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When looking at the paper-is-cited-by-paper binary relation in an undirected manner: Is all research connected to each other or will there be many connected components?

In other words: Is there a way from a, say, computer science paper to a chemistry paper by walking along the citation graph? Let's say we are only looking at publications in established, well-known conferences conferences/journals.

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    It's like playing the Wikipedia game. You go from one article to something really random just by clicking the in-text (anchor) links. If a paper has a citation to one paper that has a lot of citations and so on - by the rules of large numbers you will eventually get to something random.
    – Memj
    Commented Sep 8, 2015 at 15:45
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    It is unlikely that all reearch across all fields forms a single tree. All it takes is one paper which does not share the main tree to establish that and I'd be surprised if there wasn't one such. Then again: what practical difference does that make? What's the real problem you're trying to address?
    – keshlam
    Commented Sep 8, 2015 at 15:48
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    @keshlam: Purely philosophical/interest ;-).
    – knub
    Commented Sep 8, 2015 at 15:55
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    @keshlam You seem to have missed the point I was making. You said, "It is unlikely that all research across all fields forms a single tree." That's not a matter of probability: it's a matter of clear fact because there is nothing even remotely tree-like about the citation graph. The situation where A cites B and C, and B and C both cite D is extremely common. That is not a tree.
    – David Richerby
    Commented Sep 9, 2015 at 6:28
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    @keshlam: The question explicitly says "undirected", which already implies no-one's looking for a tree. As such, all it takes is one instance which neither cites anything else, nor is cited by anything else. While this might exist, it might be difficult to identify it while still establishing it to belong to the considered class of documents in the first place.
    – O. R. Mapper
    Commented Sep 9, 2015 at 8:27

5 Answers 5

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When looking at the paper-is-cited-by-paper binary relation in an undirected manner: Is all research connected to each other or will there be clusters?

First, it's possible to have an unconnected paper. Unlikely, but possible, to write something that has no citations and is never cited. This is clearly an edge case, but you did say all.

Second, all research can connect, but there can still be clusters. The two are not mutually exclusive - fields will cluster, presumably, but there will then be some links between fields. My suspicion, from personal experience, is that these papers will be methodological in focus primarily.

In other words: Is there a way from a, say, computer science paper to a chemistry paper by walking along the citation graph?

Third, yes, you could get from a CS paper to a Chemistry paper. I know this because I can trace that tree in my own work, and the work my work cites. The path isn't even all that convoluted or exciting.

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    I understand cluster in the strict sense, i.e. a cluster consists of everything that is connected. In that case, the two are mutually exclusive.
    – knub
    Commented Sep 8, 2015 at 20:23
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    @knub in the strict sense, that's a clique, not a cluster.
    – OrangeDog
    Commented Sep 9, 2015 at 10:42
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    @OrangeDog What knub describes is a connected component. A clique would be a subset of papers that all cite each other directly, ie. one step.
    – Peter Bloem
    Commented Sep 9, 2015 at 10:50
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    A clique is fully connected. The term would be connected component, which I just edited in the question.
    – knub
    Commented Sep 9, 2015 at 10:51
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    Doh. Well at least we all agree it's not a cluster.
    – OrangeDog
    Commented Sep 9, 2015 at 10:54
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Most likely not (but almost). Apart from the specific papers without references that people have already mentioned, we can look at crawls of large collections of papers. Take, for example, this dataset of all papers in the theoretical high-energy physics section of ArXiv:

Nodes 27770
Nodes in largest WCC 27400 (0.987)

The largest weakly connected component (WCC) is what you're after: the largest subset of papers that are connected to each other by a path of citations (ignoring direction). While the largest WCC is almost as big as the entire graph, there are papers outside it. Usually, with graph like this, these form little clusters of their own.

For a more cross-domain dataset, consider the citeseer graph, again a small proportion of papers, outside the largest WCC.

Now, of course, these datasets don't contain all of academia, and adding more papers would mean connecting some islands to the WCC, but I'd say adding more papers also adds new little islands. No matter what rule you use to decide which papers count and which don't, I think you always end up with disconnected islands.

Of course, if your question is whether any randomly chosen paper in domain A is likely to be connected with a random paper in domain B, the answer is yes. There will be a large WCC encompassing all domains, and a few tiny islands. I've seen visualizations to this effect, but unfortunately I can't find them at the moment.

