Mapping the physics research space: a machine learning approach

The embedding of topics into a high dimensional research space allows us to use their spatial positions to infer the value of their pairwise similarities by measuring the topics relative closeness. In particular, the topic vector space can be used to compute the similarity between two research topics as the cosine similarity between their vectors: where \(\mathit{vec}_{i}^{t}\) and \(\mathit{vec}_{j}^{t}\) are the 200-dimensional embeddings for topics i and j, respectively.

The similarity measure can be used to generate the research space network (RSN) that considers the similarity as the weight of the connections and preserves only the most important links by removing the ones associated to negative or small values of the cosine similarity. The resulting RSN is visualized in Fig. 2, where we show that our methodology successfully groups together research topics belonging to the same section as taxonomized in the PACS 2010 Regular Edition of the Physics and Astronomy Classification Scheme [59]. Although our methodology is completely general and does not make use of the PACS hierarchical classification, we can use the latter as an external validation of the quality of the obtained embedding and classification. Indeed, our approach treats each 6-digits PACS code as a mere keyword and the information regarding the hierarchical structure of the classification scheme is not used to train the vector embeddings. In other words, our algorithm is unaware of the existence of the ten Sections of the PACS classification. However, when we do look at the resulting research space by coloring the nodes according to their PACS Section we notice that the General section is correctly placed at the center of the research space network, along with the Interdisciplinary Physics section, as one would expect. On the other hand, we note that Physics of Gases, Plasmas, and Electric Discharges, Condensed Matter, and Nuclear Physics seem to be populating three different boundary areas of the research space (as also observed in previous studies [3]). Overall, the position of the topics is consistent with the information codified in the PACS codes but—in addition—our approach also allows us to understand the relative position of each topic and PACS section with respect to each other, therefore enabling us to quantitavely measure their degree of relatedness.

Research activities in the context of the research space can be analyzed at different geographical scales. More precisely, we can fingerprint scientific production at the level of individual authors, institutions, cities, or countries by geolocating scientific publications (Fig. 3). This can be achieved by considering all the articles published in American Physical Society’s (APS) journals in the period 1986–2009 and by associating to each publication: (a) the information contained in the authors’ affiliation; and (b) the set of research topics (i.e. the PACS codes) used in the paper. In the following, we focus on geographical units constructed by first parsing the city names from the affiliation strings for each article, and then clustering together neighboring cities to obtain distinct urban areas. More specifically, we follow the same procedure used in [60]: first, we infer the country in which each affiliation-city pair is located; second, for each country, we compute a geographic distance matrix (using Vicenty’s formula) connecting each pair of cities; and lastly we use hierarchical clustering to define the different urban areas with the additional constraint that the maximum distance within each cluster has to be less than 50 km. Once we have the geographical units defined, we count how many publications have been produced in each PACS code by each distinct urban area.

In order to provide a specific fingerprinting for the degree of specialization of each geographical unit we extend to scientific production [47] the concept of Revealed Comparative Advantage (RCA, [61]). RCA has a long history in the economic literature where it has been used to study the level of specialization of nations and regions in terms of industrial production, technological production, and trade exports (see for example [62‐75]). The RCA [61] is defined as: where \(X_{c,k}^{t}\) denotes the number of publications produced in urban area c in PACS code k in the time window t. In practice, the numerator represents the percentage share of papers published in PACS code k by location c; while the denominator represents the percentage share of papers published in PACS code k across the world. By comparing these two figures, we can assess whether a given geographical unit is relatively more specialized in a certain research topic.

By using the above definition, we consider a geographical unit to be a specialized scientific producer in PACS code k at time t if \(\mathit{RCA}_{c,k}^{t}>1\). Using the RCA, we can generate location-specific specialization profiles that allow us to fingerprint the structure of the scientific production system of each geographical area. In particular, we can create a (time varying) fingerprint matrix \(F_{ck}\), where c is a geographical unit and k is a topic, and assign non-zero entries only if \(\mathit{RCA}_{c,k}^{t}>1\). We can visualize the matrix \(F_{ck}\) to have a general understanding of the different specialization patterns, a sort of research DNAs, that characterize the knowledge production of geographical units, as shown in Fig. 4. As an example, we also show the scientific fingerprints of three different urban areas: Darmstadt (Germany), Cambridge (MA, USA), and Pittsburgh (PA, USA). This let us appreciate how different locations might specialize into different parts of the research space. For instance, Darmstadt has a relative comparative advantage on \({\sim}70\%\) of all the PACS code in Nuclear Physics (PACS section 20). On the other hand, Pittsburgh is specialized in Physics of Elementary Particles and Fields (PACS section 10) while its specialization in Nuclear Physics is particularly low. Lastly, we observe that Cambridge is the only city among the three considered with a more homogenous pattern of specialization (with the exception of Nuclear Physics). Overall, by looking at Fig. 4, we can start to appreciate how different cities might cluster their scientific expertise around distinct areas of the research space, even though exceptions—as in the case of Cambridge—do exist.

