Tag Archives: Network Analysis

Ghost Edges and References

Snippet from the Acta Eruditorum, June 1711 issue, p. 269.

Friday n° 29, April 25th, 2019

If being remembered or forgotten is a function of reference frequency, of circulating information, an obvious conclusion seems to be that if you want to be remembered, you yourself should start circulating information lest you get forgotten. In scholarly contexts, this basically means spreading the word about what one is doing or has produced. This might in turn trigger references to you and your publications, discoveries, theories or other achievements which in turn might provide starting points for other references. Self-advertisement, for this and other, more directly visible reasons, has been and is part and parcel of academic communication. In network analysis, the reasons why such attempts at self-promotion were successful or not is often explained or even predicted by the structural features of the individual’s networks.

Shadowy networks

But what about the networks we are only partially able to reconstruct because of source loss? In some cases, I know that there were connections but can’t say much more about depth and nature of these connections because the source documents necessary to judge this have been lost. Any network reconstructed under such circumstances will be distorted, because the parts of it traces of which have survived as documents will be privileged over those parts where this is not the case. So what to do with the parts of the network which can only be traced as shadows, as ghosts of nodes and edges that once were?

Self-advertisement, done successfully

Let me start with a small piece of circumstantial evidence with throws one such ghost edge in my network of letters into sharper relief. In June 1711, the Acta Eruditorum published a small piece of seven pages titled “On the manuscript commentary of Blessed Jerome which exists in the library of Marcus Meibom in Amsterdam”.[1] Most of the text was composed of excerpts from the manuscript in question, but ahead of this the editorial board of the Acta Eruditorum lost a few words on how they got the paper in form of a short introduction:

“Lately the illustrious Adriaan Reland whom we already have given honourable mention in these Acta more than once has sent us some excerpts from a manuscript commentary on Job by [St.] Jerome, whishing them to be included in our Acta with the intention that scholars may by this specimen pass judgment on whether it is a genuine work by Jerome or not.”[2]  

[Mencke]: De b[eati] Hieronymi commentario m[anu]s[cri]pto in Jobum, AE 30, June 1711, p. 269.

This passage now not only fits in quite well with my overall framework of references and their valorisation in scholarly circles. It also explicitly states what I – based on studies such as that of Huub Laeven on the networks of Otto Mencke (1644–1707) and his son Johann Burchard Mencke (1674–1732), the successive editors of the Acta Eruditorum[3] – already had suspected: that either Mencke sr. or jr., or both, were in direct correspondence with Reland. Until now I just had no tangible evidence for such a connection, as the letters of all parties involved, Otto Mencke, Johann Burchard Mencke, and Adriaan Reland, have only fragmentarily survived. The letter concerning the codex containing the work in question here, the commentary on the Old Testament book of Job supposedly written by St. Jerome, the church father, does not exist anymore (at least not to my knowledge). But the easiest way to account for the passage just quoted is to assume that it indeed did exist.

Ghost edges

As glad as I am to finally have made sure that this particular ghost edge really existed, I am nevertheless aware that the basic problem context underlying this discovery has just become a bit more serious at the same time.

For on the one hand some of his surviving letters already point to Reland being a conscious and very active self-promoter who had a keen eye and good hand in picking opportunities to distribute his publications, and to this another distribution channel – that of the Acta Eruditorum – has just been added now. As there are quite a few “honourable mentions” of Reland by the Acta Eruditorum, like Johann Burchard Mencke wrote (or let write: as leading editor he has to be held responsible for anonymous texts in his journal), this prompts the question of how they were caused in the first place. As there are at least two letters by Reland to Jean-Paul Bignon (1662–1743), editor of the Journal des Savants in Paris, which deal with Reland sending Bignon his publications for distributing them amongst his French connections including review copies,[4] there is no reason to assume that something similar might not have taken place in his correspondence with the leading German scholarly journal as well as with its French model. This might seem to be supported by the rising reference frequency in the Acta Eruditorum concerning Reland between 1701 and 1711:

(Only the text pieces containing references to Reland are counted, not the total of references.)

The upward trend visible here might be taken as just the depiction of a young scholar’s rise of fame while making his way through academia. Reland had been awarded his first professorial post in Harderwijk in 1698/99, only to move to a more prestigious chair at Utrecht in 1700/01, publishing continuously. Or it might be an illustration of a correspondence successfully feeding the editorial board at Leipzig with relevant news and thus ensuring continuous reference to oneself. As I cannot say much more about the ghost edge than that it existed, but not for how long and how intensively it was used, the question has to remain open for now.

