Community-centric modeling of citation dynamics explains collective citation patterns in science, law, and patents

TL;DR

The study uncovers universal patterns in citation dynamics across science, law, and patents, including exponential growth, linear preferential attachment, aging/recency, and sleeping beauties. It introduces the Community Citation Model (CCM), which situates publications in a high-dimensional knowledge space and links citation probability to the proximity between citing and cited items, thereby capturing collective dynamics absent in publication-centric models. The CCM, fitted via Noise Contrastive Estimation and EM, reproduces observed patterns and more accurately forecasts future citation trajectories, particularly for highly impactful papers. These findings suggest that knowledge formalization is governed by robust, cross-domain mechanisms and that a collective, space-based perspective can enhance predictive understanding and forecasting of scientific, legal, and technological impact.

Abstract

Many human knowledge systems, such as science, law, and invention, are built on documents and the citations that link them. Citations, while serving multiple purposes, primarily function as a way to explicitly document the use of prior work and thus have become central to the study of knowledge systems. Analyzing citation dynamics has revealed statistical patterns that shed light on knowledge production, recognition, and formalization, and has helped identify key mechanisms driving these patterns. However, most quantitative findings are confined to scientific citations, raising the question of universality of these findings. Moreover, existing models of individual citation trajectories fail to explain phenomena such as delayed recognition, calling for a unifying framework. Here, we analyze a newly available corpus of U.S. case law, in addition to scientific and patent citation networks, to show that they share remarkably similar citation patterns, including a heavy-tailed distribution of sleeping beauties. We propose a holistic model that captures the three core mechanisms driving collective dynamics and replicates the elusive phenomenon of delayed recognition. We demonstrate that the model not only replicates observed citation patterns, but also better predicts future successes by considering the whole system. Our work offers insights into key mechanisms that govern large-scale patterns of collective human knowledge systems and may provide generalizable perspectives on discovery and innovation across domains.

Figures (17)

• Figure 1: Universal patterns of citation dynamics across three knowledge systems (law, science, and patents). A. Exponential growth in publications. B. Degree distribution of the publications. C. Linear preferential attachment, where the number of new citations acquired in a year is proportional to the accumulated number of citations. D. Aging function representing the probability that a publication receives the $x+1$th citation at age $\Delta t$, where $x \in \{25, 50, 100\}$. E. Skewed distribution of the Sleeping Beauty coefficient ($B$).
• Figure 2: The Community Citation Model (CCM) reproduces universal citation patterns. A. A publication is represented as a point in a vector space. A new publication is placed near a randomly chosen existing publication and cites nearby publications according to Eqs. \ref{['eq:adj_prod']} and \ref{['eq:CCM']}. B--D. The CCM reproduces preferential attachment, recency, and the skewed sleeping beauty coefficient distribution. In C we show the aging function for publications with 50 references. We compare the CCM to the preferential attachment model (PAM) and the long-term citation model (LTCM).
• Figure 3: Embeddings learned by the CCM reveal community structures that help explain universal citation dynamics. A. 2D projection of the 128-dimensional embedding by UMAP. The projection is generated with ten nearest neighbors and cosine similarity as the proximity metric. B. The evolution of the embedding space in the neighborhood of the top sleeping beauties (SB). The brighter yellow areas represent dense areas of knowledge space in the five years leading up to the date displayed, which we approximate via kernel density estimation. We define the neighborhood as the 3,000 publications that are closest to the focal SBs marked by stars. The red circles represent other SBs in the vicinity. The embeddings are projected into 2D using Principal Component Analysis.
• Figure 4: Prediction of citation counts for individual publications published in 2000 using initial citation trajectories of 5 years. A. Citation prediction errors for the CCM, PA, and LTCM over time. The error is calculated by $\hat{c}(t)$, i.e., $|\log(c(t) + 1) - \log(\hat{c}(t) + 1)|$, where $c(t)$ and $\hat{c}(t)$ are the actual and predicted citations in year $t$ from the unseen publications published after the training window, respectively. The 95% confidence intervals are estimated by bootstrapping. B. Citation prediction errors for the top-10% highly-cited publications. The CCM has the lowest prediction errors overall, especially for young publications and highly cited publications. C. Precision for identifying the top-10% highly-cited publications. A precision of 0.1 corresponds to random guessing.
• Figure 5: Dynamics of the APS citation network. A. Growth of publications. B. Growth of citations. C. Growth of the average number of references. D. Degree distribution. E. Linear preferential attachment. F. Aging function. G. Skewed distribution of the Sleeping Beauty coefficient.