The RCA in the context of the research space has been introduced by Guevara et al. [47] to explore the principle of relatedness [57, 58, 76] in the process of scientific production: i.e. it is easier to specialize and work in related research areas requiring a set of common skills/knowledge. Indeed, relatedness has been found to play an important role in explaining the patterns of future development of industries and research production at the level of cities, regions, and nations [47, 57, 77‐83]. This is due to the fact that different sets of capabilities and skills might be needed to grasp the depth and complexity of different research topics, therefore affecting the ability of researchers to move and develop a competitive edge across different disciplines. This observation builds upon the idea that congnitive proximity [84, 85] is required to successfully absorb and use new knowledge [86].

Our analysis provides support to the principle of relatedness and the fingerprint matrix shows patterns of specialization that are indeed not random. Some of these patterns can be appreciated in Fig. 5 where we plot each PACS code in which a city has a comparative advantage using the spatial coordinates identified using the research space mapping. Also in this case, we can appreciate how spatial ontologically consistent clusters of competences emerge. In other words, it appears that urban areas tend to develop around their current domain of expertise implying that scientific relatedness does play a role in explaining the structure of knowledge production of a city. In order to quantify the relatedness of a specific PACS code to the overall domain of expertise of a given geographical unit, we use the knowledge density (as proposed in [57]). The knowledge density \(\omega _{i,c}^{t}\) around PACS code i in urban area c at time t is defined as: where \(\phi _{i,j}\) is the level of knowledge similarity between PACS codes i and j. In our case, we use cosine similarity to measure the similarity between two research topics. Given this definition, for a location c and time window t, the closer topic i is to other topics in which c has a relative comparative advantage, the higher its knowledge density. To understand how this metric works, let us consider what happens when the index—which varies between zero and one—takes its extreme values. For a given PACS code i and urban area c combination, the value of \(\omega _{i,c}^{t}\) is equal to zero if c has no comparative advantages in topics related to i; while it has a knowledge density equal to one if it has an advantage in all the topics related to i. In other words, the closer is i to the current domain of expertise of c, the denser the knowledge space will be around PACS code i.

For each geographical unit, we can associate the knowledge density \(\omega _{i,c}^{t}\) with four different types of transitions that characterize the time evolution of the comparative advantage of a PACS code. We look at the distributions of \(\omega _{i,c} ^{t}\) when: (1) a PACS code that is inactive (i.e. \(\mathit{RCA}=0\)) at time \(t-1\), remains inactive at time t; (2) a PACS code that is inactive at time \(t-1\) becomes active but with no comparative advantage (i.e. \(0<\mathit{RCA}\leq 1\)) by the city at time t (i.e. inactive to active but not specialized); (3) a PACS code remains active but with no comparative advantage by the city at both \(t-1\) and t (i.e. remained not specialized); and lastly (4) a PACS code that is active but with no comparative advantage by the city at time \(t-1\), while a comparative advantage (i.e. \(\mathit{RCA}>1\)) emerges at time t (i.e. from not specialized to specialized). Looking at these distributions we observe that PACS codes that normally remain inactive are the ones in which the knowledge density at the previous time step was the lowest, while the opposite holds for the codes in which urban areas become specialized (a visualization of the results is reported in the swarm plot in Fig. 6). In other words, it is easier to develop a stronger comparative advantage in research topics that are related—in the research space—to the ones in which a location is already specialized in.

