Long term citation model

updated: 2022-11-18

#network-science/citation-dynamics

Model

The long-term citation model characterizes citation dynamics of a paper during by a set of time moments , where represents the time of th citation, and is the total number of citations during . One models as an inhomogeneous Poisson process with rate function given by

where is the fitness and is accumulated number of citations up to time . Variable accounts for offset attention, which is also called initial attractiveness (see Preferential attachment model). Function characterizes the aging effect given by the log-normal distribution:

shaped by paper-specific parameters .

The long-term citation model predicts citations by the following differential equation:

with boundary condition . By solving the differential equation, we obtain the prediction of citations by the long-term citation at time :

Note that I follow the derivation by a follow-up study, which amends a mathematical leap in the original paper.

Fitting algorithm

Regarding the fitting algorithm, the original paper employes a least-squared fit (Equation 3 in the original paper). Alternatively, the maximum likelihood method can be used (see here).

Tthe long-term citation model is overparameterized. Each paper is characterized by three parameters, resulting in 3N parameters to be estimated. Consequently, the long-term citaiton tends to overfit, especially for papers with few citations.

Another approach is to formulate the estimation from a Bayesian perspective and introduce a prior that regulates the model's excessive fitting to data. See a follow-up study.

Critiques

The long-term citation model demonstrated an excellent predictive capacity of citations, which is later questioned Comment on “Quantifying long-term scientific impact” | Science.

Furtheremore, the derivation of the model contains a critical error and unjustified assumptions (see here) .

Code

References

Interactive graph
Links to this page
Network Science
Preferential attachment model