WoS-based co-authorship [all_nodes – edges of unlimited length]
As a first step we here show the analysis of the co-authorship network between 1925 and 1970 created by considering only the WoS-generated set of papers in Set F, which we call the general relativity publications space (see Table 1). We report the results of this analysis considering that all nodes and edges remain permanently after they first entered the network, as is usually assumed in most studies of dynamical co-authorship networks (the unlimited edge length and all_nodes criteria in “Construction of the multilayer social network” section). The network thus created is labeled WoS_AllNodes_UnlimitedLength.Footnote 20
The edges of this quite dispersed network remain fewer than nodes over the entire period. The dimension of the network steadily increases without any major shift, aside from rapid growth in the total number of nodes and edges after World War II, and more precisely after 1950 (Fig. 2a). In spite of the growth in the number of connections, the largest connected component remains always significantly smaller than the entire network. A possible major development in the formation of the largest connected component seems to occur between 1964 and 1965, when both the numbers of nodes and edges of the largest components increase rapidly and considerably. This pattern appears more evident in the diagram of the percentage of the nodes and edges of the largest component over the respectively total number of nodes and edges: in 1965 there is a clear reversal in the trend of relative number of edges of the largest connected component, which constantly decreases from the late 1940s. This change, however, does not seem to have a strong impact on connectivity, as the dimension of the largest component remains significantly lower than the dimension of the network, less than 20% at its apex in 1970 (Fig. 2b).
The picture that emerges from the analysis of WoS co-authorship network is that scholars working on general relativity formed a very small network, which remained weakly connected. In spite of its smallness, the diameter of the largest component grows monotonically in this period, which is a further indicator of the dispersion of the network and its instability (Fig. 2c).
Co-authorship extended [all_nodes—edges of unlimited length]
The second step involved analyzing the co-authorship network generated by considering the extended-simplified co-authorship edges—described in “Construction of the multilayer social network” section as the merged set of three different co-authorship record sources, and not just of the WoS-derived co-authorship edges—and using the same criteria adopted in the previous section: edges are considered to be of unlimited length, and the nodes are not restricted to those still active at the specific year of the analysis. We call this network Co-authorExtended_AllNodes_UnlimitedLength.
This network contains a greater number of nodes and edges, but still represents a small and disconnected network where the number of nodes remains always greater than the number of edges. However, the network also clearly displays change in structure and trend, contrary to what could be seen in the analysis of the WoS-based co-authorship network in “WoS-based co-authorship [all_nodes – edges of unlimited length]” section. There are some moments of rapid change in the slopes of the curves of nodes and edges of the largest connected component that did not appear in the previous analysis (Fig. 3a). What is relevant from a historical perspective is the identification of these moments of change and the lag between the change in the slope of the total number of nodes over time and that of the edges of the largest component over time. To analyze and visualize these shifts we calculated the discrete first derivative of the total number of nodes and edges of the largest component with respect to year.Footnote 21 The increased rate of growth in the number of nodes starts right at the end of World War II and continues at an ever-increasing rate until the early 1960s, after which the rate seems to slow down. This does not correspond to any similar modification in the largest connected component, as identified by the first derivative of the curve of the edges of the largest connected component. That transformation starts later, after 1958, and continues growing up to the end of the period under consideration, with a first major peak in 1964 (Fig. 3c, d). Following the hypothesis put forward by Bettencourt et al. (2008, 2009) that the formation of a scientific field is related to the stabilization of the diameter of the largest component, while the number of nodes in the components continues growing, one sees that, a few years after the drastic change between 1961 and 1962, the largest component stabilizes and the diameter stops growing in 1963 (Fig. 3b).Footnote 22
The stabilization does not seem to represent, however, the formation of a small-world network. The size of the largest connected component reaches 329 units at its peak in 1970, corresponding only to about 40% of the dimension of the network, significantly smaller than giant components identified in the scientific literature, both in relative and absolute terms (Newman 2004). The evolution of the average path length and of the clustering coefficient shows no robust sign of small-world network behavior either. If one, however, compares these values with random graph models such as the Erdős-Rényi model or the Barabási–Albert model, some kind of small-world behavior of this network seems to emerge (see “Appendix 1”, Fig. 22).