It is interesting to explore the possibility of using the knowledge density as a predictor of the emergence of a comparative advantage of a city in a specific PACS code in the future. Operationally, we follow the same methodology proposed in [47] and postulate that the order in which each urban area will become specialized should closely follow the list of PACS codes ranked according to their associated value of knowledge density. We can test this hypothesis against a null one assuming that, instead, specialization occurs independently of the current level of knowledge density. In other words, the alternative hypothesis would suggest that an urban area develops a comparative advantage at random, regardless of its previous level of expertise and specialization. The predictive performance of the knowledge density \(\omega _{i,c}^{t}\) can be evaluated using a statistic which is normally used in the machine-learning community to measure the accuracy of a model: the area under the so-called Receiver Operating Characteristic (ROC) curve. The ROC curve is used to plot the true negative rate of a model (for example of a classifier) against its true positive rate. That is, the share of correctly classified negative values against the share of correctly classified positive values. If the value of the area under the ROC curve is greater than 50%, then the accuracy of our prediction using the knowledge density is greater than the one we would have from a random prediction where PACS would not have been ranked by their knowledge density. In our case, we actually have a distribution of such values since—for a given time period—we can compute the accuracy of our model for each geographical unit. In other words, we test our ability to predict the research trajectories of each city in each distinct time window. The results, reported in Fig. 6, show that the accuracy is higher than 50%, confirming that the structure of the research space can be used to predict how research trajectories evolve over time. While beyond the scope of the presented work, it is possible to envision the use of current estimate of the knowledge density to forecast the physics research areas in which specific urban areas will be able to specialize in future years.

In our analysis, we use the APS Data for Research (2010 release) data collection which comprises information about more than 400,000 articles published by the American Physical Society. In particular, in this work, we limit our attention to the years 1986–2009 and we perform our analysis dividing our sample in 3-years non-overlapping time windows. However, the results we provide are robust with respect to the exact choice of the time window size and on whether or not overlapping intervals are considered. Papers are geo-located parsing the information contained in the authors affiliations following the procedure detailed in [60], authors are disambiguated following the procedure detailed in [94], while research topics are assigned considering the first 6 digits of the PACS classification scheme [59]. Overall, our dataset includes 2307 urban areas and 5800+ PACS codes. However, in our analysis, we restrict our attention to cities that have at least 6 publications in each time window. This restricts our original sample to 402 urban areas and 854 PACS codes.

In order to produce the PACS code embeddings, we employ the StarSpace model proposed by [54]. Starspace is a general-purpose embedding model that aims at creating embeddings for a variety of entity types (e.g. words, sentences, documents, images, etc.) by associating to each entity an N-dimensional vector. In our case, the vector size is set to \(N=200\) and the vectors are obtained by minimizing a loss function that simultaneously maximizes the (cosine) similarity between embeddings of PACS that are used by the same author, and by minimizing the (cosine) similarity between embeddings of PACS that do not appear together when looking at the career of scientists. In other words, once PACS codes are mapped into this new 200-dimensional space, PACS that frequently appear together in the list of publications of a scientist will tend to be close, while PACS that rarely appear together will belong to different areas of the embedding space. More specifically, the model minimizes the following loss function: where \(E^{+}\) denotes the set of positive entity pairs (i.e. PACS that often appear together), \(E^{-}\) denotes the set of negative entity pairs (i.e. PACS that rarely appear together), \(\kappa =50\) is the number of negative pairs used for each batch update (i.e., this model uses a K-negative sampling strategy as in [50]), \(\operatorname{sim}(\cdot)\) denotes the cosine similarity between two embeddings, and \(L^{\mathrm{batch}}\) denotes the batch specific loss function that compares the positive pair \((a,b)\) with the negative pairs \((a,b^{-}_{i})\) using a margin ranking loss of the form \(\max (0,\mu -\operatorname{sim}(a,b)+\sum_{i\in [1,\kappa ]}(a,b^{-}_{i}))\). The loss function is then minimized using stochastic gradient descent [95]. The value of N has been chosen after examining the prediction performance of our model when trying to reconstruct the bag-of-topics of the authors. In particular, we computed the percentage of correctly predicted PACS codes in the top k predictions made by the algorithm. This metric is commonly denoted by hit@k [96] and it is the same performance metric used also in [54]. In Fig. 9 we show how its value varies with the size of the embedding dimension N and \(k=50\). In light of this analysis, we decided to set N equal to 200 since it provided a good compromise between the training time required to fit the model and its overall prediction quality.