Modeling scientific-citation patterns and other triangle-rich acyclic networks

We propose a model of the evolution of the networks of scientific citations. The model takes an out-degree distribution (distribution of number of citations) and two parameters as input. The parameters capture the two main ingredients of the model: the aging of the relevance of papers and the formation of triangles when new papers cite old. We compare our model to three network structural quantities of an empirical citation network. We find that unique point in parameter space optimizing the match between the real and model data for all quantities. The optimal parameter values suggest that the impact of scientific papers, at least in the empirical data set we model, is proportional to the inverse of the number of papers since they were published.

Figure 1

(Color online) An example of a citation network—the citation network of articles cited by this paper (with the indices being the indices of the reference list).

Figure 2

(Color online) The number $T_i$ of triangles as a function of the order the vertex is added $i$. The solid, dotted, and dashed lines correspond to the empirical citation network of high-energy physics papers, the KN model, and the extended KN model, respectively. The values of $T_i$ for the two theoretical models are averages of over 200 independent samples.

Figure 3

(Color online) Network statistics for our model. Solid lines correspond to the citation network of high-energy physics papers and the dashed lines represent our model data. In (a) and (b), we use the model parameters $\alpha =-1$ and $\beta =0.99$. (a) shows the average number $T_i$ of triangles as a function of the index of the added vertex $i$. (b) displays the number of arcs passing $i$, ${\lambda }_i$ as a function of $i$. (c) and (d) correspond to (a) and (b) but for $\alpha =0$ and $\beta =0.99$. (e) and (f) show the same for $\alpha =\beta =0$. (g) and (h) also correspond to (a) and (b) but for parameters $\alpha =-1$ and $\beta =0$.

Figure 4

(Color online) The in-degree distribution $P\left(k_{\text{in}}\right)$ for our four parameter combinations (indicated in the panels).