Local topological features of robust supply networks

Abstract

The design of robust supply and distribution systems is one of the fundamental challenges at the interface of network science and logistics. Given the multitude of performance criteria, real-world constraints, and external influences acting upon such a system, even formulating an appropriate research question to address this topic is non-trivial. Here we present an abstraction of a supply and distribution system leading to a minimal model, which only retains stylized facts of the systemic function and, in this way, allows us to investigate the generic properties of robust supply networks. On this level of abstraction, a supply and distribution system is the strategic use of transportation to eliminate mismatches between production patterns (i.e., the amounts of goods produced at each production site of a company) and demand patterns (i.e., the amount of goods consumed at each location). When creating networks based on this paradigm and furthermore requiring the robustness of the system with respect to the loss of transportation routes (edge of the network) we see that robust networks are built from specific sets of subgraphs, while vulnerable networks display a markedly different subgraph composition. Our findings confirm a long-standing hypothesis in the field of network science, namely, that network motifs-statistically over-represented small subgraphs-are informative about the robust functioning of a network. Also, our findings offer a blueprint for enhancing the robustness of real-world supply and distribution systems.

Supplementary information: The online version contains supplementary material available at 10.1007/s41109-022-00470-2.

Keywords: Minimal model; Network motifs; Spatial networks; Supply chain management.

Fig. 1

Examples of model setup and possible networks with a low (a) and high (b) robustness. Values of $r$ are indicated below each network. In (a), $E_r=(1,4),(5,4)$, M=6. In (b), any edge can be removed without reducing the demand stisfaction, $|E_r|=M=10$

Fig. 2

An example of a typical Pareto front of an ($c$, $r$) optimization

Fig. 3

Scheme of the optimization process. First, $G^0$ is created as random networks that satisfy the optimization constraints (M, edge length, etc.). Then, to create the new generation, the algorithm picks the best networks from the previous generation and creates new networks by mutating and recombining the best networks. The new generation is finally created by combining the best networks from the previous generation, mutations and recombinations of randomly selected best networks, and purely random networks that satisfy the optimization constraints. Then, the procedure is applied to the new generation. The process is repeated for $G^N$ generations

Fig. 4

Pareto fronts in a series of $(c,r)$ optimizations with $N=20$ and varying M boundaries. Each figure combines the results of 50 model setup runs that have different spatial distribution of nodes

Fig. 5

We generate 50 different network setups with $N=20$ and for each setup perform a $(c,r)$ optimization with restriction on the allowed number of edges M (shown below each figure). From the resulting Pareto fronts, we take 50 vulnerable networks with given $r$ and compute their motifs. For a single network, the result of motif computations are 13 z-scores that indicate how over- or underrepresented each subgraph is in the original network, compared to its randomized versions. This gives 50 z-score values for each of the subgraphs that form a distribution drawn with shaded vertical violin plots. Blue circles in the figures are the mean values and vertical lines with ticks are the standard deviations of these distributions

Fig. 6

Motif patterns of robust networks in $c$, $r$ optimization with $N=20$. Similarly to Fig. 5 we take 50 networks with high $r$ from the Pareto fronts

Fig. 7

The target network motif pattern typical to the robust biological systems, reproduced from Milo et al. (2004). X axis represents different subgraphs, Y axis shows normalized z-score of the corresponding subgraph

Fig. 8

Distribution of $r$ values as a function of $\sigma$ for the results of $(c,\sigma )$ optimization. Sets of networks that correspond to given robustness are plotted as violin plots to demonstrate their distributions, while the solid line shows the behavior of their means

Fig. 9

Histograms of the average increase in robustness for high and low changes in one topological metric. We take 500 random networks that have full demand satisfaction and some robustness. For each combination of a network and an edge that is not in the network, we compute r, motif signature, and three-node subgraph counts. Then, for each network, we separate combinations into two groups: the ones that yield the highest increase in a simple metric (counts of feedforward loop (a), z-score of feedforward loop (b), and $\sigma$ (c))—denoted as high, and the remaining combinations—denoted as low. After this we compare the average increase in r in these two groups

Fig. 10

Application of the model to industrial data. a shows a subnetwork of the full European supply network of an automotive company. The subnetwork is generated by selecting the routes used for transporting one product category and taking a neigborhood of these routes with size parameter $t=0.6$. In b all product subnetworks are analysed together using Spearman correlation (upper panel) and the corresponding p-value (represented as $-\text{log}(p)$; lower panel) of their $r$ and $\sigma$ values for different neighborhood sizes t. The dashed lines indicate zero correlation (upper panel) and $-\text{log}(0.05)$ (lower panel), respectively