Triadic motifs are the smallest building blocks of higher-order interactions in complex networks and can be detected as over-occurrences with respect to null models with only pair-wise interactions. Recently, the motif structure of production networks has attracted attention in light of its possible role in the propagation of economic shocks. However, its characterization at the level of individual commodities is still poorly understood. Here we analyse both binary and weighted triadic motifs in the Dutch inter-industry production network disaggregated at the level of 187 commodity groups, using data from Statistics Netherlands. We introduce appropriate null models that filter out node heterogeneity and the strong effects of link reciprocity and find that, while the aggregate network that overlays all products is characterized by a multitude of triadic motifs, most single-product layers feature no significant motif, and roughly 80% of the layers feature only two motifs or less. This result paves the way for identifying a simple "triadic fingerprint" of each commodity and for reconstructing most product-specific networks from partial information in a pairwise fashion by controlling for their reciprocity structure. We discuss how these results can help statistical bureaus identify fine-grained information in structural analyses of interest for policymakers.

翻译：三元组动机是复杂网络中高阶相互作用的最小构建单元，可通过与仅存在成对相互作用的零模型相比的过度出现频率来检测。近年来，生产网络的动机结构因其可能在经济冲击传播中的作用而受到关注。然而，其在单个商品层面的特征化仍鲜有研究。本文利用荷兰统计局数据，基于187个商品组分解的荷兰产业间生产网络，分析了二元和加权三元组动机。我们引入能过滤节点异质性和链接互惠性强效应的适当零模型，发现：当叠加所有产品的聚合网络呈现多种三元组动机时，大多数单一产品层不存在显著动机，约80%的产品层仅包含两个或更少动机。这一结果不仅为识别每个商品的简单"三元组指纹"铺平道路，还表明可通过控制互惠性结构，以成对方式从部分信息重建大多数产品特定网络。我们讨论了这些结果如何帮助统计机构在政策制定者关注的结构性分析中识别精细化信息。