostly based out of American universities and grows from an early connection between Wheeler’s group and Bergman’s group, which in our previous analysis did not appear at all (Fig. 17). This edge was created by physicist Arthur Komar’s going from Princeton, where he was a PhD student of John Wheeler’s, to Syracuse University as a junior associate of Peter Bergmann in 1957. In this picture the role of junior researchers around 1959 is especially evident as Komar and Pirani have a particularly high betweenness centrality comparable, or even higher than, those of research group leaders at the time (3th and 4th highest values, see Table 5 in “Appendix 2”).
The sudden change between 1959 and 1960 is, in this case, due to the merging of these two previously disconnected largest components, which makes Sachs and Robinson emerge as the brokers of this dynamic, as suggested in the previous analysis and as easily seen by their closeness and betweenness centrality measures (Table 6 in “Appendix 2”). The analysis also indicates the presence of other large groups, such as the French groups in Paris around mathematician André Lichnerowicz and theoretical physicist Marie-Antoinette Tonnelat, who, however, remain significantly isolated, separated from both each other and the largest component. Both Lichnerowicz and Tonnelat have, in 1960, a high degree centrality (2nd and 4th respectively) but their complete disjunction from the largest component clearly conveys their marginalization within the relativity community that was being established. Only in 1965 do both Lichnerowicz and Tonnelat enter the largest connected component. However, while they maintain a relatively high degree centrality, and assume a high betweenness centrality in 1965, their distance from the center of the network is made evident by the low closeness centrality, which indicates, that while having many students, they remain marginal to the community that was being established around other major centers and scientists (see Table 7 in “Appendix 2”).
Copresence at institution + Influence + Collaboration + ExtendedCo-author network [only_nodes—8-year length edges]
We completed our analysis by including the fourth layer of our multilayer network introduced in “Construction of the multilayer social network” section copresence at institution. This kind of edge is, as discussed in more detail in “Collaboration edges: definition and criteria” section, purely hypothetical, but has the potential to reveal some hidden connections that were not accessible from other collaboration edges. It constitutes the space of possible collaborations and, as such, is of great utility for historical studies. While we are here taking into consideration hypothetical collaboration edges in addition to those previously analyzed, there is substantial evidence that, in a field still as small as relativity research, practitioners tended to meet and discuss with persons working in the same or geographically closer institutions. In this section, we report the result of the analysis using the 8-year rule for the length of edges. As far as the nodes are concerned, we report the results obtained using the only_nodes rule, excluding thereby those who were not active in general relativity research in that year. This was done to limit the bias created by the large increase of uncertain connections between scientists. This choice implies that the only_nodes rule is consistently applied to all the four layers forming the merged network. We call this network AllRelations_OnlyNodes_8Years.
The number of edges increased enormously, as one might expect in view of the fact that we are including a new layer of connections to the pre-existing three layers and in spite of the fact that the number of nodes is slightly less than in previous analyses (as we are using only_nodes rather than the all_nodes rule). The comparison between the total number of nodes and edges of the AllRelations_OnlyNodes_8Years network with respect to the previous three layers of collaboration edges (Influence_AllNodes_8Years) shows both the effect of the war in disrupting existing institutional connections and the relevant impact of post-WWII changes in the physics landscape and career practices. The general image is of a two-step change occurring in the 1950s. The first step occurs in 1951, when the total number of edges becomes greater than the total number of nodes, and the second one occurs in 1958 when the pace of increase of edges quickens (Figs. 18a, b).
A comparison between the largest connected component in the influence and all-layers networks displays similar post-1950s growth, with a first major shift between 1950 and 1951, a second one between 1956 and 1958 and a third one between 1961 and 1962. The number of nodes in the largest component also shows a similar pattern after 1950. Even Einstein’s disappearance from the network in 1955 does not provoke a major disruption in the largest component of AllRelations_OnlyNodes_8Years. This is interpreted as a measure of how robust the socio-institutional network had become by the mid-1950s (Fig. 19a). The number of nodes and edges of the largest connected component over the total number of nodes and edges conveys even more clearly the picture that a giant component started forming between 1950 and 1951 and continued growing, as far as the number of nodes are concerned, while remaining stable as far as the relative number of edges of the largest connected component over the total number of edges is concerned (Fig. 19b).
In the case of the AllRelations_OnlyNodes_8Years network, we can also see more indications of small-world network behavior. Between 1957 and 1962 the average path length of the largest connected component stops growing and after 1962 starts decreasing in spite of the growth of the number of nodes (Fig. 20a). Secondly, in spite of the 8-year rule this network does show a stabilization of the field at the end of the 1950s (more specifically from 1958), when the diameter stops growing and starts decreasing in spite of the large growth of nodes in the largest component, contrary to what occurred in all the other networks that used the 8-year rule (see Fig. 20b). This picture is confirmed by comparing the AllRelations_OnlyNodes_8Years network with the values of the Erdős-Rényi and BA models (see Fig. 24 in “Appendix 1”).