To better understand the change occurring between 1961 and 1962 with a sudden increase in the size of the largest connected component, we have closely analyzed the networks in these 2 years. This scrutiny shows that the transformation is due to the creation in 1962 of new edges between the largest connected component of the 1961 network and a smaller component mostly composed of West German scientists. This connection was established by the career move toward the United States of one scholar. In 1961, West German astronomer and physicist Engelbert Schucking started working as a research associate at Syracuse University with Peter Bergmann, who had established a center focusing on general relativity research back in 1947 (Fig. 4). The two scientists published a paper together with mathematician Ivor Robinson in 1962 (Bergmann et al. 1962). Schucking might then have played the role of broker between previously separated research groups (Granovetter 1973).
The last indicator we used to understand relevant historical changes in the network structure is the pattern of the centrality measures of particularly relevant scientists. As is very easily seen in the network images, Einstein is still present in our analysis of the early 1960s network in spite of the fact that he had already passed away in 1955. This is a consequence of our choice of having considered a node in the network as lasting forever. While this is questionable for obvious reasons, this choice gives the possibility to see how scientists continued to be intellectually relevant in shaping the structure of the developing social network of authors. In the Co-authorExtended_AllNodes_UnlimitedLength, in spite of the fact that Einstein did not publish any co-authored paper since the late 1940s, he still maintained the highest betweenness centrality, as well as the highest degree and closeness centrality, well after his death.
Figure 5 shows the ranked diagram of Einstein’s closeness centrality for the co-authorship network here analyzed (see “Method of analysis” section). The diagram shows that Einstein reached a high level of centrality only after his move to Princeton in 1933 when he started publishing co-authored papers with his assistants at the Institute for Advanced Study. Einstein was clearly in a dominant position in the largest connected component that started slowly growing after mid-1930s, and he maintained this central position thanks to some of his close associates in the 1930s, such as Peter Bergmann and Leopold Infeld, who played a major role in general relativity research in the 1950s and 1960s and both helped increase the largest connected component with their students. The pattern of Einstein’s closeness centrality again suggests that the period between 1961 and 1964 was particularly important for the structure of the network.Footnote 23 We have already shown that the period around 1961–1962 appears as the moment in which the largest connected component started growing at a more rapid pace. Exactly at the same time, in 1962, Einstein lost his place as the most central actor in the network. This might mean that the network started taking shape around actors who gradually became more and more emancipated from Einstein’s intellectual influence, as shown by the comparisons of the ranked closeness centrality measures of some of the most central actors in that period. In 1962, former Einstein’s assistant, Bergmann took the highest value, especially thanks to the dominant position of the Syracuse University research center on general relativity in attracting postdocs, and a strong collaborative connection with other centers of growing relevance, such as King’s College London around Hermann Bondi, who also became a central actor in the 1960s (Fig. 5).
The increasing intellectual distance from Einstein is visible in changes in the centrality measures of John A. Wheeler. He started a research group on general relativity before the mid-1950s at Princeton University, and Wheeler’s center has come to be considered a posteriori a major center in the renaissance phenomenon (Thorne 1994; Misner 2010; Rickles 2018; Blum and Brill 2019). In fact, all through the 1950s, Wheeler and his co-authors remained disconnected from the largest connected component up to the mid-1960s, forming an increasingly larger separated connected component. As a consequence, Wheeler had very low-ranked closeness centrality until the late 1960s (Fig. 5), when he rapidly became the most central actor. Figure 6 shows the network in 1969 when Wheeler and his co-authors have been included in the giant component and Wheeler immediately reached a central position. Not only did he have the second highest betweenness and degree centrality measures in 1969 (after Bergmann), but he is also the most central figure in the largest, purple cluster, as identified by the Girvan–Newman algorithm (see “Appendix 2”, Table 2).
Table 2 Centrality measures of the seven scholars who have the five highest centrality measures of one of the three following centrality measures: betweenness centrality (%), closeness centrality (%) and degree in the Co-authorExtended_AllNodes_UnlimitedLength network in 1969 Co-authorship extended [only_nodes—edges of unlimited length]
Assuming that edges have unlimited length results in the unlimited persistence of the nodes, which is highly unrealistic in long-lasting networks. In order to focus on active scientists, we have analyzed the co-authorship network using the only_nodes criteria (see “Construction of the multilayer social network” section), which gives a more realistic picture of the scientists working on general relativity in a specific period. While the previous network might be interpreted as providing measures of who contributed to the intellectual base of the field in a given period, the present network shows the actors in a given time and their connections with other active. We call this network the Co-authorExtended_OnlyNodes_UnlimitedLength.
