The classical origin of modern mathematics

F. Gargiulo1, A. Caen2, R. Lambiotte1, T. Carletti1

1. Department of Mathematics and Namur Center for Complex Systems - naXys2. DICE, Inria, Lyon, France

The aim of this paper is to study the historical evolution of mathematical thinking and itsspatial spreading. To do so, we have collected and integrated data from different online academicdatasets. In its final stage, the database includes a large number (N ∼ 200K) of advisor-studentrelationships, with affiliations and keywords on their research topic, over several centuries, from the14th century until today. We focus on two different topics, the evolving importance of countriesand of the research disciplines over time. Moreover we study the database at three levels, its globalstatistics, the mesoscale networks connecting countries and disciplines, and the genealogical level.

Introduction

The statistical analysis of scientific databases, including those of the American Physical Society, Scopus, the arXivand ISI web of Knowledge, has become increasingly popular in the complex systems community in recent years.Important contributions include the development of appropriate scientometric measures to evaluate the scientificimpact of scientists, journals and academic institutions [11–15] and to predict the future success of authors [18, 19]and papers [20]. In parallel, the structure of collaboration has attracted much attention, and collaboration networkshave become a central example for the study of complex networks, thanks to the high quality and availability of thedatasets [10]. From a dynamical point of view, different papers [16, 17] studied the mobility of researchers duringtheir academic career, showing that the statistical properties of their mobility patterns are mainly determined bysimple features, such as geographical distance, university rankings and cultural similarity.

Limitations of the aforementioned datasets include their relatively narrow time window extending, at best, over100 years and the difficulty to disambiguate author names, and thus to distinguish career paths across time. Theoriginal motivation of this paper was to address these issues by performing an extended study of The MathematicsGeneaology Project, a very large, curated genealogical academic corpus [21]. The dataset, whose basic statistics havebeen analysed elsewhere [4, 7], extends over several centuries and contains pieces of information allowing us to retrievethe direct genealogical mentor-student links, but also university affiliations at different points of a career and and theresearch domains. Data from the same website have already been used to assess the role of mentorship on scientificproductivity [5] and to study the prestige of university departments [6].

Our main goal is to analyse the history of modern mathematics, through the processes of birth, death, fusion andfission of research fields across time and space. In particular, we focus on the temporal evolution of the roles andimportance of countries and of disciplines, on the structure of “scientific families” and on the impact of genealogyon the development of scientific paradigms. As is often the case when performing a data-driven analysis of historicalfacts [8, 9], the data set is expected to be incomplete and to present biases, mainly for the more ancient data. In thepresent case, the website collects the data in two ways: a participative method, based on the spontaneous registrationof scholars (who can also register their students and their mentors), and a curated method, based on historical factsand performed by the creators of the web site.The presence of biases calls for the use of appropriate statistical measures, in preference based on ranking insteadof absolute measures. In this work, we have also introduced data-mining methods to correct and enrich the datastructure. A first contribution of this work is thus methodological, with the design of a methodological setup that couldbe applied to other systems. We have then performed an analysis of the system at three levels of granularity. First,a global one investigates the fully aggregated “demography” (population in terms of countries and disciplines) of thedatabase, with the aim to classify countries and disciplines according to their normalized activity behavior. Trackingthe evolution of the rankings helps identify transition points in the mathematical history, associated to emerging fieldsof research. Second, we have constructed directed weighted networks where nodes are scholars endowed with a set ofattributes (thesis defense date, thesis defense location, thesis disciplines) and linked to other nodes using the genealogyassociated to the mentor-student relation. This “mesoscale” network allows us to investigate the relationships betweenthe attributes and to identify a strong hierarchical structure in the scientific production in terms of countries as wellas its evolution in the course of time. Finally, using an approach typic of kinship networks studies [24, 25], we focus onthe statistical properties of the tree structure of the genealogy in terms of family structures. We conclude by showingthe presence of strong memory effects in the network morphogenesis.

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Dataset and associated networks

The core of our dataset has been extracted from the website ”mathematical genealogy project”. It is one of thelargest academic genealogy available on the web, consisting of approximatively 200K not–isolated scientists (186505)with information on their mentors and students. The data cover a period between the 14th century until nowadays.For a majority of mathematicians, we have detailed information about his/her PhD, including the title (for the 88%of the scholars), the classification according to the 93 classes proposed by the American Mathematical Society[1] (forthe 43% of the scholars), the University delivering the degree as well as the year of its defense. However, becausea large part of the database is spontaneously filled by the scientists, the data is imperfect and attributes may bewrong or missing. A first step has thus consisted in comparing the database with additional data form Wikipedia[2] and, for more recent entries, with the Scopus profiles of scientists [3]. After this preliminary phase, we haveenriched the information available for the authors, by developing algorithms aimed at correcting the dates based onthe genealogical structure and the available statistics, assigning to each thesis a discipline, based on the thesis title,and by disambiguating universities names. As previously stated, the algorithms, summarized in the supplementarymaterial, are general and could be applied in other contexts. After the enrichment, all the scholars of the databasehave a corrected date, the 88% has an associated discipline and the 94% and associated country. As a next step,we have exploited the enriched database in order to study the geographical and temporal evolution of mathematicalscience. Different representations, described below, have been adopted.

