Commodity-specific triads in the Dutch inter-industry production network

Abstract

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 analyze both binary and weighted triadic motifs in the Dutch inter-industry production network disaggregated at the level of 187 commodity groups, which Statistics Netherlands reconstructed from National Accounts registers, surveys and known empirical data. 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 $85\%$ 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.

Figures (6)

• Figure 1: (a) Graphical representation of the Dutch multi-layer production network. For illustrative purposes, we represent three industries/firms $i$, $j$ and $k$ as nodes, all placed on three commodity group layers, namely (from top to bottom): Cereals, Beer/Malt, Bread and other bakery products. The connections between the same three nodes are different in the different layers. (b) The possible 13 types of connected triadic subgraphs. Each triple of Industries/Firms can trade different products by forming, on each commodity-specific layer, either one of the 13 possible connected subgraphs or one of the remaining subgraphs where at least one node is disconnected (not shown).
• Figure 2: Normalized Triadic Occurrences (a) and Fluxes (b): the Aggregated Network ($-\bullet-$) presents a high occurrence of subgraphs $m=1$ and $m=13$, representing open-Vs and completely reciprocated triads, respectively. The latter covers most of the total amount of money traded. The Cereals commodity layer ($-\bullet-$), with a high occurrence of subgraph $m=1$. A relatively high amount of money is distributed across $m=1$, $m=4$ and $m=6$. Gas/Hot Water/City Heating layer ($-\bullet-$) with a predominant occurrence and flux in subgraph $m=1$. Agricultural Services layer ($-\bullet-$), with a highly heterogenous spectrum of occurrences and fluxes. Completely cyclical triads have a high occurrence in the aggregated network, but break apart when passing to single commodity layers as G.H.C and Cereals, if not for rare cases such as Agricultural Services. In single commodity layers $m=1$ receives the highest concentration of money, signalling a large amount of money flows over structures that greatly depend on a limited number of suppliers, which control the market.
• Figure 3: Triadic binary motif analysis: DBCM ($\bullet$) vs RBCM ($\bullet$). (a) Analysis of the aggregated network with a single representative commodity. Numerous motifs and anti-motifs are present using DBCM and RBCM as null models. (b-d) Commodity groups where RBCM reproduces all the triadic structures, and they are, respectively, Cereals, Electrical Components, and the Construction of Tunnels, Waterways, and Roads. (e-f) Commodity groups with one network motif, namely Bread and Gasoline. (g) Commodity group with two network motifs, namely Beer/Malt. The CIs are computed by extracting the $2.5$-th and $97.5$-th percentile from an ensemble distribution of $500$ graphs. The numerous motifs and anti-motifs in the aggregated network can be seen as the aggregation of commodity groups presenting very few characteristic patterns.
• Figure 4: Comparison DBCM ($\bullet$) vs. RBCM ($\bullet$): (a) Empirical Counter Cumulative Distribution Function $ECCDF$ of the number of deviating binary triadic motifs and anti-motifs across commodity layers. (b) Number of commodities $c_{h}(m)$ having a $m$-type motif (overoccurrence). (c) Number of commodities $c_{l}(m)$ having a $m$-type anti-motif (underoccurrence). RBCM explains more triadic structures than DBCM, as shown in the difference of their $ECCDF$. Passing from DBCM to RBCM reduces the number of $m$ motifs across commodities, with the exception of $m=6$, and anti-motifs, with the exception of $m=8$. The deviation of those triads is, hence, due to three-node correlations that go beyond directional and reciprocal tendencies of supply/use among industries. RBCM, hence, signals an increased vulnerability to demand shocks originating from the bankrupcty of industries of type $k$ in sub-types $m=6$ and an increased resiliency to supply/demand shocks of industries of type $j$ in triadic formations $m=8$.
• Figure 5: Triadic weighted motif analysis: DBCM+CReM$_{A}$ ($\bullet$) vs RBCM+CRWCM ($\bullet$). (a) Analysis of the aggregated network with a single representative commodity. A large number of motifs and anti-motifs are present when using DBCM+CReM$_{A}$, while three motifs are present when using the RBCM+CRWCM. (b-d) Commodity groups where RBCM+CRWCM reproduces all the triadic structures, and they are, respectively, Seeds, Metal Components for Doors & Windows, and Airline Services. (e-f) Commodity groups with one network motif, namely Coffee/Tea and Textile raw materials and products. (g) Commodity group with two network motifs, namely Shipping Services. The CIs are computed by extracting the $2.5$-th and $97.5$-th percentile from an ensemble distribution of $500$ graphs. Passing from the aggregated network to the disaggregated product layers unveils the presence of a few commodity-specific motifs and anti-motifs.