Table 1 Theoretical definitions of the network measures and metrics used in this study and relevance for livestock movements and veterinary epidemiology.

Assortativity (mixing pattern)

The assortativity coefficient measures the proportion of links between and within groups of nodes sharing identical attributes. Its value lies between \(-1\) and +1, negative values mean that nodes are likely to connect with other nodes showing different attributes; a positive value shows homophily (i.e. assortative mixing)⁹⁸.

Reflects the preferential (trade) connections of an epidemiological unit, e.g. a holding or a district, with other epidemiological units which have a similar attribute, e.g. number of trade links (degree assortativity) or location (district assortativity and federal state assortativity)^99,100.

Average path length

The average number of steps along the shortest paths for all possible pairs of nodes. The shortest path is the smallest number of edges from node i to node j¹.

The average number of steps needed for an infection to spread from a random holding to another randomly chosen holding⁶.

Betweenness centrality

For a node k, the number of shortest paths between nodes i and j that pass through k¹. Captures the capacity of a node to propagate an information or infection across the network³¹.

Holdings with high betweenness can act as gatekeepers for controlling or altering the spread of pathogens; removing these holdings will fragment the network⁶.

Degree centrality

For a node, the number of edge(s) connected to it, i.e. the number of neighbor(s) it has. In directed graphs, each node has both, an in-degree, \(k^{in}\) (number of incoming edges) and an out-degree, \(k^{out}\) (number of outgoing edges). Nodes with a high degree, compared to other nodes in the network, are considered well connected and called hubs^1,6,101.

For a holding, the numbers of sources (or trading partners) providing animals (in-degree) or that receive animals from it (out-degree)⁶. Livestock holdings with a high degree centrality are at higher risk of infection and more likely to infect a large number of other holdings in the network⁸¹.

Density (connectance)

Proportion of edges, among the maximum possible number of edges in the network, that are actually existing. The proportion ranges from 0 to 1, where 0 means that all nodes are isolated while a network with density of 1 displays maximal cohesion³.

Represents the fraction of all possible trades among all livestock holdings that are actually present⁶.

Diameter

Length of the longest shortest path, i.e. among all shortest paths between every pair of nodes in the network for which a path exists, the diameter is the length of the longest one. The smaller the diameter, the more connected the network is¹.

Longest geodesic distance between any pair of holdings in the network. A shorter diameter means that the number of generations for a disease to spread throughout the network is reduced^6,100.

Global clustering coefficient (CC)

Fraction of closed triplets, i.e. three nodes linked to each other and forming a closed triangle, among all possible triplets. It lies between 0 and 1^3,5.

Reflects holding-to-holding interactions (or trade links). High clustering coefficient induces a fast spread of diseases in the network⁵.

Range

For a node, the number of nodes that can be reached from it through a path of random length²².

Measure the potential of a holding to spread an infectious disease in the network²².

Strongly connected component (SCC)

A strongly connected component is the subset of nodes in a directed graph in which a directed path exists in both directions between every pair of nodes in the subset¹.

A subset of holdings in a livestock network in which all holdings are mutually accessible by following the direction of the trades⁶. The size of the largest SCC can be used to estimate the lower bound of the maximum epidemic size¹².

Weakly connected component (WCC)

In a directed graph, a subset of nodes in which every pair of nodes is connected by one or more paths, ignoring the direction of edges¹.

A subset of holdings in which a link exists between every pair of holdings, ignoring trade direction. The size of the WCC can be used to estimate the upper bound of the maximum epidemic size¹².