The mesoscale networks

As a first step, we have considered a network where the nodes are attributes of researchers (i.e. universities, cities,countries and discipline), and directed links, from A to B, correspond to situations when a scientist with attributeA was the PhD supervisor of a student with attribute B. In the case of universities, and under the assumption thata supervisor is in the same university as his/her PhD student, directed links are therefore a proxy for the mobilityof a scholar, but also of the flow of knowledge between different places. In the case of disciplines, directed linkscorrespond to a transfers of knowledge from one scientific discipline (the one of the mentor) to another one (the one ofthe student), from one generation to another, and how disciplines at a certain time may inherit, in terms of ideas andmethods, from research fields at previous times. Let us note that the network allows for self-loops. The procedure isillustrated, for the case of flows between countries, in Fig 1.

Figures

In the following, we will perform a longitudinal study of the system, by considering the evolution of networksobserved in different time windows. Note here that the data are not uniformly distributed across time, with a strongbias towards recent times.

The genealogical tree and its partitions into families

The genealogical graph is the most obvious representation of our dataset, consisting in an oriented acyclic graph[28] linking a mentor to her/his students. This defines automatically the structure of hierarchical generations. Noticehowever that the structure of our data is not simply a tree due to the several cases where a student has two advisors.A very common process in kinship is to cut the genealogical directed acyclic graphs into linear trees (alliances) whereeach individual has a single progenitor (the mother for representing the uterine links and the father for the agnaticones). In this representation, the links between alliances represent the matrimonial structures between the differentalliances in the society. In our context, when a scientist has more than one advisor, it is not clear which links should becut to retrieve the original ancestors (no gender–like roles exist in this framework and moreover our dataset preventsus from identifying the principal supervisor from the secondary one, if any). We thus propose a method to reproducethe optimal ancestry lines and to identify the important families in the genealogy. The method, fully described inthe supplementary material, is based on the decomposition of the network into pure linear trees, and their statisticalclustering based on probabilistic arguments; roughly speaking given two nodes A and B that can be linked in morethan one way, thus implying the presence of non-trivial loops, we assign to every link in such paths the probabilitythat A and B will be disconnected if the link is removed. We thus select links to be removed by maximising theprobability that A and B are still linked. The resulting partition of the graph into families identifies 84 families;

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remarkably, the 24 mot populated families cover the 65% of the scientific population in the database. Let us observethat alternative methods for family identification do exist, see for instance [26, 27].

Results

Global statistics

Let us define the relative abundance profile of each country in different periods fI(t) = NI(t)/N(t), where N(t) isthe total number of scientists whose country is known in the database at time t and NI(t) is the number of scientistsin country I at time t. Notice that, at this stage, the genealogical information is not used. From such profileswe can asses the evolution of the importance of the countries had in the history of mathematics due to the theirdifferent historical dynamics. In order to compare the profiles of different countries independently from their totalproduction, we normalize each of these profiles by their “volume”(f̃I(t) = fI(t)/

∑t fI(t)) and then we classify them

based on their Kolmogorov-Smirnov distance (see Fig. 2 where the results are reported using a dendrogram, andthe SI where we reported the profiles for the top 10 countries in the database). We observe different prototypicalbehaviors: countries with a central role in the ancient history whose centrality has decreased in the last centuries(for instance Italy, France and Greece), countries with a central role before the world wars (e.g. central Europecountries), countries emerging after the world wars (such as Japan and India), countries recently emerging (amongwhich China and Brazil). Because of the normalization procedure we used, we obtain a cluster where USA is linkedwith ex-URSS countries and show a similar decreasing behavior in the latest decades (impossible to observed in anon–normalized context).