And on the other hand, the fragmentary state of the Reland correspondence has by now turned up quite a handful of such connections where there are either indications of direct correspondence and no surviving letters or one or two letters surviving, indicative of a communication channel which must originally have accommodated many letters more. As I already pointed out in one of my earlier posts, the correspondence between Mathurin Veyssière de la Croze (1661–1739) and Reland is only documented by three letters, all of which are no longer extant in the original. Given the close intersection of research interests between the two of them it is highly unlikely that there were not much more originally; but how could this be translated into a meaningful part of a network? The same is true for Johann Baptist Ott (1661–1744) the communication between whom and Reland is only evidenced by one printed letter in Reland’s second treatise on the Samaritan coins, as I pointed out here. It is also true for Marcus Meibom (1630–1711), the scholar whose manuscript was the reason for the note in the Acta Eruditorum which caused me to write this post in the first place. It is even true for another of my protagonists, Johannes Braun, as there are a few letters between him and Reland still extant but the bulk of Braun’s correspondence is also lost. This is by far no complete list, but I am determined to draw one up as far as this will be possible.  

The Chicken-and-Egg of Loss, Forgetting, and References

The question posed by ghost edges is of course how they relate to forgetting. They are clearly indicative of structural forgetting taking place, but in which way? Their presence could be seen as a natural effect of processes of structural forgetting: As someone fades from structural remembrance, his papers or letters become devalued and thus more likely to be sold off, discarded, or altogether lost. But their presence could also be the cause rather than the effect of becoming forgotten: As the papers and letters of someone become sold off, were discarded, or otherwise lost, materials are removed from the archival record which might have triggered new references had they still been there, which in turn leads to a drop in reference frequency and thus to structural forgetting. This is a new variant of the ages-old chicken-and-egg problem, so I have go to searching for additional factor which might help me figuring out if a particular shadowy part of the network is a ghost edge chicken or a ghost edge egg.

[1] [Johann Burchard Mencke]: De b[eati] Hieronymi commentario m[anu]s[cri]pto in Jobum, qui Amstelodami in Bibliotheca Marci Meibomii exstat, in: Acta Eruditorum 30, June 1711, pp. 269-275.

[2] Ibid, p. 269: „Misit nuper ad nos Vir Cl. Hadrianus Relandus, cujus non semel honorificam in his Actis fecimus mentionem, Excerpta quaedam ex Commentario MS. Hieronymi in Jobum, eaque Actis nostris inseri cupivit eo consilio, ut eruditis hoc e specimine iudicandi copia fieret, sitne is genuinus Hieronymi foetus nec ne.”

[3] Cf. Huub Laeven: Otto Mencke (1644–1707): The Outlines of his Network of Correspondents, in: C. Berkvens-Stevelinck, H. Bots, J. Häseler (eds.): Les grands intermédiaires culturels de la république des lettres. Études de réseaux de correspondances du XVIe au XVIIIe siècles, Paris: Champion 2005, pp. 229–256 ; —: “Dies ist wol ohne Streit die größte unter denen Holländischen Public-Bibliotheken“. Johann Burkhard Mencke’s bezoek aan Leiden in 1698, in: Omslag. Bulletin van de Universiteitsbibliotheek Leiden en het Scaliger Instituut 4, 1/2006, pp. 1–3.

[4] UB Leiden BPL 885 – 052 (Rabault-Risseeuw): Adriaan Reland to Jean-Paul Bignon, Utrecht 13 June 1714 (19th century copy), and KB Den Haag 72 D 37, 11 A: Adriaan Reland to Jean-Paul Bignon, 23 June 1714.

And the winner is…

Snippet of the network visualization for the 502 publication sample described in this post; mind the prominent position of Reland’s “Palaestina Illustrata” (big green dot left on top) and the unconnected component in the lower right corner.

… Adrien Reland’s 1714 Palaestina ex monumentis veteribus illustrata!

And so the winner is me, too, for I guessed it correctly.