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    Interestingly (by the citeseer stats), for any two papers that are both in the WCC, you have an expected shortest path of 6.35 citations. So while there may be disconnected papers, most of academia is pretty well connected.
    – Peter Bloem
    Commented Sep 9, 2015 at 0:18
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    Just as a comment, for me the best network graph visualization of the arxiv database is this: paperscape.org. You can clearly see clusters and clusters close to each other resemble very well the connection between research fields. If you look for famous big physicists you will find them connected everywhere.
    – Santiago
    Commented Sep 9, 2015 at 10:52
  • @Santi The best network graph visualization of the physics subset of the arxiv database, please.
    – JeffE
    Commented Sep 9, 2015 at 11:59
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    @JeffE it is actually the whole arxiv, which includes also quantitative finance, quantitative biology, math and computer science. Of course, as you can see, they are just very faint clusters in the lower left corner, because physics dominates the landscape.
    – Santiago
    Commented Sep 9, 2015 at 12:17
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In short, no. For starters, and perhaps surprisingly to many folks, a large number of papers in the scholarly literature have neither incoming nor outgoing citations. These obviously won't be connected to anything else. Let's throw these out and look only that set of papers that do cite journal articles or do receive citations from journal articles. Even then, not all articles are connection by citation relationships.

Depending on which data set one uses, the fraction of papers in the giant component of a citation graph will vary, but in my experience typically 90-95% of papers will be in this giant component and the rest will be singletons or members of small connected components.

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    Note that the OP explicitly asks for undirected paths. So even if one paper has no citations at all, it will still be connected to the larger academic world if it is itself cited.
    – Stephan Kolassa
    Commented Sep 8, 2015 at 16:12
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    I should have been clearer in my answer. I propose throwing out all articles with no outgoing citations and no incoming citations, and then looking to see if the remainder are all members of single connected component. One will inevitably find that they are not.
    – Corvus
    Commented Sep 8, 2015 at 16:55
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    @Corvus Can you give an example of a paper that is in not in the giant component and a paper that is in the giant component?
    – JiK
    Commented Sep 8, 2015 at 19:29
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    @JiK hey, that is a trap! As soon as he does it, we have both articles connected here in this page and then both going to the giant component.
    – arivero
    Commented Sep 9, 2015 at 2:29
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    @arivero: Indeed - this highlights how it makes little sense to examine this question without establishing what set of documents is considered (e.g. only peer-reviewed publications). Otherwise, we could, for instance, argue that a large number of otherwise disconnected papers are indeed connected ... via appearing in the Google Scholar database, which forms one huge document full of outgoing citations.
    – O. R. Mapper
    Commented Sep 9, 2015 at 14:15
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I don't see how this can be answered short of actually generating the relevant graph and analyzing it. All answers here are simply conjecture. Very reasonable conjecture, based on valid assumptions and reaching conclusions that I find very plausible, but conjecture nonetheless.

I would expect most papers to be connected by a (sometimes very long) path, but not all. However, there is no way of demonstrating this unless we have the graph. I'd be interested to check this, by the way. If anyone has an idea of how to get the relevant nodes and edges, I'd be happy to give it a go.

Until then, I'm afraid this question has to remain unanswered.

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I'm sure, each paper is connected to all others. The question is, how long is this citation way from one paper to the other. This is the sort of the 5 handshakes rule thing.

I bet, one can even prove that all academic texts are interconnected through the works of, for example, Sigmund Freud.

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  • This is a bet I would be very happy to take! It would not take long to provide a counter example given any particular bibliometric dataset, given that you can find the connected components of a graph in linear time (Hopcroft, J.; Tarjan, R. (1973). "Algorithm 447: efficient algorithms for graph manipulation". Communications of the ACM 16 (6): 372–378.) and that you will never find everything contained in a single giant component.
    – Corvus
    Commented Sep 8, 2015 at 16:59
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    @Corvus, then, take it. A cool reference does not mean you are automatically right
    – ikashnitsky
    Commented Sep 8, 2015 at 17:11
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    You could in principle publish a completely spurious paper that cites one paper in the giant component plus everything you want to pull into it. Therefore any example Corvus gives is liable to be nobbled by someone who wants it to be wrong, and it is impossible to publish a paper that actually cites an example of each kind of paper ;-)
    – Steve Jessop
    Commented Sep 8, 2015 at 21:02
  • @stevejessop "A reductionist approach to unifying the citation graph", J. Bibliometrics (2017). I can see it now...
    – Andrew is gone
    Commented Sep 9, 2015 at 7:43
  • @SteveJessop: I hope my question is not that important, that someone tries to publish such a bogus paper to an established conference, and it gets accepted.
    – knub
    Commented Sep 9, 2015 at 17:22

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