The same procedure is applied to disciplines and results reported in Fig. 3 allow to identify three main blocksof disciplines: the disciplines that were more central during the industrial revolution (before 1900) are associatedto physical applications (such as thermodynamics, mechanics and electromagnetism). The disciplines reaching theirmaximum of expansion around the 1950 are more abstract, even if several links exist to applied topics, such astelecommunication and quantum physics. Finally, the last decades have witnessed the emerging dominance of appliedmathematics (e.g. statistics, probability) and computer science.To capture the rise and fall of countries or disciplines, we have compared the rankings of the top 10 countries anddisciplines in different time periods. Standard indicators for rank comparison, such as the Kendall- Tau index, cannot

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the average prevalence profile for the cluster 〈f̃C(t)〉 =∑

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be applied here since the elements in the top-k lists are not conserved in time [22]. For this reason, we have used adistance measure based on a modified version of the Jaccard index allowing to compare ranked sets, J(rank1, rank2)(more information is provided in the Supplementary material). As for the original Jaccard index the modified versionis such that J(rank1, rank2) = 1 when the rankings rank1 and rank2 are completely equivalent, and gives a value 0when these latter are not correlated at all. This information is then transformed in a distance by taking dJ = 1− J .Increases in the distance measure, dJ , indicate major reshaping of the rankings.

As we can observe in the upper plot of Fig. 4, corresponding to countries, we observe several transition points;for example a transition can be observed during the first World War, with the decreasing centrality of Austria andHungary due to the end of the Austro-Hungarian emperor and the entering in the ranking of Russia. Anothertransition is connected with the European political reshaping during the second World War, this is the period atwhich for the first time, USA surpasses Germany in the ranking. A third transition, around the 1960s, shows theincrease of centrality of the Soviet Union (testified by the presence of several east-european countries in the ranking).Finally, more recently, we observe the decline of Russia and the emergence of new countries such as Brazil.

A similar analysis can be performed for disciplines. Results reported in Fig. 5 show the presence of three tipping

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points. The first one is connected to industrial revolution and to the emergence of disciplines related to the physicsof machines (such as thermodynamics and electromagnetism). The second one is connected to the emergence of fieldslinked to telecommunication and cryptography (e.g. number theory, spectral functions) during the second World Warperiod. Finally the third one, in the 80’s, concerns the emergence of computer science and statistics.

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FIG. 4: Modified Kendall-Tau index comparing the countries’ rankings in different periods.

Mesoscale networks

Network of countries

The countries network can be used to represent the knowledge flows from one country to another one, associated tothe transition of a student in a country, becoming a professor in another country. The network presents few importanthubs, that are the gravity centers of the scientific research (USA, Germany, Russia, UK). Each of these hubs tendsto be surrounded by a community of countries. These communities can be associated to historical divisions, forinstance a large block connected to USA scientific production, the Commonwealth nations, the ex-Sovietic block,the central European countries. The betweenness of countries allows to detect countries at the infercace betweendifferent communities, such as France connecting the central European countries with the USA-centered communityor Poland connecting European research and the ex-Sovietic area.

Another important index of the countries network is the weighted in(out)–degree of the nodes and the numberof self loops. The in–degree of a country represents the number of scientists obtaining their PhD elsewhere andmentoring a PhD student in that country. Therefore a high in–degree is associated to a country with a strongcapacity of attracting scholars and absorbing knowledge from abroad. On the contrary, a high out–degree representsa country producing scholars and exporting knowledge elsewhere. Each country can be therefore characterized bythree normalized quantities, the fraction of scientist formed inside the country and remaining there, the fraction ofscientist formed inside and leaving and the fraction formed abroad and absorbed by the country. In Fig. 6A wedisplay the different positions of the most important countries with respect to these indexes, the closer a country isto one of the triangle vertices the larger is the associated index. The size of the dot is a measure of the productionof a country, i.e. estimated by its number of PhDs. The most productive countries tend to be the most selfishones, with a large fraction of self-loops. The most important exporters are Russia and the UK. Countries withsmall scientific production show a tendency for importing scholars. Observe that these indexes evolve in time (seeFig. 3 of the SI). An important inversion point between kin(t) and the kout(t) is often observed around the sec-ond World War. Moreover, a key signature of emerging scientific countries seems to be the presence of kin(t) > kout(t).

To characterize the mobility of scientists across countries, we show in Fig. 6B the fraction of the total productionand the total absorption of migrant scientists for the first r countries respectively in the in and out–degree ranking.

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FIG. 5: Modified Kendall-Tau index comparing the disciplines’ rankings in different periods.

The top 7 countries produce 80% of the total international scholars. On the contrary, the curve concerning thetotal absorption has a lower slope, depicting a larger worldwide spreading around the world. This shows a stronghierarchical structure in academic research where few countries ensure a large share of the worldwide diffusion ofscientific knowledge. This scenario obviously evolved in time, as we observe in Fig. 7. Remark that scientific leaderschanged at different points in time, but also that the scientific leadership group (countries producing the 80% of thewhole scientists production) is more restricted in recent times. It is interesting to notice that the minimal size ofthe scientific elite has been reached in the sixties during the world bi–polarization resulting from the cold war. Sincethen, the size increased again with the emergence of globalization.

