Fig 1. Schematics of the fractional counting method for publication and collaboration counts.

Two matrices, the fractional publication counts by countries A_t and the topic distributions of papers B_t, were extracted from the document set of year t. (1) $A_t^T{B}_t$ represents the fractional publication counts by topics at year t. For further analysis, based on the hierarchical tree of clusters in Fig 2, the fractional publications by 41 topics are grouped into five topical clusters: material, plasma, device, diagnostics, and simulation. R_t is the aggregated matrix and is transposed in the figure to match with the hierarchical tree of 41 clusters. (2) The country profile of a paper is transformed into a collaboration matrix W₁, which was distributed over the five topical clusters by weights. For each year, by aggregating the collaboration matrices of all published papers, we had five fractional collaboration matrices.