Citation dynamics

It has been found that citation distributions are best described by a shifted power-law.[1] The probability that paper ${\displaystyle i}$ is cited at time ${\displaystyle t}$ after publication as:

${\displaystyle \Pi _{i}(t)=\eta _{i}c_{i}^{t}P_{i}(t)}$

where ${\displaystyle \Pi _{i}(t)}$ serves as the outcome variable for each particular paper ${\displaystyle i}$ at time ${\displaystyle t}$. Fitness, ${\displaystyle \eta _{i}}$, captures the inherent differences between papers, accounting for the perceived novelty and importance of a discovery. ${\displaystyle c_{i}^{t}}$ represents the cumulative number of citations acquired by a paper ${\displaystyle i}$ at time ${\displaystyle t}$ and ${\displaystyle P_{i}(t)}$ is a log-normal survival probability.[2] The probability is equal

${\displaystyle P_{i}(t)={\frac {1}{{\sqrt {2\pi }}\sigma _{i}t}}exp\left\lbrack -{\frac {(lnt-\mu _{i})^{2}}{2\sigma _{i}^{2}}}\right\rbrack }$

where ${\displaystyle t}$ is time; ${\displaystyle \sigma }$ is longevity, capturing the decay rate; and ${\displaystyle m}$ indicates immediacy, governing the time for a paper to reach its citation peak. The ultimate impact ${\displaystyle c_{i}^{\infty }}$ represents the total number or citations that the paper receives during its lifetime.

${\displaystyle c_{i}^{\infty }=m(e^{\lambda _{i}}-1)}$

Where ${\displaystyle m}$ is a global parameter that has the same value for all publications. ${\displaystyle \lambda _{i}}$ represents the relative fitness of the paper. From the above formula, we can see that the total number of references that the paper can receive during its lifetime depends only on its relative fitness which is very hard to quantify.

• Citation index