More information about this network, in particular the properties of the aggregated transition networks concerningthe whole historical period, can be found in the Supplementary Information.

The transition network of disciplines

The transition network of disciplines represents transfers of knowledge from one scientific discipline (the one ofthe mentor) to another one (the one of the student). The structure of this graph is quite homogeneous in termsof degree and four major communities can be identified: computer science, geometry, analysis and physics. Each

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FIG. 6: Panel A: Relative position of the countries between selfish/exporting and absorbing behaviors. Panel B: Fraction ofscientists produced and absorbed from the first countries in the rankings respectively of kin and kout. In the boxes are displayedthe countries producing and absorbing the 80% of scientists. Both the panels concern the temporal aggregate of the network.

community represents the disciplines exchanging more knowledge between them then with other research fields,and therefore can be interpreted as the scientific paradigms (according to Thomas Kuhn definition) at a certain period.

In Fig. 8 we show the normalized mutual information (NMI) between the community structures obtained fromdifferent temporal slices of the network. The NMI index varies between one, when the two partitions are equal, andzero, when the two classifications are completely disjoined. A low value of the NMI indicates a “revolution” in the senseof a strong reorganization of the knowledge structures, previously non interacting research fields start to exchangeknowledge. The figure shows two important points where the NMI is low. The first transition, observed between1930 and 1940, can be associated to the period when Statistics and Probability merged together, attracting thenmore applied disciplines like information theory, game theory and statistical mechanics, and leading to the emergenceof the field of applied mathematics. The second transition is between 1970 and 1980, where computer science andstatistics form one community, together with dynamical systems and applications in other fields of science. The latesttransition is expected to be a spurious effect, due to a lack of data in recent years (the last time window starts in2010 and therefore can contain data only for 5 years). Another potential approach, alternative to the measure of theNMI, to identify the structural changes in these structure could be the one proposed in [29].

The genealogical structure

This last section is devoted to the study of the genealogy tree reconstructed from our data and of its relevance inthe evolution of the history of mathematical science.

The first result is the presence of a strong memory effects in the network morphogenesis, as students very often doresearch in the same discipline their mentor did. To quantify this idea we analyzed the genealogical chains where

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FIG. 7: Temporal network. Fraction of countries producing (and absorbing) the 80% of the scientists in different historicalperiods.

the “filiation” link connects a mentor with a student maintaining the same discipline of the mentor. We call theseobjects iso-discipline chains. In Fig. 9 (left panel) we show results concerning the conditional probability of having achain of length n+ 1 given a chain of length n, in other words the probability to have one more descendant workingin the same research field of the whole chain, aggregating data over all the disciplines. The first point thus representsthe probability for a student to have the same discipline of his/her mentor, one can clearly appreciate that as thechains get longer the probability to continue the same iso-discipline increases. This very marked memory effect inthe network can be associated to the existence of “schools” where a long tradition in a discipline exists such thatnew students are attracted and continue the tradition. Observe however (Fig. 9 right plot) that this phenomenonstrongly depend on the discipline .

As previously explained, we have partitioned the network into disjoint families of scholars. Fig.10A shows that the65% of the scientists can be divided into 24 macroscopic families with size S > 500. The largest family is the oneoriginated in 1415 by the Italian medical doctor, Sigismondo Policastro. The second one, is the family originated bythe Russian mathematician Ivan Petrovich Dolby, at the end of the 19th century. The large size of this family, bornmore recently than other families and geographically located mostly around Russia, is due to a high “fecundity rate”in the Russian school of Mathematics.

The aggregated network between the families, reminiscent of kinship of the alliance networks defined in [23, 24],can be described using some typical topological indicators [25]: 1) the endogamy index, �0 describing the fraction ofloops in the network (links between the same family); 2) The concentration index cx denoting the heterogeneity of theconcentration of links between pairs of families (cx = 1 when all links are concentrated on a single pair and cx = 1/n

2

when links are homogeneously distributed among the n families); 3) the network symmetry index sx that varies from

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0 in case of total link unbalance, namely the outgoing flux and the ingoing one are very different each other, to 1 in

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SigismondoPolcastro JeanLeRondd'Alembert

WalterWilliamRouseBall CatoMaximilianGuldberg

WallaceL.Cassell Jean-DominiqueCassini RenéJustHaüy

ViktorLvovichKirpichov CornelisBenjaminBiezeno

NathanielBowditch HenryBracken CassiusJacksonKeyser

SebastianoCanovai WilliamSharpey AugustOttoFöppl

FrankMorley FriedrichLeibniz IvanPetrovichDolbnya

ElissaeusJudaeus NikolauskolausBodaPodavonNeuhaus

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FriedrichLeibniz IvanPetrovichDolbnya ElissaeusJudaeus

