Abstract Higher-order networks effectively represent complex systems with group interactions. Existing methods usually overlook the relative contribution of group interactions (hyperedges) 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 methodology to precisely measure the contribution of different orders to topological network properties. First, we propose the order contribution measure, which quantifies the contribution of hyperedges 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 hyperedges 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 hyperedges of other orders. Third, to account for labels, we propose a metric of label group balance to assess how hyperedges of different orders connect la
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