Citation between papers can be treated as a causal relationship. In addition, some citation networks have a number of similarities to the causal networks in network cosmology, e.g., the similar in-and out-degree distributions. Hence, it is possible to model the citation network using network cosmology. The casual network models built on homogenous spacetimes have some restrictions when describing some phenomena in citation networks, e.g., the hot papers receive more citations than other simultaneously published papers. We propose an inhomogenous causal network model to model the citation network, the connection mechanism of which well expresses some features of citation. The node growth trend and degree distributions of the generated networks also fit those of some citation networks well.

The features of collaboration patterns are often considered to be different from discipline to discipline. Meanwhile, collaborating among disciplines is an obvious feature emerged in modern scientific research, which incubates several interdisciplines. The features of collaborations in and among the disciplines of biological, physical and social sciences are analyzed based on 52,803 papers published in a multidisciplinary journal PNAS during 1999 to 2013. From those data, we found similar transitivity and assortativity of collaboration patterns as well as the identical distribution type of collaborators per author and that of papers per author, namely a mixture of generalized Poisson and power-law distributions. In addition, we found that interdisciplinary research is undertaken by a considerable fraction of authors, not just those with many collaborators or those with many papers. This case study provides a window for understanding aspects of multidisciplinary and interdisciplinary collaboration patterns.

The increasing use of mathematical techniques in scientific research leads to the interdisciplinarity of applied mathematics. This viewpoint is validated quantitatively here by statistical and network analysis on the corpus PNAS 1999–2013. A network describing the interdisciplinary relationships between disciplines in a panoramic view is built based on the corpus. Specific network indicators show the hub role of applied mathematics in interdisciplinary research. The statistical analysis on the corpus content finds that algorithms, a primary topic of applied mathematics, positively correlates, increasingly co-occurs, and has an equilibrium relationship in the long-run with certain typical research paradigms and methodologies. The finding can be understood as an intrinsic cause of the interdisciplinarity of applied mathematics.