The analysis of the network of active nodes connected by edges of unlimited temporal length leads to very similar results as in the previous analysis, while revealing moments of historical change more clearly. Figure 7a, b show clearly the radical shift occurring between 1961 and 1962 in the process of the creation of a social network working on general relativity research agendas. The increasing rate of growth of both the nodes and edges in the largest connected component between 1961 and 1962 and the tendency of the component to become more and more connected (number of edges over number of nodes) appear with particular evidence. While it remains a disconnected network with only 40 percent of the authors belonging to the largest connected component at its apex, the dynamic of this network shows a pattern similar to that found without excluding inactive scientists in the year-graphs in “Co-authorship extended [all_nodes – edges of unlimited length]” section. The only_node choice only makes more evident particularly disruptive changes, such as the death of Einstein in 1955, which clearly disrupted the largest component (see the drop in the number of nodes and edges between 1955 and 1956 in Fig. 7a)
A close study of the networks shows that the rapid increase occurs between 1961 and 1962 because different large connected components of the network, including the two largest ones, merge. In 1961, there is no clear giant component. The three largest connected components are grouped around central figures, whom we call research leaders. The three largest components center on three figures with the highest degree centrality: Hermann Bondi, leader of the relativity group at King’s College London since 1955; Bergmann at Syracuse University; and John Wheeler at Princeton University. The change between 1961 and 1962 is due to the merging of two of these largest components (Bondi’s and Bergmann’s) as well as other smaller components. Besides Schucking, already identified in the previous analysis, a central position is held here by the mathematician Ivor Robinson as he connects the active groups formed around Peter Bergmann at the Syracuse University with the UK-based group where Bondi had a central position (see Fig. 8).Footnote 24
Co-authorship extended [all_nodes—8-year length edges]
In previous sections we hypothesized that the edges, once created, would remain forever, a highly problematic assumption that may not capture the rapidity with which social groups of co-authors form in historical dynamics. As a final step of the co-authorship analysis, we have analyzed the effect of posing a limit to the temporal length of edges on the simplified-extended co-authorship network, hypothesizing that each edge lasted 8 years (see “Construction of the multilayer social network” section). In this case we report only the result of the all_nodes, because the 8-year rule gradually eliminates all those nodes that were no longer active in the field, with no further need to put more constraints for the presence of the nodes in the year-graphs. We call this network the Co-authorExtended_AllNodes_8Years.
The picture is very similar to the one offered in the previous analysis apart from the fact that the edges are fewer and the network much less stable as edges between any two authors disappear 8 years after not having co-authored any paper in the meantime. This assumption makes even more evident the radical change occurring between 1961 and 1962, shown with great clarity by the first derivative of the number of nodes and edges of the largest connected component (Fig. 9).
However, the 8-year rule produces a significant difference if one looks at those indicators that might show a stabilization of the field in the 1960s. The 8-year network, in fact, shows no sign of stabilization during the 1960s, as the diameter continues growing until 1969 (Fig. 10).Footnote 25
The 8-year rule, moreover, allows a study of the centrality of scientists in a changing scientific environment. It shows, for example, the predominant role Wheeler, and one of his former PhD students, Charles Misner, come to occupy in the last years of the 1960s in the Co-authorExtended_AllNodes_8Years network with respect to the other central figures of the network, such as Bondi and Bergmann (Fig. 11).
Discussion on the dynamics of co-authorship networks
The four co-authorship networks above discussed have some commonalities and, also, significant differences, which might also lead to different historical interpretations. Compared to other scientific co-authorship networks analyzed in the scientometric literature, the relativity co-authorship network remained small and sparse in all four analyses (see, e.g., Barabási et al. 2002; Newman 2004; Mele et al. 2006; Fatt et al. 2010). Average degree remains less than one in all these networks throughout the entire period, which results in great uncertainty on the insights obtained with in this analysis.
