Table 4 Network level metrics and their SCN implications

\( <k>=\frac{\sum_i{k}_i}{N} \)

where N is the total number of nodes in the network

Indicates, on average, how many connections a given firm has. Higher average degree implies good inter-connectivity among the firms in the SCN, which is favourable in terms of efficient exchange of information and material.

In directed networks, the in-degree characterises the number of supply channels while the out-degree indicates the number of sales channels, of a given firm.

\( \mathrm{diameter}=\underset{i,j}{\max }\ l\left(i,j\right) \)

where l is the number of hops traversed along the shortest path from node i to j.

The diameter of a SCN is the largest distance between any two firms in the network, in terms of number of intervening links on the shortest path. More complex manufacturing processes can include large network diameters (i.e. many stages of production) indicating difficulty in governing the overall SCN under a centralised authority.

\( \mathrm{D}=\frac{<k>}{N-1} \)

where <k > is the mean degree of all the nodes and N is the total number of nodes, in the network

Density of a SCN indicates the level of interconnectivity between the firms involved. SCNs with high density indicate good levels of connectivity between firms which can be favourable in terms of efficient information exchange and improved robustness due to redundancy and flexibility (Sheffi and Rice, 2005).

\( \mathrm{C}=\frac{N}{N-2}\left(\frac{\max (k)}{N-1}-\mathrm{D}\right) \)

where N is the total number of nodes in the network and max(k) is the maximum degree of a node within the network. Density is determined as per the equation above.

Network centralisation provides a value for a given SCN between 0 (if all firms in the SCN have the same connectivity) and 1 (if the SCN has a star topology). This indicates how the operational authority is concentrated in a few central firms within the SCN. Highly centralised SCNs can have convenience in terms of centralised decision implementation and high level of controllability in production planning. However, highly centralised SCNs lack local responsiveness since relationships between firms in various tiers are decoupled (Kim et al., 2011).

\( \mathrm{H}=\frac{\sqrt{variance(k)}}{<k>} \)

where <k > is the mean degree and variance (k) is the variance of the degree, of all the nodes in the network.

Heterogeneity is the coefficient of variation of the connectivity. Highly heterogeneous SCNs exhibit hub firms (i.e. firms with high number of contractual connections). In extreme cases, there may be many super large hubs (winner take all scenario, indicating centralised control of the overall SCN through a single firm or a very few firms).

\( <C>=\frac{\sum_i{C}_i}{N} \)

where N is the total number of nodes in the network and C _i is the number of triangles connected to node i divided by the number of triples centered around node i.

Clustering coefficient indicates the degree to which firms in a SCN tend to cluster together around a given firm. For example, it can indicate how various suppliers behave with respect to the final assembler at the global level (Kim et al., 2011). Therefore, the higher the clustering coefficient, the more dependent suppliers are on each other for production (Brintrup et al., 2016).

The degree distribution P _k of a scale free network is approximated with power law as follows;

P _k  ∼ k ^−γ

where k is the degree of the node and γ is the power-law exponent (also known as the degree or scale free exponent).

SCNs with γ < 2 include very large hubs which acquire control through contractual relationships with other firms at a rate faster than the growth of the SCN in terms of new firm additions. As γ continues to increase beyond 2, the SCNs include smaller and less numerous hubs, which ultimately leads to a topology similar to that of a random network where all firms have almost the same number of connections.

Note that in directed networks, two γ values are generally reported – one for the in-degree and another for the out-degree.

Assortativity is defined as a correlation function of excess degree distributions and link distribution of a network.

For undirected networks, when degree distribution is denoted as p _k and excess degree (remaining degree) distribution is denoted as q _k , one can introduce the quantity e _j,k as the joint probability distribution of the remaining degree distribution of the remaining degrees of the two nodes at either end of a randomly chosen link.

Given these distributions, the assortativity of an undirected network is defined as;

\( \rho =\frac{1}{\sigma_q^2}\left[\sum_{jk} jk\left({e}_{j,k}-{q}_j{q}_k\right)\right] \)

where σ _q is the standard deviation of q _k .

Positive assortativity means that the firms with similar connectivity would have a higher tendency to connect with each other (for example, highly connected firms could be managing sub-communities in certain areas of production and then connect to other high-degree firms undertaking the same function). This structure can lead to cascading disruptions – where a disruption at one leaf node can spread quickly within the network through the connected hubs (Brintrup et al., 2016).

In contrast, a negative assortativity (i.e. disassortativity) indicates that it is the firms with dissimilar connectivity that tend to pair up in the given network. An unfavourable implication of disassortativity in SCNs is that since high degree firms are less connected to one another, many paths between nodes in the network are dependent on high degree nodes. Therefore, failure of a high degree node in a disassortative network would have a relatively large impact on the overall connectedness of the network (Noldus and Van, 2015).

On the other hand, disassortative networks are generally resilient against cascading impacts arising from targeted attacks – since hub nodes are not connected with each other, the likelihood of disruption impacts cascading from one hub node to another is minimised (Song et al., 2006).

\( Q=\sum_{s=1}^k\left[\frac{l_s}{L}-{\left(\frac{d_s}{2L}\right)}^2\right] \)

where k is the number of modules, L is the number of links in the network, l _s is the number of links between nodes in module s, and d _s is the sum of degrees of nodes in the module s.

To avoid getting a single module in all cases, this measure imposes Q = 0 if all nodes are in the same module or nodes are placed randomly into modules.

SCNs with high modularity contain pronounced communities – i.e. partially segregated subsystems or modules embedded within the overall SCN system (Ravasz et al., 2002; Newman, 2003). Each of these subsystems are generally responsible for a particular specialised task.

The percolation threshold for random node removal is given as;

\( {f}_c=1-\frac{1}{\frac{<{k}^2>}{<k>}-1} \)

where <k > is the mean degree and <k ² > is the second moment of the degree, of all the nodes in the network.

The percolation threshold of a SCN indicates the percentage of firms needed to be randomly removed prior to the overall SCN breaks into many disconnected components (when the giant component ceases to include all the nodes). In summary, this indicates the number of random firm failures that would drive the SCN from a connected state to a fragmented state (loss of overall interconnectivity).