The analysis of centrality measures in the period of major topological changes in the structure of this network identifies a younger scholar who seems to have played a particularly relevant role in connecting different parts of the network in the moment when the giant component started forming in the early 1950s. British physicist Felix Pirani emerges exactly in that period as a possible broker between different national communities, American and British, and more importantly between two different research topics, quantum gravity avant la lettre and the steady-state cosmology developed at Cambridge University. After having received his PhD with a dissertation on the quantization of Einstein’s field equations in 1951, he moved to Cambridge to do a second PhD with Hermann Bondi, one of the proponents of the steady-state cosmological model, an alternative to the evolving universe model. In 1951, Pirani became an early broker in the network, as shown by his high betweenness centrality, and he also maintained this particularly central position in the next few years, until 1960, anticipating other major actors who would gradually become more central in the 1960s. Centrality measures in this 4-layer network also give more relevance to Dennis Sciama at Cambridge, who appears as a central actor much earlier than in our analysis of previous networks (Fig. 21 and Table 8 in “Appendix 2”).
Discussion on the dynamics of flattened multilayer collaboration networks
The results presented in the co-authorship networks analyzed in “The dynamics of the co-authorship network” section were strongly disputable, as all the co-authorship networks remained very sparse all through the period of analysis, with an average degree lower than one until 1970. The inclusion of more layers with different kinds of collaborations significantly increased the density of the networks and allowed for a more robust analysis. The first two layers, collaboration and influence, provide similar historical pictures of the shift in the topology of the network (Fig. 16). Strikingly, the Influence_AllNodes_8Years network precisely matches the year in which the shift occurred in Collaboration_AllNodes_8Years (1959–1960). This result was contrary to our expectations, as we anticipated that by including many more edges the formation of the giant component would have started earlier. We interpret this result as a sign of the robustness of the finding that historically relevant changes occurred precisely between 1959 and 1960
We interpreted Fig. 16 in the following way. While the number of scholars working on general relativity started increasing soon after the end of World War II, this increment did not lead immediately, nor straightforwardly, to a change in the structure of the network. A topological shift occurred only about 15 years after the end of World War II, when a giant component started forming at a very rapid pace. The addition of other types of collaboration data backdates this topological shift by a few years with respect to the co-authorship networks. The delay with respect to the end of World War II undermines the view that the renaissance of general relativity was simply a by-product of the increase of the number of physicists. By taking into consideration the collaboration and influence layers it emerges more clearly that post-doc movements between research groups were essential to the formation of the giant component and that this preceded the astrophysical discoveries, thus undermining the historical narrative that sees in these discoveries the major moving force beyond the renaissance of general relativity.
Apart from a more detailed and robust analysis of the structural dynamics of the relativity collaboration network, the new layers offer also a richer understanding of the process of the renaissance of general relativity, confirming the relevance of particular groups and specifying which kinds of movement increase the connectivity of the network. Collaboration and influence edges not only anticipate relations that later appear as co-authorship relations, but effectively show connections that were not available in our previous analysis, such as those between Wheeler and his students or junior associates with the largest component built around Bondi and other European groups in the collaboration layer (Fig. 14). This implies that Wheeler might have been more influential in the forming relativity community before it becomes evident in the co-authorship network, as in the extended co-authorship networks he becomes part of the largest connected component only in 1964, namely, after the discovery of quasars.
Various parameters of the influence layer, such as the diameter of the largest component, the clustering coefficient and the average path length, also do not provide any new results concerning the topology of the network with respect to the collaboration layer. The same can be said for the comparison of the parameters of the Influence_AllNodes_8Years network with the Erdős-Rényi and BA models, which does not show any relevant new insights with respect to the previous analysis. The influence network, however, reveals some central persons, and entire groups, such as the French groups of students of Lichnerowicz and Tonnelat, which were completely absent in our previous analyses, most probably because of different national or disciplinary traditions of co-authorship between PhD supervisor and students, as well as a different research environment as concerns the career path of relativity experts.
The AllRelations_OnlyNodes_8Years provides, instead, a somewhat different picture, as should be expected by the fact that the number of edges more than doubles with respect to the Influence_AllNodes_8Years. The emergence of a giant component appears much earlier, in the early 1950s, only a few years after the end of World War II. Further, the diameter and other parameters stabilize by the late 1950s, in any case before 1962.Footnote 27 The effect of World War II appears quite distinctly in this network both as a disruption of the socio-institutional network emerging in the 1930s and in the fact that a giant component started forming early after the war contrary to all our previous analyses. All these elements seem to indicate the establishment of the giant component started after the war and stabilized by the late 1950s—a foundation that might be called the socio-institutional preconditions of the renaissance of general relativity, where institutional relations clearly preceded, and most probably favored, actual collaborations. The introduction of this layer in the analysis also gives a different indication of who might have been particularly central individuals, leading to the evaluation of Felix Pirani as a particularly central scientist in this early phase of the construction of a giant component.