## 3 Background
### Affinity Poisson Process  
Figure 2a shows an example that a paper \(p_{1}\) accumulates citations over time, which can be considered as each citing paper's arrival at its publication time. Prior work [6] modeled such arrivals as a Poisson process with one intensity function. That is, paper citations are indistinguishable and homogeneous regardless of a citing paper's research field. However, as research become more interdisciplinary, an individual paper is likely cited by both internal and external publications from the same and other research fields respectively. Thus, the single horizontal timeline in Figure 2a cannot differentiate varying citation intensities generated by numerous fields over time. In other words, the prior work has neglected disproportionate influences of different fields on individual papers' citations, exposing limitations of understanding interdisciplinary citation dynamics across fields of sciences. In this respect, a new framework has been proposed to model popularity dynamics by incorporating heterogeneous nature of a social system consisting of subgroups interacting one another (i.e., intra- and interfield interactions) and view the paper citations as the superposition of multiple Poisson processes for different citing fields [9]. As shown in Figure 2b, the paper citations are now decomposed into different timelines depending on the research fields of citing papers, and each timeline is modeled as an independent Poisson process. Since the superposition of Poisson processes is still a Poisson process [32], the new framework extends the prior work [6] from a homogeneous citation process to cross-field popularity dynamics. Figure 2: Citations across different fields. (**a**) Each square \(p_{i}\) represents the \(i\)-th paper in its field (specified at top and color-coded). The paper \(p_{1}\) in the cited field \(f_{1}\) receives citations from multiple citing fields (\(f_{1}\), \(f_{2}\) and \(f_{3}\)) over time. Here, \(t_{n}\) (\(n=1,...,9\)) denotes the publication time of the \(n\)-th citing paper. (**b**) The citations are decomposed into different timelines according to the citing fields. As the superposition of Poisson processes is also a Poisson process, this is a generalization of the citation arrival process in (**a**) by considering interdisciplinary citation flow. **Problem Statement. ** In more detail, let us first define a set of research fields \(\mathbf{F}\) in sciences and focus on the paper citations in one specific field \(f_{\text{cited}}\in\mathbf{F}\). Suppose that the \(i\)-th paper \((i=1,...,I)\) in the cited field \(f_{\text{cited}}\), published at time \(t=0\), has received \(N_{i}^{f}\) citations from a citing field \(f\in\mathbf{F}\) during a time period \([0,T]\). When the citations came from the same field, they are called _internal citations_ (i.e., \(f_{\text{cited}}=f\)), otherwise called _external citations_ (i.e., \(f_{\text{cited}}\
eq f\)). Then, paper \(i\)'s citation timestamps from each citing field \(f\), \(\mathbf{D}_{i}^{f}\) consist of the publication times of the citing papers such that  
\[\mathbf{D}_{i}^{f}=\left\{t_{n}^{f}\mid n=1,...,N_{i}^{f},0\leq t_{n}^{f}\leq T \right\}. \tag{1}\]  
By aggregation, the citation timestamps of the cited field receiving from each field \(f\), \(\mathbf{D}^{f}\) and from all citing field \(\mathbf{D}\) are respectively,  
\[\mathbf{D}^{f}=\left\{\mathbf{D}_{i}^{f}\mid i=1,...,I\right\},\quad\mathbf{D}=\left\{\mathbf{ D}^{f}\mid f\in\mathbf{F}\right\}. \tag{2}\]  
Note that the timestamps are shifted by the publication time of the cited paper for simplicity. The fundamental objective is to predict individual paper \(i\)'s citation volume \(c_{i}(t)\) at time \(t\), by learning model parameters from all the citation histories \(\mathbf{D}\) received by \(i\)'s research field. In a more general statement, it is aimed to predict the popularity of each information item at an arbitrary time based on its popularity histories. **Citation Intensity.