Topological Structure of Manufacturing Industry Supply Chain Networks
Table 5
Network-level metrics used and their SCN implications.
Mathematical representation
SCN implication
Average degree
For an undirected network:
For a directed network:
where and are the total number of links and nodes in the network.
Indicates, on average, how many connections a given firm has. A higher average degree implies good interconnectivity among the firms in the SCN.
Network diameter
where is the number of hops traversed along the shortest path from node to .
The diameter of a SCN is the largest distance between any two firms in the network (i.e., the maximum shortest path length). 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.
Network density ()
where is the mean degree of all the nodes and is the total number of nodes, in the network.
Density of a SCN indicated 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 [50].
Network centralisation ()
where is the total number of nodes in the network and is the maximum degree of a node within the network. Density is determined as per the equation below.
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 [23].
Network heterogeneity ()
where is the mean degree and 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 very few firms).
Average clustering coefficient ()
where is the total number of nodes in the network and is the number of triangles connected to node divided by the number of triples centered around node .
The 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 [23]. Therefore, the higher the clustering coefficient, the more dependent suppliers are on each other for production [46].
Characteristic (or average) path length ()
The characteristic path length is
where is the total number of nodes in the network and is the shortest topological distance between nodes and .
Characteristic (or average) path length is the average topological distance between all pairs of firms (along the shortest path) in a SCN. It measures how efficiently information can be transferred between pairs of firms within a SCN.
The degree distribution of an undirected scale-free network is approximated with power law as follows:
where is the degree of the node and is the degree exponent (also known as the power law or scale-free exponent). Directed networks generally include two separate degree distributions, one for the in-degree and another for the out-degree. In such cases, there will be two separate degree exponents, i.e., and .
SCNs with 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 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. In particular, when is less than or equal to 2, the network topology is referred to as a “hub and spoke” topology; when is higher than 2 but less than 3, the network topology is referred to as scale-free; and when is higher than 3, the network topology is random.
Assortativity is formally defined as a correlation function of excess degree distributions and link distribution of a network. For undirected networks, when degree distribution is denoted as and excess degree (remaining degree) distribution is denoted as , one can introduce the quantity 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
where is the standard deviation of which is given as
Assortativity, is a value between −1 and 1. For , the network is assortative; for , the network is neutral; and for , the network is disassortative.
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 subcommunities 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 [46]. In contrast, a negative assortativity indicates that it is the firms with dissimilar connectivity that tend to pair up in the given network. Note that assortativity can also be defined in terms of node attributes other than the degree.
