The relevance of higher-order ties

Higher-order networks effectively represent complex systems with group interactions. Existing methods usually overlook the relative contribution of group interactions (hyperlinks) of different sizes to the overall network structure. Yet, this has many important applications, especially when the network has meaningful node labels. In this work, we propose a comprehensive methodology to precisely measure the contribution of different orders to the overall network structure. First, we propose the order contribution measure, which quantifies the contribution of hyperlinks of different orders to the link weights (local scale), number of triangles (mesoscale) and size of the largest connected component (global scale) of the pairwise weighted network. Second, we propose the measure of order relevance, which gives insights in how hyperlinks of different orders contribute to the considered network property. Most interestingly, it enables an assessment of whether this contribution is synergistic or redundant with respect to that of hyperlinks of other orders. Third, to account for labels, we propose a metric of label group balance to assess how hyperlinks of different orders connect label-induced groups of nodes. We applied these metrics to a large-scale board interlock network and scientific collaboration network, in which node labels correspond to geographical location of the nodes. Experiments including a comparison with randomized null models reveal how from the global level perspective, we observe synergistic contributions of orders in the board interlock network, whereas in the collaboration network there is more redundancy. The findings shed new light on social scientific debates on the role of busy directors in global business networks and the connective effects of large author teams in scientific collaboration networks.