A two weeks’ post

But one thing after the other. First, the date: this is my blog post for

Saturday, March 9th, 2019, for Friday no. 22;

I seem to acquire a bad habit of posting on Saturdays rather than Fridays. For today, my excuses are, in chronological order, a) I had to help out at my son’s school, fixing some shelves; b) I had to figure out how to clear some financial matters regarding how to pay my salary from this project in a way that some of it actually ends up with me in the end; c) my youngest daughter stubbornly refused to go to sleep this evening. Obviously I am not very good at scheduling my home office work hours, for although I started out at 8:00 in the morning, by 21:30 I had got down to a meagre seven hours of working time. And d) I made a tiny configuring mistake, messed up my calculations, and had to start all over again, ruining some two hours of work. I suspect I better had not started calculating only at 21:30, but a), b), and c) forced me to.

Going for some metrics, step 1: Gathering data

But the calculations are crucial, because that’s what I originally intended to show you. As I announced in my last post, I now have finished three-quarters of my projected sample of scholarly journals, which has taken me two weeks to complete, clean, and analyse now.

That is to say, I have gone through the digitized versions of the Journal des Savants, the Philosophical Transactions, and the Maandelyke uittreksels, of Boekzaal der geleerde waerelt (more info here), between 1700 and 1799. I have added all issues of these journals referencing one of my four protagonists in any way to my database. In numbers, this means 117 issues of the Journal des Savants, 9 of the Philosophical Transactions (yes, only nine, but I have already argued they are somewhat special here), and 89 of the Maandelyke uittreksels – or 205 journal issues in total. In going through these issues I have examined each page on which at least one of my protagonists was referenced, identified all other scholars and all publications on that page, and stored these data, too, as a basis for co-citation analysis to identify perceived epistemic communities and their changes over time. This netted me another 287 quoted publications, and another 739 scholars referred to and/or connected to quoted publications (as authors, editors, translators etc.). I must confess that I am a bit proud of having managed to completely identify close to 690 of these, or more than 90%. But today everything will be about publications, so the really interesting figure is that of 502 publications in this sample.

Going for some metrics, step 2: Setting a framework for analysis

The problem that I now got was how to deal with these data. Plain visualizing did not work out anymore to really map out the intriguing details. So it needed to be done the rough way, by calculating metrics for the sample, hoping for something interesting to turn up. The visualizations did help me, in the way of pointing out which ways to structure the relations to be investigated would likely not yield good results. I first thought I might try something like mapping ‘shared contributors’, that is, connecting publications by way of the same scholars contributing to them (as authors, editors, or translators). This turned out to be pretty useless because it only favoured large edited collections who assembled texts from many different authors. And that’s the reason why this post is not about persons but publications only; the real co-citation analysis still has to wait for a bit. So it seemed better to bypass persons for the time being and only to link publications to publications, and I did so by way of quotations, that is, two publications are seen as connected if at least one of them quotes the other. That brought some interesting results about.

Going for some metrics, step 3: Finally going for metrics!

With the framework set up to network this publication sample, it was high time (half past nine already!) to start calculating metrics. To facilitate calculation, I started with assuming the network edges to be undirected, that is, a link from publication A via a quote in A to publication B works both ways, from A to B and from B to A. I know that this might be a questionable simplification; but as this may be tested by comparison, I will set this matter aside for a rainy day when I don’t have any idea what to write in this blog for the week to have the opportunity to at least redo these calculations for directed edges. (And as I once started tracking time in this post, it is now 4:00 in the morning and high time that I get to bed to have at least some sleep. I’ll continue around noon.) Moreover, as the structure of connections within what one takes to be a network depends on what the question is, and I am interesting in tracing perceived epistemic communities here – that is, publications seen as belonging to shared domains of knowledge – edge directionality is less important than it might be for other questions to ask. (This is a piece of midnight reasoning, but it still looks sound now at 10:30 on the day after.) I could as well also have collapsed the journals into edges connecting the rest of the publications to more explicitly claim this, but I wanted to also get an impression of the position of the individual journal issues, and so I kept them as nodes. Admittedly, this model does have the drawback that relying on quotations only provides only a quite simplified picture of the interconnections structuring domains of knowledge, but it should serve to give a good first impression of the sample’s general structure. Having sample and model settled, I used the opportunity to test Nodegoat’s analysis functions and calculated some metrics.

