Model for the number of citations an article receives over time

Question

Is there a standard model for the number of citations an article will receive over time? If not, is there a good source for data on citation counts for articles divided up over time? Furthermore, does this vary by field

It seems plausible to me that the number of citations decays approximately exponentially, with the rate of decay depending on the article, but I have empirical evidence for this conjecture.

Such a model would be useful because a few years after an article was published, one could use the MLE method to estimate the number of citations the article would receive over any time period. This would allow us to count citations in a way that does not bias older articles, one of the chief problems with straight citation counts. Additionally, it would give us a better idea of the inaccuracies of metrics like impact factor.

Answer 1

I imagine you'd have an initial publication lag where citations might be lower for example in year 1 than in year 2 and that then aggregate citation counts over time would have to be monotonically decelerating. You'd also have substantial variability in terms of individual articles. For example, I've heard of articles which accelerate in their citation count over time as they become the standard citation for a particular topic.

Independent of overall number of citations received, journals and fields differ dramatically in the rate at which citations accumulate. A good starting point is to look at "citation half life" (i.e., the median time it takes a journal or field to acquire half of its total citations). If you can access ISI you can examine how different fields and journals differ in citation half life.

For example, I just had a quick look at a few fields from ISI 2012 which yielded the following data:

Field                       Citation Half-Life (in years)
PSYCHIATRY                  7.6
ASTRONOMY & ASTROPHYSICS    6.9
MATHEMATICS                 >10.0
PSYCHOLOGY                  >10.0

So clearly psychiatry and astronomy tend to cite more recent articles, whereas mathematics and psychology often cite older articles.

The implication is that impact factors (which are based on the last 2 or possibly 5 years post publication) should be higher in psychiatry/astronomy than in mathematics/psychology. It also means that early career researchers in psychiatry/astronomy should accumulate a better h-index more quickly than researchers in mathematics/psychology, all else being equal.

Answer 2

I think the model for all kinds of papers is very unlikely to exist. Certainly, the trajectory will be different for very strong and fundamental contributions vs. small incremental improvements, that often become more or less irrelevant in a few years. Further, looking through Google Scholar data (already mentioned by Anonymous Mathematician), I would hypothesize that there is also a difference between fields. In my rather hype-driven research field (services / software engineering), citations often seem to roughly follow a bell curve:

Example 1

Example 2

Example 3

These papers, if they are timely, generally need one or two years to pick up steam, then there are a few years with lots of citations, after which interest in the presented ideas dies down again.

More fundamental work often seems to have a less clear trajectory:

Example 4

And then of course you have publications which are just "weird":

Example 5

Note that all examples I have provide have a not-too-small number of citations on them. I would say for publications < 50 citations on Google Scholar (i.e., most of them), all bets go out through the window anyway. Fluctuations are too large for any sort of statement over the shape of their trajectory.

Answer 3

Google Scholar will plot the number of citations over time for papers. I don't know whether it will do so for arbitrary papers, but it will for any paper listed on an author's page (e.g., Terry Tao).

I haven't studied the data systematically, but some things are clear from browsing:

1. Citation statistics for individual papers can be incredibly noisy. For example, a paper may have 5 citations one year, 15 the next, then 7, 2, 12, etc., with no obvious pattern or explanation. Furthermore, different papers written by the same person on similar topics and with similar citation rates sometimes show visibly different levels of noise, and I don't know why. For papers with particularly high citation counts (e.g., 50+ citations per year), noise is not as big a factor, but such papers are uncommon.

2. Different papers can take radically different trajectories. For example, some of my papers have a consistent growth in their yearly citation counts over ten or more years. Others seem relatively stable, probably decreasing a little over time but with yearly fluctuations that are substantially larger than the decrease. Some have fallen completely off the map and only get cited occasionally (perhaps a Poisson process). These different scenarios are correlated with how good I consider the paper, but only loosely, and I think the trajectory would be tough to predict from the first few years of citation counts.