NikolauskolausBodaPodavonNeuhaus René-LouisBaire

KarlFriesach FrançoisDubois GustavConradBauer

SigismondoPolcastro JeanLeRondd'Alembert

WalterWilliamRouseBall CatoMaximilianGuldberg

WallaceL.Cassell Jean-DominiqueCassini RenéJustHaüy

ViktorLvovichKirpichov CornelisBenjaminBiezeno

NathanielBowditch HenryBracken CassiusJacksonKeyser

SebastianoCanovai WilliamSharpey AugustOttoFöppl

FrankMorley FriedrichLeibniz IvanPetrovichDolbnya

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A

OBS 0.95 0.25 0.99EXP 0.27 0.07 0.99

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FIG. 10: Panel A: Relative size of the different families and family’s initiator name. Panel B: Table with the values of thetopological indicators for the real (observed) network in the first row and for the randomized model (expected) in the secondrow.

case of perfect symmetry of fluxes. To asses the relevance of such indicators computed for our genealogy network, wecompared them with the expected values for a random multinomial reshuffling - null model - (see Fig. 10B), we canobserve that, while the symmetry is a structural property, being unchanged by the reshuffling, the endogamy and theconcentration are typical signatures of this network and moreover they are much higher than in traditional kinshipnetworks [25]. These results imply that the obtained scientific families are structurally very distant between themand that their relationships are very hierarchical (being these mediated by the largest families).

Finally, we studied the distribution of families across countries and disciplines. As shown in Fig. 11A, with theexception of few cases, the most important countries (in term of production) are present in all the families, while theremaining countries are represented in a very low number of families (from 1 to 3). This feature implies a strongcorrelation between the genealogical structures and the geography. A similar behaviour can be observed for disciplines(Fig. 11B) even if, in this case, the curve describing the number of families with members working in a given disciplineis smoother. We can therefore conclude that the genealogical families are strongly specialized in terms of geographyand epistemic content.

Conclusions

In this paper, we have presented a data-driven study of the history of mathematical science, based on the Math-ematical Genealogy Project. A first important aspect has been the cleaning and correction of the incomplete andsometimes inaccurate dataset. This operation was performed by means of machine-learning and by incorporating

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FIG. 11: Panel A: countries are set on the horizontal axis, ranked by the relative presence in the database, while in the verticalaxis, we report the families ranked by their size. A point at the intersection of the country-column and family-row indicatesthat the country is present in this family. The upper plot, is the column-marginal of the matrix represented in the central plot,representing the number of families where each country appears. The right plot is the row-marginal of the matrix representedin the central plot, representing the number of countries present in each family. Panel B: On the horizontal axis we put thedisciplines, ranked by the relative presence in the database, in the vertical axis, the families ranked by the size. A point at theintersection of the discipline-column and family-row indicates that the discipline is present in that family. The upper plot, isthe column-marginal of the matrix represented in the central plot, representing the number of families where each disciplineappears. The right plot is the row-marginal of the matrix represented in the central plot, representing the number of disciplinesdeveloped in each family.

data from other sources, including Wikipedia.We have then considered three different approaches to analyse the data: a demographic approach analyzing the time

evolution of the prevalence of certain attributes (i.e. country or disciplines); a mesoscale network approach focusingon the connections between these attributes; a “kinship” approach based on the clustering of genealogical trees.Our analysis reveals important transition points in the history of mathematics and allows us to categorize countriesaccording to their capacity to attract, export and self-maintain knowledge. Moreover, the community structures ofthe network of disciplines allows us to better describe the transformation of knowledge across time. Finally, we havealso identified important scientific families, associating them to their founder, and described their geographical anddisciplinary distribution.

Interesting lines of research for the future include the integration of additional datasets, based on different method-ologies, to extend te scope of this work beyond the mathematical sciences.

[1] http://www.ams.org/msc/msc2010.html[2] http://www.wikipedia.org[3] http://www.scopus.com[4] http : //people.maths.ox.ac.uk/porterm/research/priyathesisf inal.pdf[5] Dean MR, Ottino JM, Nunes Amaral LA (2010) The role of mentorship in protégé performance. Nature 465.7298: 622-626.[6] Myers SA, Mucha PJ, Porter MA (2011) Mathematical genealogy and department prestige. Chaos-Woodbury 21.4: 041104.[7] Engin A, Gunes MH, Yuksel M (2011) Analysis of academic ties: a case study of mathematics genealogy. GLOBECOM

Workshops (GC Wkshps), 2011 IEEE. IEEE.[8] Schich M, Song C, Ahn YY, Mirsky A, Martino M, Barabási AL, Helbing D (2014). A network framework of cultural

http://www.ams.org/msc/msc2010.htmlhttp://www.wikipedia.orghttp://www.scopus.comhttp://people.maths.ox.ac.uk/porterm/research/priya_thesis_final.pdf

13

history. Science, 345(6196), 558-562.[9] Sinatra R, Deville P, Szell M, Wang D, Barabási AL (2015). A century of physics. Nature Physics, 11(10), 791-796.