A lot of numbers

To keep the scores comparable across different metrics, I chose to normalize them where possible. A short look at the visualization (see header for larger picture) made clear what was to be expected: the network is disconnected. Not all publications are referred to as situated in overlapping domains of knowledge; some are distinctly separate from the rest. This is important here because it directly impacts the calculations of the closeness metric. Closeness serves to indicate how close any given node is to all other nodes in the network. In mapping out shared domains of knowledge this might be quite interesting, as nodes with low closeness scores might be supposed to partake of many fields at once being referenced in differing contexts, and thus be important to look at. But closeness measures path lengths to calculate the average distance from node X to all other nodes, and this does not work in disconnected networks (because two nodes between which there is no path would be at undefined = infinite distance from each other). Nodegoat provides a workaround to that by substituting a maximum distance for path length between unconnected nodes, this maximum being equal to the total number of nodes in the network.[1]

First: Betweenness

The first measure I calculated was betweenness[2] (because of the problems with closeness). Betweenness works on disconnected networks and thus was the obvious first choice. Roughly put, betweenness serves to indicate whether a certain node may be considered a gatekeeper, that is, if it is situated at a vital connection point in the network. In assuming an undirected graph, for mapping shared domains of knowledge this resembles closeness: it should point to publications situated at the intersection of several domains, and thus of potential importance for each of these. I must qualify this as ‘potential importance’ because only being situated between different fields of knowledge does not equal making important contributions to any of these fields; this will have to be cross-checked with other metrics, then.

Top 10 Betweenness

These are the top ten results in terms of betweenness for the whole of the network:

And those are the respective results for the top-scoring publications of/connected to:

Adrien Reland, by Betweenness

Eusèbe Renaudot, by Betweenness

Johannes Braun, by Betweenness

Thomas Gale, by Betweenness

What becomes directly apparent is that there are a lot Reland’s publications in the network, not only on vital connection points but also in total. In fact, 37 out of 287 non-journal publications have Reland as their main author/editor, or 12,9 %; the number does matter because I only added publications to this sample when coming across a quotation within the sample. Compared to 2.8 % for Eusebè Renaudot (8 publications), 1.7 % for Johannes Braun (5 publications), and 1.4 % for Thomas Gale (4 publications) this looks even more impressive. But a large output as such does not tell anything about the quality or the reception of that output. There also is one publication by Eusebè Renaudot among the top ten in betweenness, too.

Second: Closeness

As already said, the closeness scores for this sample are to be taken with a grain of salt because of the disconnectedness workaround implemented in Nodegoat. But the prediction I theoretically made when thinking about betweenness in this network – that it would line up quite closely with closeness because pointing to structurally similar positions within the network – did come true in looking at the results for closeness, so I am tempted to regard these results as usefull still. Important: The first score in the “A” for Analysis column is always the one discussed!

Top 10 by Closeness

The top ten publications of the network in terms of closeness scores are:

This again underscores the relevance of Reland’s publications within the network as a whole, and especially of his Palaestina Illustrata.

And those are the respective results for the top-scoring publications of/connected to:

Adrien Reland, by Closeness

Eusèbe Renaudot, by Closeness

Johannes Braun, by Closeness

Thomas Gale, by Closeness

Third: Degree

In undirected graphs such as this, Degree just counts the number of connections any given node in the network has. To be precise, Nodegoat counts the number of all edges starting from and ending at any given node, allowing for duplicates edges, thus producing high scores on average.[3] For an analysis of shared domains of knowledge as proposed here Degree should be relevant as a corrective. Whereas Betweenness and Closeness both point to structural positions of publications in regard to their location between domains of knowledge, Degree points to the importance of a given publication at this structural position. Or, to put it more precisely, to the attention generated by the publication in question, as I am focusing on quotations made by scholarly journals to this publication; and many such quotations do not necessarily point to the scholarly but in any case to the public impact a work made.

Top 10 by Degree

Looking at the Degree scores from this perspective makes clear that while Reland’s publications might share in more domains of knowledge, within their respective domains Renaudot’s publications obviously attracted comparable attention. And looking at the respective results for the top-scoring publications of/connected to Johannes Braun and Thomas Gale in comparison makes clear that they were not only tied more closely to special domains of knowledge but also attracted less attention by the journals in question, perhaps because of this more stringent specialisation. So here are the Degree scores for publications divided by individuals:

Adrien Reland, by Degree

Eusèbe Renaudot, by Degree

Johannes Braun, by Degree

Thomas Gale, by Degree

A cautionary note: It does not pay to put too much trust in Degree scores, because they tend to push publications situated within certain reference patterns. Consider the example of Mathurin Veyssière de la Croze’s Lexicon aegyptiaco-latinum, the Coptic dictionary edited and published by Karl Gottfried Woide in 1775 which has already featured in one of my posts.