In spite of this sparseness, the methodology here applied to look at the dynamics of different kinds of co-authorship relations with different hypotheses on the permanence of nodes and edges in the network provides, on the one hand, some tentative insights on the development of the field of general relativity and, on the other, allows for a comparison of the impacts of different analytical assumptions on the historical interpretations. The co-authorship network retrieved automatically from WoS (WoS_AllNodes_UnlimitedLength) shows a relevant shift only in 1965. One might then be tempted to interpret the small but evident change in the trend occurring in 1965 as the result of the discovery of quasars. This would support the common view that general relativity started to coalesce into a more defined physical discipline as a consequence of astrophysical discoveries in the 1960s, starting from the discovery of quasars (Schmidt 1963). Accordingly, the so-called renaissance of general relativity could be interpreted as a result of the establishment of the field of relativistic astrophysics. As a final finding, there is no quantitative signature of the establishment of a scientific field before 1970, as the diameter continues growing all through the timeframe of our analysis.
The adding of other co-authorship relations changes this perspective and makes this interpretation untenable. The analysis of Co-authorExtended_AllNodes_UnlimitedLength shows that the end of World War II clearly has a strong quantitative influence on the growth of the network with a sudden increase of the total number of nodes and edges, but this does not lead immediately or directly to the formation of a giant component. This starts to happen more than 15 years later. A first, small change occurs only between 1958 and 1959, but a more evident shift happens between 1961 and 1962. Many of the indicators employed show that a giant component starts forming at a rapid pace around 1962, while at the same time Einstein is displaced as the most central actor. At the same time the diameter of the largest component stabilizes, which might be interpreted as the formation of a more closely connected field of enquiry with respect to a previous dispersion in different agendas. This means that there was a meaningful change in the network structure at the beginning of the 1960s, before the astrophysical discoveries starting form 1963. Finally, while the structure of the co-authorship network does not follow the preferential attachment model, it is also considerably different from an Erdős-Rényi graph, as it has a very high clustering coefficient showing, at least in part, small-world features. This analysis seems to allow the identification of a particularly relevant passage in the topology of the network and therefore also for the identification of particularly central actors in this passage.
Imposing some restrictions on the co-authorship network created two new networks: in the first only nodes active in the year of the analysis are included; the second modifies the temporal length of the edges. In spite of these restrictions, the historical picture concerning the moment of the shift in the network structure is not affected. Rather, the restriction to active nodes makes this shift even more evident and also shows the specific connections that modify the social collaboration structure of active scientists. Einstein’s death clearly disrupts this network and the shift in the largest connected component between 1961 and 1962 emerges more clearly. In this case, a study of centrality measures cannot be used to evaluate Einstein’s later influence in the structure of the largest component as the node suddenly disappears in 1955, but it gives a more realistic perception of the role of authors in their social network. The co-authorship network in 1962 shows the first meetings of various groups or components as the result of early-career scientists moving from one place to another, an effect studied by historian of science David Kaiser (2005), who terms the phenomenon “postdoc cascade.” Centrality measures of betweenness centrality and closeness centrality here have also a striking predictive power. In 1963, Robinson, Schucking and Peter Bergmann would be three of the four organizers of the first conference on the newly born relativistic astrophysics, which the meeting itself helped establish: the Texas Symposium on Relativistic Astrophysics (Robinson et al. 1965; Schucking 2008). The three above-mentioned scientists were the only US-based scientists at the time who were within the first five highest values of both betweenness centrality and closeness centrality measures in the recently established largest connected component in 1962 (see Table 3 in “Appendix 2”). This provides further ground to the thesis that, contrary to the idea that the conference itself and the field of relativistic astrophysics created the basis for the explosion of the general relativity field, an already established group in general relativity embedded the new discovery within a field that was already being formed by other means and around other topics.
Table 3 Centrality measures of the nine scholars who have the five highest centrality measures of one of the three following centrality measures: betweenness centrality (%), closeness centrality (%) and degree in the Co-authorExtended_OnlyNodes_UnlimitedLength network in 1962 While providing the same general picture of the previous two co-authorship networks, the temporal restriction on the length of edges provides some granularity in the comprehension of the historical changes by more clearly distinguishing relevant and sudden processes, and identifying well the most central figures in these processes. However, contrary to the second case, the restrictions on the nodes and the temporal length of edges result in the diameter not stabilizing.