[10] Newman ME (2001) The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA 98[11] Ramasco JJ, Dorogovtsev SN, Pastor-Satorras R (2004) Self-organization of collaboration networks. Phis. Rev. E 70,

036106[12] Pan RK, Kaski K, Fortunato S (2012) World citation and collaboration networks: uncovering the role of geography in

science. Sci. Rep. 2[13] Bergstrom C T, West JD, Wiseman MA (2008) The Eigenfactor metrics. J. Neurosci. 28.45[14] Radicchi F, Fortunato S, Markines B, Vespignani A (2009) Diffusion of scientific credits and the ranking of scientisandts.

Phis. Rev. E 80, 056103[15] Radicchi F, Castellano C (2011) Rescaling citations of publications in physics. Phis. Rev. E 83, 046116[16] Gargiulo F, Carletti T (2014) Driving forces of researchers mobility. Sci. Rep. 4, 4860[17] Deville P, Wang D, Sinatra R, Song C, Blondel VD, Barabási AL (2014). Career on the move: Geography, stratification,

and scientific impact. Scientific reports, 4[18] Wang D, Song C, Barabási AL (2013). Quantifying long-term scientific impact. Science, 342(6154), 127-132.[19] Acuna DE, Allesina S, Kording KP (2012) Future impact: Predicting scientific success. Nature 489.7415 , 201-202.[20] Shen HW, Wang D, Song C, Barabási AL (2014) Modeling and predicting popularity dynamics via reinforced poisson

processes. In AAAI 2014, 291-297.[21] The mathematical genealogy: http://genealogy.math.ndsu.nodak.edu/index.php[22] Fagin R, Kumar R, Sivakumar D (2003). Comparing top k lists. SIAM Journal on Discrete Mathematics, 17(1), 134-160.[23] Hamberger K, Houseman M, White D (2011) Kinship network analysis. In: Car- rington, P., Scott, J. (Eds.), The Sage

Handbook of Social Network Analysis. Sage Publications, London, pp. 533–549.[24] White D, Jorion P (1992) Representing and analyzing kinship: a new approach. Current Anthropology 33, 454–462.[25] Roth C, Gargiulo F, Bringé A, Hamberger K (2013). Random alliance networks. Social Networks, 35(3), 394-405.[26] Karloff H, Shirley KE (2013). Maximum entropy summary trees. In Computer Graphics Forum (Vol. 32, No. 3pt1, pp.

71-80). Blackwell Publishing Ltd.[27] Clough JR, Gollings J, Loach TV, Evans TS (2015). Transitive reduction of citation networks. J. Complex Netw. 3(2),

189-203.[28] Karrer B, and Newman M (2009). Random acyclic networks. Physical review letters 102(12)[29] Rosvall M, Bergstrom CT (2010). Mapping change in large networks. PloS one, 5(1), e8694.

Appendix A: Methods

In this document we describe the details of the statistical methods that we used in the paper.

1. Rank comparison

The Jaccard index is a statistical measure used to define the similarity between two sets A and B:

J(A,B) =#(A

⋂B)

#(A⋃B)

, (A1)

where #A is the number of elements in the set A. By its very first definition it is thus a measure based simply onthe content of the set, returning the fraction of elements that are common between two sets. To measure the distancebetween ordered sets, on the other side it is often used the Kendall Tau distance, that is a measure of the number ofpermutations that happen between two rankings of the same set of elements. In our case, the rankings in differenttime windows can contain (and it is usually the case) different sets of elements: a country, for example, can enter theranking only at a certain time, being absent before. Therefore, in our case also the Kendall Tau index cannot be asuitable solution.For this reason we introduced a new index allowing us to compare ordered sets with unequal entries, because it is basedon the Jaccard index we decided to named it the ”extended Jaccard”, J̃ . Given a set A composed by L elements and

given an ordered classification of it, rA, we define the unordered extended set Ã, of cardinality L̃ =∑L

i=1 i = L(L+1)/2where each element of A is repeated as many times as the complementary of its rank. For instance let A = {a, b, c, d},thus L = 4 and assume given the following ranking rA = {1 : a, 2 : c, 3 : d, 4 : b}, that is a is the first element, c thesecond followed by d and finally b, then the unordered extended set à is given by:

à = {a, a, a, a, c, c, c, d, d, b} , (A2)

http://genealogy.math.ndsu.nodak.edu/index.php

14

namely the set where each element i ∈ A appears exactly (L− ri + 1) times, being ri the rank of i ∈ rA. The modifiedJaccard index is simply the Jaccard index applied to the unordered sets generated by this procedure:

J̃(rA, rB) = J(Ã, B̃) (A3)

In such a way we can take into account the importance of a permutation with respect to the ranking of the exchangedelements and also consider new elements in the set because it can be directly applied also in the case where the entriesof the rankings are not the same.