With a Degree of 33 it scores shortly below a top 10 place in Degree which, following the reasoning laid out above, should indicate its importance for its particular domain(s) of knowledge – it attracted a lot of attention, because it has a lot of quotes. But all of these quotes are in fact derived from one singe journal issue, the June 1774 issue, part one, of the Journal des Savants, as becomes visible from the cross-references section of its database entry.  

And there it assembles all of these quotes because it features in many different respects in the announcement written by Woide himself to highlight his upcoming publication, which makes the reference section of this piece look a bit monothematic:

Snippet from the references contained in Woide’s publication announcement

So while Degree may be taken as an indicator of importance within a given field by capturing public attention, this is easily manipulated, and has to be cross-checked against another metric for validation.

Fourth: Pagerank

Nodegoat allows for calculating Pagerank scores, following the original Google algorithm. As Pagerank was originally invented to determine the relative importance of information sources (in this case, websites) in a network through which a user might randomly move, this model might also be applied to domains of knowledge in a web of publications, treating quotations as paper-borne hyperlinks. Random movement through the graph is facilitated by its undirectedness, so please keep in mind that modelling it as undirected may be a questionable decision, and don’t trust these metric too much. The good thing about Pagerank is that allows for determining both the relative importance of quoted publications and of quoting journals in this particular configuration, so I would like to use it to counterbalance the shortcomings of Degree pointed out above. In running Pagerank, Woide/la Croze’s dictionary disappears from the leading places all of a sudden, so this seems to work. The top publications according to Pagerank thus are:

Top 10 by Pagerank  

Surprise, surprise: The main hubs for distributing quotes – and thus sorting domains of knowledge – are journals! Makes me almost wonder why I chose to go by them in the first place… But more interesting in here is that there one publication from the 287 quoted works in the sample which managed to get a place in the top 10, and that is – surprise again! – Reland’s Palaestina Illustrata. This in turn is due by it being quoted by a number of journal issues (from all three journal series) which are themselves important quotation hubs, as becomes evident when looking at the cross-references to Palaestina Illustrata, that is, the publications in the sample linking to it.

This looks as if Pagerank indeed might be a good tool to determine the importance of a publication at its respective structural position, at best in combination with Degree (and Palaestina Illustrata scores high on both). So what about the top scores for the rest of the field?

Adrien Reland, by Pagerank

Eusèbe Renaudot, by Pagerank

Johannes Braun, by Pagerank

Thomas Gale, by Pagerank

Combining metrics

Well, that was a lot of tables, numbers, and scores. Well done! Only a few more to go. Now the last thing to do is to find a way to purposefully combine the different metrics results assembled so far to draw something relating to my larger research question from it. To do so, I ventured for a first try to identify the top 5 publications of each of my protagonists by comparison of the results I presented above. This yielded the following table[4]:

Top 5 by: Betweenness Closeness Pagerank Degree
Reland Palaestina ex monumentis veteribus illustrata Palaestina ex monumentis veteribus illustrata Palaestina ex monumentis veteribus illustrata Palaestina ex monumentis veteribus illustrata
Reland De nummis veteris Hebraeorum De nummis veteris Hebraeorum Antiquitates sacrae vet. Hebr. (4th ed.) De nummis veteris Hebraeorum
Reland De religione mohammedica De religione mohammedica Antiquitates sacrae veterum Hebraeorum De religione mohammedica
Reland Antiquitates sacrae veterum Hebraeorum Antiquitates sacrae veterum Hebraeorum Verhandeling van de godsdienst Antiquitates sacrae veterum Hebraeorum
Reland Encheiridion studiosi Dissertationum Miscellanearum Oratio de galli cantu Hierosolymae Decas exercitationum […] nomine Jehovae
Braun Commentarius in epistolam ad Hebraeos Commentarius in epistolam ad Hebraeos Leere der verbonden (4th ed.) Doctrina foederum
Braun Leere der verbonden (4th ed.) Vestitus sacerdotum Hebraeorum Avertissement necessaire aux eglises Leere der verbonden (4th ed.)
Braun Avertissement necessaire aux eglises Doctrina foederum Commentarius in epistolam ad Hebraeos Commentarius in epistolam ad Hebraeos
Braun Doctrina foederum Avertissement necessaire aux eglises Vestitus sacerdotum Hebraeorum Avertissement necessaire aux eglises
Braun Vestitus sacerdotum Hebraeorum Leere der verbonden (4th ed.) Doctrina foederum Vestitus sacerdotum Hebraeorum
Renaudot Historia patriarcharum Alexandrinorum Historia patriarcharum Alexandrinorum Liturgiarum Orientalium Collectio Liturgiarum Orientalium Collectio
Renaudot La perpetuité de la foy, 4 Liturgiarum Orientalium Collectio La perpetuité de la foy, 4 Defense de l'”histoire des patriaches”
Renaudot Liturgiarum Orientalium Collectio La perpetuité de la foy, 4 Anciennes relationes des Indes Anciennes relationes des Indes
Renaudot La perpetuité de la foy, 5 La perpetuité de la foy, 5 La perpetuité de la foy, 5 La perpetuité de la foy, 4
Renaudot Anciennes relationes des Indes Anciennes relationes des Indes Historia patriarcharum Alexandrinorum Genadii patriarchi homiliae
Gale Antonini iter Britanniarum commentarius Antonini iter Britanniarum commentarius Antonini iter Britanniarum commentarius Antonini iter Britanniarum commentarius
Gale Jamblichii de mysteriis liber Jamblichii de mysteriis liber Historiae poeticae scriptores antiqui Historiae Anglicanae scriptores V.2
Gale Historiae Anglicanae scriptores V.2 Historiae poeticae scriptores antiqui Historiae Anglicanae scriptores V.2 Historiae poeticae scriptores antiqui
Gale Historiae poeticae scriptores antiqui Historiae Anglicanae scriptores V.2 Jamblichii de mysteriis liber Jamblichii de mysteriis liber

Creating a Shortest Path Matrix

To fuse this into something more informative I thought I might utilize a special function of Nodegoat’s, and that is its ability to calculate shortest paths between given selections of nodes. In this case, this meant that I first settled on a matrix of four times four publications: The top 4 publications after comparing the scores of all metrics for each of my four protagonists. I chose to use the top four because only four of Gale’s publications made it into the sample, and this provides better comparability then. Now I used the Shortest Path function to calculate the length of the shortest path through this network from each of these publications to each other, and set the results down in form of a matrix.[5] It looks like this.


What does this tell me now? Well, first of all it highlights the isolated position of some of these publications, which obviously do not share domains of knowledge with others; this is true for the five marked with grey lines which have no connections to the rest of the matrix. It moreover points to the broader diversity of Reland’s publications compared to the rest of the network: His four publications all have connections within the matrix, while for Renaudot one publication does not and for Braun and Gale two publications each. Reland’s and Renaudot’s oeuvres and positions are similar in that both are internally coherent – the smallest paths to those of their own publications connected to the rest of the matrix are two each – and well-connected: their publications are not only connected to more of the rest of the matrix, they are also connected by shorter paths, making their works appear to be more central. Braun and Gale on the other hand are similar in having less many publications connected to the rest of the matrix, and in those publications being situated in strongly differing fields, as indicated by their larger internal distance from each other compared to Reland and Renaudot. And both of them are on the outskirts of the network rather than in the centre as indicated by the long shortest paths to other publications in the matrix.

This buttresses the claims I have brought forward based on other impressions already: That Reland and Renaudot form a comparable pair of actors, as Gale and Braun do; and that Reland seems to be the most versatile of all four, which might be the reason why he was the last of the four to get forgotten. And, as I already suspected last week, that his Palaestina Illustrata might be more relevant for this process than his other works, even if they are better known today (as his 1705 De religione mohammedica for instance).

But this has been a fairly static picture of a phenomenon spanning one century, so the next task will be to dynamize it. Work to do!

[1] So in looking at the closeness scores in the following, always keep in mind that the maximum path length taken into consideration is 502.

[2] With duplicated edges taken into account and weighted according to distance, that is, duplicates add edge length.

[3] And Nodegoat does not automatically normalize Degree centrality. But this is an easy operation: If you want to do so, just divide any absolute Degree score by the total number of edges in the network, in this case, 2577.

[4] Publications marked in red only appear once in the table and are therefore discounted from further  comparison.

[5] Please keep in mind that shortest paths are elongated in my setting because two quoted publications A and B are (almost) always connected via a journal in between, so the usual shortest path possible between A and B is two links (A –link one– Journal  –link two– B).