Following my post on citations in academic journals, I wanted to go one step further in the understanding of the dynamic of citations. So here, the dataset looks like that: for each article, we have the name of the journal, the year of publication (also the title of the article, but here we do not use it, as well as the authors), and more interesting, the number of citations in journals (any kind of academic journal) published in 1996, 1997, …, 2011. Of course, articles published in 1999 might have their first citation only starting in 1999.

base[1000:1002,]
     Publication.Year
7188             1999
7191             1999
7195             1999
     Document.Title
7188 Sequential inspection 
7191 On equitable resource approach
7195 Method for strategic  
                                        Authors     ISSN       Journal.Title
7188                         Yao D.D., Zheng S. 0030364X Operations Research
7191                                    Luss H. 0030364X Operations Research
7195 Seshadri S., Khanna A., Harche F., Wyle R. 0030364X Operations Research
     Volume Issue X139 DEV1996 DEV1997 DEV1998 DEV1999 DEV2000 DEV2001 DEV2002
7188     47     3    0       0       0       0       0       1       0       2
7191     47     3    0       0       0       0       0       0       2       0
7195     47     3    0       0       0       0       0       0       0       0
     DEV2003 DEV2004 DEV2005 DEV2006 DEV2007 DEV2008 DEV2009 DEV2010 DEV2011
7188       0       0       0       1       0       0       0       0       0
7191       3       4       1       4       4       8       4       6       1
7195       0       1       2       2       1       0       1       0       0
     X130655 X0 X130794
7188       4  0       4
7191      37  0      37
7195       7  0       7

The first step is to aggregate data, not to look at each article, but to look at all paper published in 1999 (say). And then, we look at the number in citations the year of publication, the year after, two years after, etc. It will appear in a triangle since if we look at articles published in 2010, there is only on possible year for citations (2010, since I removed 2011).

VOL=rev(unique(base$Volume))
VOL=VOL[is.na(VOL)==FALSE]
TRIANGLE=matrix(NA,16,16)
for(v in VOL){
k=k+1
sb=base[base$Volume==v,9:24]
sb=sb[is.na(sb[,1])==FALSE,]
TRIANGLE[k,1:(17-k)]=apply(sb,2,sum)[k:16]}

Then, a standard idea (at least in insurance business, for claims payment development) is to consider that data are Poisson distributed, and the number of citations should depend on the year of publication of the article (a row effect) and the development (how many years after are we looking at, i.e. a column effect). More formally, let denote the number of citations of articles published year during year (or after years). And we assume that

TRIANGLE=TRIANGLE[-16,]
TRIANGLE=TRIANGLE[,-16]
Y=as.vector(TRIANGLE)
YEAR=rep(1996:2010,15)
DEV =rep(1:15,each=15)
baseT=data.frame(Y,YEAR,DEV)
reg=glm(Y~as.factor(YEAR)+as.factor(DEV),
data=baseT,family=poisson)

Since those are incremental values, in order to look at the paper of distribution, we need to sum them on a line. Thus, we can plot


(because we used factors, the first component has been replaced by the constant in the regression) or a normalized version to compare among journals. For instance, we would like to get 100 citations over 15 years.

DYN=exp(c(reg$coefficients[1],reg$coefficients[1]+
reg$coefficients[16:29]))
DYNN=cumsum(DYN)/sum(DYN)
plot(0:15,DYNN)

And this is what we get, for several academic journals,
The pattern is rather different. For instance, in Health Economics, citations is a quick process: more than 40% of citations obtained over 15 years, were obtained during the first 4 years. On the other hand, in the Journal of Finance, it is much smaller: less than 15% of the citations were obtained during the first 4 years (on average). So it means that comparing citation based index (namely g or h) is a difficult exercise, especially with you researchers in different areas. The same g or h index for young researcher, publishing either in Stochastic Processes and their Applications or Annals of Statistics, means that after 3 years, it can be 50% higher.
Now it is possible to look more into details, with below JRSS-B (on applied statistics). Note that here, citations come extremely slowly… to it might not be a good “strategy” (assuming that a researcher’s target is simply to get – quickly – a high citation index) for a young researcher to publish in JRSS-BOn the other hand, Biometrika is much faster (both are on applied statistics, but we’ve seen here that they were not in the same cluster)We can also observe that Annals of Probability
and Stochastic Processes and their Applicationshave (almost) similar patterns (SPA might be a bit faster). Anyway, I have been surprised to see that in theoretical journals citations are extremely fast. Especially if we compare with the Journal of Finance for instancewhere I though citations were extremely fast. But I might have a non-correct interpretation: it might simply mean that in the Journal of Finance it is common to cite old papers (published 10 or 15 years ago), maybe more common that in stochastic processes…
Anyway, all suggestions about the interpretation are welcomed !