2. Correcting the dates

In the database we observed that 0.32% of the couples mentor-student were characterized by an error in the reporteddates, e.g. year(mentor) > year(student). Moreover 6% of the scientists are not associated at all with a date. Tocorrect these errors and to fill the missing data we perform a stochastic iterative algorithm that modifies the dates inself consistent way using an heuristics method based on the distribution of the time distances between mentor-studentand between students of the same mentor. The same procedure is also used to infer the missing data.

3. Associate a discipline to a scientist

A fraction high as 88% of the scientists presents the information about the title of the thesis. On the contrary only43% of the scholars has an associated disciplinary code (one of the 63 classes of the AMS classification). We thusdeveloped an algorithm able to infer from the title thesis, the most probable discipline. After a preliminary step whereall thesis titles have been translated into english, we used a learning process based on the know thesis classificationsto detect the main keyword in a title and the most probable discipline.

4. Determine the families

As already stated, in some case a student can have more than one mentor, in order to split the genealogical tree intodisjoint families we developed an algorithm, based on the topology of the network, able to assign to each node withmore than one ”parent” its more probable familiar membership. The basic idea of the method is to generate randombinary trees (namely random cutting one of the parents links for each scientist) and to look, for each realization ofthe network, if the detached parent and the son still fall in the same connected family. Repeatedly applying thisprocedure on all the possible random trees, we can extract for each node with multiple parental links, the probabilityfor each link that its removal will not influence the positioning of the removed parent in the same family of the son.

Since this procedure is computationally heavy, we performed a hierarchical grouping of the nodes in super-nodesstructures: the super-nodes are the connected subgraphs of the network with a tree structure, in other words a super-node is obtained merging together all the nodes of the subtree under scrutiny. The super-nodes are then connectedwith weighed links obtained summing the number of links connecting the nodes part of each super-nodes. Becausethe total number of links is highly reduced, the calculation time of the most probable links is now strongly reduced.

Appendix B: The historical patterns

In this section we show some prototypical ”participation” patterns for different countries and disciplines, namely theprofiles of the relative presence in each time window, for the 10 most significant countries (see Fig. 12) and disciplines(see Fig. 13).

From Fig. 12 we can clearly notice the loss of centrality of countries like Italy and Netherlands, and partially France.The rise of US and Russia coincide with the decline of Germany after the Second World War. Finally we can evincethe fast growing trends for emerging countries like China, India and Brazil.In the case of disciplines, one can appreciate from Fig. 13 the growth and fall of quantum theory and the parallelgrowth of group theory that is strongly inherent the physics of the 60’s. Other disciplines like number theory andlogic are quite uniformly distributed in the time.

15

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FIG. 12: Relative abundance profiles for the 10 most significant countries in the database.

Appendix C: The mesoscale network structures

In Fig. 14 we report the structure of the time-aggregated static mesoscale networks. As one can evince from thegraph, the strengths of the country network are strongly heterogeneous, varying from 6500 (USA) to 1 for severalperipheral counties appearing just once in the database. On the other hand the heterogeneity is less marked in thecase of disciplines. As we can observe, both for countries and disciplines, the degree centrality top list is not equivalentto the betweenness. For the countries, the comparison between the top-10 countries in terms of degree and in term ofbetweenness put in evidence the very strategical role of France and est-European countries (Russia, Ukraine, Poland)in connecting information flows between the network.

The difference between the two rankings is still more astonishing for the disciplines where just two disciplines(combinatorics and statistics) appear in the two top-10 lists. This scenario suggests the existence of very well structuredepistemic communities around the top connected disciplines, where the connections between the communities isperformed by peripherals nodes of each community.

In Fig. 15 we represent the time evolution of the fraction of self-loops, in- and out-links, with respect to the totalnumber of links for a given country. As we can observe the emerging countries are characterized by a significantlyhigh in-degree and a very small out-degree. The case of Italy is quite interesting showing a strong exodus during thesecond world war, followed by an inversion around the 50s due to the scientific politics of favoring the coming backof the emigrated scientists. The same maximum of the out-degree due to the second world war can be observed in

16

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Computer science

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f̃(t

)

FIG. 13: Relative abundance profiles for the 10 most representative disciplines in the database.

