How are topics born? Understanding the research dynamics preceding the emergence of new areas

Clique-based method

We measure the collaboration pace of a graph by analysing the diachronic activity of triangles of collaborating topics. To this end, we first extract all 3-cliques from the five sub-graphs associated to each topic under analysis. A 3-clique, as shown in Fig. 3, is a complete sub-graph of order three in which all nodes are connected to one another and is employed for modelling small groups of entities close to each other (Luce & Perry, 1949).

To study the dynamics preceding the debut of each topic, we analyse the evolution of the same 3-clique in subsequent years. Figure 4 summarizes the process. Considering a 3-clique having nodes $\left\{A,B,C\right\}$, we quantify its collaboration index μ_Δ in a year by taking into account both node weights $\left\{W_a,W_b,W_c\right\}$ and link weights $\left\{W_{ab},W_{bc},W_{ca}\right\}$. (3)${\displaystyle \begin{array}{c}\hfill {\mu }_{A-B}=\mathrm{mean}\left(P\left(A|B\right),P\left(B|A\right)\right)\hfill \\ \hfill {\mu }_{B-C}=\mathrm{mean}\left(P\left(B|C\right),P\left(C,B\right)\right)\hfill \\ \hfill {\mu }_{C-A}=\mathrm{mean}\left(P\left(C|A\right),P\left(A|C\right)\right)\hfill \\ \hfill {\mu }_{\Delta }=\mathrm{mean}\left({\mu }_{A-B},{\mu }_{B-C},{\mu }_{C-A}\right).\hfill \end{array}}$

The index μ_Δ is computed by aggregating the three coefficients μ_A−B, μ_B−C and μ_C−A as illustrated by Eq. (3). The strength of collaboration μ_x−y between two nodes of the topic network, x and y, is computed as the mean of the conditional probabilities $P\left(y|x\right)$ and P(x|y), where P(y|x) is the probability that a publication associated with a topic x will be associated also with a topic y in a certain year. The advantage of using conditional probabilities instead of the number of co-occurrences is that the value μ_x−y is normalised with respect to the number of publications associated to each topic. Finally, μ_Δ is computed as the mean of the strengths of collaboration of the three links in a 3-clique. This solution was adopted after testing alternative approaches during the preliminary evaluation, as discussed in the Results section.

The evolution of the 3-clique collaboration pace can be represented as a timeline of values in which each year is associated with its collaboration pace, as in Eq. (4). We assess the increase of the collaboration pace in the period under analysis by computing the slope of the linear regression of these values. (4)${\displaystyle {\mu }_{\Delta \text{time}}^{\mathrm{clique}-i}=\left[{\mu }_{\left(\Delta \text{yr}-5\right)},{\mu }_{\left(\Delta \text{yr}-4\right)},{\mu }_{\left(\Delta \text{yr}-3\right)},{\mu }_{\left(\Delta \text{yr}-2\right)},{\mu }_{\left(\Delta \text{yr}-\text{1}\right)}\right].}$ Initially, we tried to determine the increase in the collaboration pace exhibited by a clique by simply taking the difference between the first and last values of the timeline (μ_Δyr−5 − μ_Δyr−1). However, this method ignores the other values in the timeline and can thus neglect important information. For this reason, we applied instead the linear interpolation method on the five measures using the least-squares approximation to determine the linear regression of the time series f(x) = a⋅x + b. The slope a is then used to assess the increase of collaboration in a clique. When a is positive, the degree of collaboration between the topics in the clique is increasing over time, while, when it is negative, the number and intensity of collaborations are decreasing.

Finally, the collaboration pace of each sub-graph is measured by computing the mean of all slopes associated with the 3-cliques.

To summarize, for each input topic we select a subgraph of related topics in the five years preceding the year of debut (or analysis for topics in the control group). We then extract the 3-cliques and associate each of them with a vector representing the evolution of their pace of collaboration. The trend of each clique is computed as the angular coefficient of the linear regression of these values. Finally, the increase in the pace of collaboration of a subgraph is obtained by averaging these values.

Triad-based method

The triad-based method employs the triad census (Davis & Leinhardt, 1967) to measure the change of topology and the increasing density of the subgraphs during the five year period. The triad census of an undirected graph, also referred to as global 3-profiles, is a four-dimensional vector representing the frequencies of the four isomorphism classes of triad, as shown in Fig. 5.

The triad census summarises structural information in networks and is useful to analyse structural properties in social networks. It has been applied to several scenarios, such as identifying spam (Kamaliha et al., 2008; O’Callaghan et al., 2012), comparing networks (Pržulj, 2007), and analysing social networks (Faust, 2010; Ugander, Backstrom & Kleinberg, 2013).

In this study, we use triad census to describe all the sub-graphs associated to a input topic in terms of frequencies of H_i (see Fig. 5) and we then evaluate how the frequencies of empties (H₀), one edges (H₁), two-stars (H₂) and triangles (H₃) change in time. Figure 5 illustrates the four classes of triads for an undirected graph in the case of topic networks. An increase in the number of triangles suggests the appearance of new collaboration clusters among previously distant topics.

In contrast with the 3-cliques approach, the triad census does not consider the weight of the links, but only their existence. Hence, it is useful to assess how the inclusion of links with different strengths affects the analysis. To this end, we performed three experiments in which we considered only links associated with more than 3, 10 and 20 topic co-occurrences.

We initially perform the triad census over the five graphs associated to each input topic. For example, Table 1 shows the results of the triad census over the five sub-graphs associated with the debutant topic Artificial Bee Colonies.

Next, we check whether the co-occurrence graph is becoming denser by analysing the change of frequencies associated with H_i (see Fig. 6). We first calculate the percentage growth of each H_i (Eq. (5)) and then compute their weighted summation ( Eq. (6)). We label the resulting metric the growth index. We empirically tested other solutions for aggregating the various contributions (e.g., considering only H₃, summing the values, weighting the sum in a variety of ways) and found that this definition of growth index provides the best discrimination between the two classes of graphs. (5)${\displaystyle \text{\%}\mathrm{Growth}{H}_i=\frac{\left(H_i^{Yr-1}-H_i^{Yr-5}\right)\ast 100}{H_i^{Yr-5}}}$ (6)${\displaystyle \mathrm{Growing}{\mathrm{Index}}_{\mathrm{topic}}=\sum _{i=0}^{3}i\cdot \text{\%}\mathrm{Growth}{H}_i.}$

The growth index takes into account the contributions from H₁, H₂ and H₃. Although, the number of triangles (H₃) can by itself be a fair indicator of the density, previous studies showed that all four classes of triads are useful for computing network properties, including transitivity, intransitivity and density (Faust, 2010; Holland & Leinhardt, 1976). Taking into consideration only H₃ might fail to detect some subtler cases, characterized for example by a contemporary increase of H₂ and decrease of H₁.

To summarize, the triad-based method receives the same input as the clique-based method. For each of the five subgraphs associated to a topic, we perform the triad census obtaining the different frequencies, H_i, in different years. We then analyse them diachronically to quantify the increase in density.