Abstract: Convexity in a network (graph) has been recently defined as a property of each of its subgraphs to include all shortest paths between the nodes of that subgraph. It can be measured on the scale [0, 1] with 1 being assigned to fully convex networks. The largest convex component of a graph that emerges after the removal of the least number of edges is called a convex skeleton. It is basically a tree of cliques, which has been shown to have many interesting features. In this article the notions of convexity and convex skeletons in the context of scientific collaboration networks are discussed. More specifically, we analyze the co-authorship networks of Slovenian researchers in computer science, physics, sociology, mathematics, and economics and extract convex skeletons from them. We then compare these convex skeletons with the residual graphs (remainders) in terms of collaboration frequency distributions by various parameters such as the publication year and type, co-authors' birth year, status, gender, discipline, etc. We also show the top-ranked scientists by four basic centrality measures as calculated on the original networks and their skeletons and conclude that convex skeletons may help detect influential scholars that are hardly identifiable in the original collaboration network. As their inherent feature, convex skeletons retain the properties of collaboration networks. These include high-level structural properties but also 2 the fact that the same authors are highlighted by centrality measures. Moreover, the most important ties and thus the most important collaborations are retained in the skeletons. Keywords: convexity, co-authorship, convex skeletons, centrality, weak links. The notion of network convexity has been defined only recently by Marc and Šubelj (2018) . By their definition, a convex network is a connected (undirected) graph where every connected subgraph includes all shortest paths between its nodes. Such a network takes generally the form of a tree of cliques. Furthermore, by removing the least number of edges from a graph to obtain its largest convex subgraph, we extract a convex skeleton of the original graph with many interesting properties (Šubelj, 2018). One of these properties is the fact that the resulting convex skeleton is a generalized spanning tree with potential cliques retained, which leads to a network structure with a similar degree distribution, clustering, and node distances like in the original graph but with unique shortest paths between nodes. Convex skeletons can thus be regarded as a network abstraction technique with possible applications in network modelling and visualization. Particularly for scientific collaboration networks since they turn out to be rather convex (Marc and Šubelj, 2018) . The goal of the present study is to build upon the two above analyses and investigate the concept of convexity and convex skeletons in the context of scientific collaboration (or co-authorship) networks at the level of authors (i.e. here we are not interested in country-level or institutional collaboration at all). The questions we would like to answer by this analysis is whether convex skeletons extracted from author collaboration networks can be used as their meaningful abstraction, whether author rankings by centrality measures calculated on convex skeletons differ substantially from those calculated on the original networks, and what kind of edges are removed when convex skeletons are extracted. The motivation for studying convexity in co-authorship networks is at least threefold. Firstly, as already mentioned above, different collaboration networks turn out to be rather convex (Marc and Šubelj, 2018; Šubelj, 2018) , which is in contrast to paper citation and other bibliographic networks. Convex skeletons should therefore represent their meaningful abstraction, which is not the case for the latter. Secondly, since co-authorship networks are projections of bipartite graphs of authors and papers, these are unions of cliques by construction. Yet, how these cliques are connected together is unknown. Convex skeletons propose possibly the simplest configuration in which cliques are connected in a tree and thus 3 provide a very simple framework for understanding the structure of scientific co-authorship. Lastly, although convexity is a well understood property of different mathematical objects, and a key component in mathematical optimization, it has not been explored in networks until only recently. Different collaboration networks likely represent the most obvious setting to bridge this gap. The article is organized in the following way. After recalling the published literature on collaboration networks in Section 2, we introduce the methods dealing with network convexity and convex skeletons in Section 3, and present the data underlying our analysis in Section 4: co-authorship networks of Slovene researchers in computer science, physics, sociology, mathematics, and economics. We extract convex skeletons from these collaboration networks and, due to space limitations, show the main results for the first three disciplines in Section 5 with a special focus on the differences between the skeleton and the residual network. (A residual network or remainder consists of the same set of nodes as the original network and of a subset of edges from the original network after the edges forming a convex skeleton have been removed). We then take conclusions, discuss the limitations of this study and propose future research directions in Section 6. Related work Due to the lack of collaboration-based network convexity studies, we will now only recall some fundamental and recent publications on scientific collaboration (or co-authorship) networks. One of the reasons to study research collaboration is the reported positive correlation between impact and collaboration (Hsu and Huang, 2011) , 1997; Melin and Persson, 1996) , for the purpose of this study we will consider those two terms as equivalent and interchangeable. In more detail, Laudel identified six types of research collaboration: collaboration involving a division of labour, service collaboration, provision of access to research equipment, transmission of know-how, mutual stimulation and trusted assessorship. Only the first type resulted systematically in co-authorships while the others were rewarded in acknowledgements (about a third of the collaborations inspected) or not at all (about a half).
Šubelj et al, REF researched on convexity in the scientific collaboration network.
Convexity in scientific collaboration networks
J. Infometr. 13(1), 10-31 (2019)
Computer Science Physics