Germany, while the opposite sign can be noticed in USA and UK. Notice also the following inversion of in and outdegree prevalence for US after the 50s. This inversion is even more pronounced for Russia.

17

Canada

Turkey

BrazilItaly

Serbia

France

CzechRepublic

Ireland

Norway

Israel

Australia

Iran

SaudiArabia

Algeria

Singapore Venezuela

Japan

Austria

NewZealand

Slovenia

ChinaChileHongKong

Germany

Spain

Netherlands

UnitedKingdomJamaica

Denmark

Finland

Morocco

Sweden

Vietnam

Thailand

Switzerland

Russia

Pakistan

MexicoSouthKorea

India

SouthAfrica

Greece

Hungary

UnitedStates

LithuaniaGeorgia

Uruguay

Belarus

Armenia

Portugal

Slovakia

Argentina

Colombia

Croatia

Peru

Scotland

LuxembourgEcuador

Benin

Belgium

Kazakhstan

BosniaHerzegovina

UkrainePoland

IndonesiaCameroon

Mongolia

Bulgaria

Romania

Philippines

Estonia

Egypt

Cuba

Uzbekistan

Tunisia

Taiwan

Bangladesh

TrinidadTobago

Nigeria

Ghana

Tanzania

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Moldova

Malaysia

Sri Lanka

Macao

AzerbaijanUganda

Jordan

Iraq

Macedonia

Latvia

Kenya

Iceland

Yugoslavia

Cyprus

Madagascar

Democratic Republic of the Congo

Senegal

BurkinaFaso

Syria

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Statistics

Combinatorics

Finite differen...

Associative rin...

Optics/ electro...

Measure and int...

Manifolds and c...

Computer science

Special functions

Differential ge...

Mechanics of pa...

General topology

Operator theory

Information and...

Sequences/ seri...

Approximations ...

Dynamical syste...

Systems theory/...

General algebra...

Algebraic geome...

Operations rese...

Integral transf...

Algebraic topol...

Partial differe...

Geophysics

Nonassociative ...

K-theory

Order/ lattices...

Functional anal...

Mathematical lo...

Mathematics edu...

Biology and oth...

Abstract harmon...

Real functions

Game theory/ ec...

Calculus of var...

Geometry

Relativity and ...

Mechanics of de...

Probability the...

Statistical mec...

Number theory

Field theory an...

Functions of a ...

Topological gro...

Category theory...

Potential theory

Convex and disc...

History and bio...

Commutative rin...

Numerical analy...

Integral equati...

Astronomy and a...

Ordinary differ...

General

Group theory an...

Linear and mult...

Fluid mechanics

Quantum Theory

Classical therm...

Fourier analysis

Several complex... Global analysis...

Degree centrality Betweeness centralityUnitedState France

0.0706796538022Germany Germany 0.0649896295984UnitedKingdom UnitedStates 0.0583775408742Canada Russia 0.0528479080743France Canada 0.0496694526785Russia UnitedKingdom 0.0471121731495Switzerland Australia 0.0433638630369Netherlands Ukraine 0.0408384239138Australia Netherlands 0.040036765459Italy Poland 0.0390721722604

Degree centrality Betweeness centralityComputer science History and biography

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Functions of a complex variable Partial differential

equations Number theory

Numerical analysis Biology and other natural sciences Functional analysis Systems theory

Quantum Theory Differential geometry Statistics Combinatorics

Operations research- mathematical

Fourier analysis Group theory and generalizations

Operator theory Combinatorics Statistics

COUNTRIES NETWORK DISCIPLINES NETWORK

FIG. 14: Static network representation for countries and disciplines. The node size is proportional to the total degree.Some centrality measures are shown in the tables. The colors of the nodes represent the community structures of the networkdetermined using the Louvain algorithm.

18

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Germany in degreeGermany out degreeGermany loops

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China in degreeChina out degreeChina loops

1700 1750 1800 1850 1900 1950 2000 2050

time

0.00.10.20.30.40.50.60.70.8

Russia in degreeRussia out degreeRussia loops

(kin

(t),

kou

t(t)

,Nlo

ops(t

))/k

TO

T(t

)

FIG. 15: Fraction of In-links,out-links and self-links for some representative countries in the dataset.

Introduction Dataset and associated networks The mesoscale networks

Figures The genealogical tree and its partitions into families

Results Global statistics Mesoscale networks Network of countries The transition network of disciplines

The genealogical structure

Conclusions ReferencesA Methods1 Rank comparison2 Correcting the dates3 Associate a discipline to a scientist4 Determine the families

B The historical patternsC The mesoscale network structures