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Table 5 Node level metrics and their SCN implications

From: Network science approach to modelling the topology and robustness of supply chain networks: a review and perspective

Mathematical representation SCN Implication
Degree (k)
The degree k i of any node i is represented by;
\( {k}_i=\sum_j{a}_{ij} \)
where a ij is any element of the adjacency matrix A.
Represents the number of direct neighbours (connections) a given firm has. For instance, in a given SCN, the firm with the highest degree (such as the integrators that assemble components) is deemed to have the largest impact on operational decisions and strategic behaviours of other firms in that particular SCN. Such a firm has the power to reconcile the differences between various other firms in the SCN and align their efforts with greater SCN goals (Kim et al., 2011).
In directed networks, the firms which have high in-degree are considered to be ‘integrators’ who collects information from various other firms to create high value products. In contrast, the firms which have high out-degree are considered to be ‘allocators’ who are generally responsible for distribution of σ s , t (n)high demand resources to other firms and/or customers.
Betweenness centrality (normalised) (Freeman, 1977)
The betweenness centrality of a node n is defined as;
\( {C}_b(n)=\frac{2}{\left(N-1\right)\left(N-2\right)}\sum_{s\ne n\ne t}\frac{\sigma_{s,t}(n)}{\sigma_{s,t}} \)
where s and t are nodes in the network, which are different from n, σ s , t denotes the number of shortest paths from s to t, and is the number of shortest paths from s to t that n lies on.
Betweenness centrality of a firm is the number of shortest path relationships going through it, considering the shortest path relationships that connect any two given firms in the SCN. Therefore, it indicates the extent to which a firm can intervene over interactions among other firms in the SCN by being a gatekeeper for relationships. Those firms with high levels of betweenness generally play a vital role in SCNs – mainly owing to their ability to increase the overall efficiency of the SCN by smoothing various exchange processes between firms.
Closeness centrality (Sabidussi, 1966)
The closeness centrality of a node n is defined as;
\( {C}_c(n)=\frac{1}{<L\left(n,m\right)>} \)
where <L(n,m) > is the length of the shortest path between two nodes n and m (note that for unweighted graphs with no geodesic distance information, each link is assumed to be one unit of distance). The closeness centrality of each node is a number between 0 and 1.
Closeness centrality is a measure of the time that it takes to spread the information from a particular firm to the other firms in the network. While it is closely related to betweenness centrality, closeness more relevant in situations where a firm acts as a generator of information (i.e. a navigator) rather than a mere mediator/gatekeeper.
For example, due to various hindrances, the market demand information can easily be distorted when it flows from the downstream firms towards upstream firms. Such distortions can lead to undue deviation between production plans of manufacturers and supply plans of suppliers, leading to a phenomenon known as the bullwhip effect in supply chains. Firms with high closeness centrality levels therefore play a major role in sharing the actual market demand information with upstream firms in the SCN, thus diminishing the adverse impacts arising from bullwhip effect (Xu et al., 2016).
Eigenvector centrality (Ruhnau, 2000)
If the centrality scores of nodes are given by the matrix X and the adjacency matrix of the network is A, then each row of matrix X, namely x, can be defined as;
\( {\displaystyle \begin{array}{l}\mathrm{x}\propto \mathrm{Ax}\\ {}\mathrm{i}.\mathrm{e}.\\ {}\uplambda \mathrm{x}=\mathrm{Ax}\end{array}} \)
The eigenvector centrality scores are obtained by solving this matrix equation. It can be shown that, while there can be many values forλ, only the largest value will result in positive scores for all nodes, so this is the eigenvector chosen.
Eigenvector centrality measures a firm’s influence in the SCN by taking into account the influence of its neighbours. The centrality scores are given by the eigenvector associated with the largest eigenvalue. It assumes that the centrality score of a firm is proportional to the sum of the centrality scores of the neighbours. A firm with a high eigenvector centrality is assumed to derive its influential power through its highly connected neighbours.