Table 1 Social Network Estimates used to describe the Node Position of Index Individuals (\(s\in \{\mathrm{1,2},..n\}\)) within the Kampala Network

Node degree,

\({k}_{s \in \mathrm{1,2},\dots n}\)

Number of adjacent edges

\(\sum\nolimits_{j = 1}^{N} {A_{s,j} }\)

Adjacency matrix, \({A}_{ij}=1\), if we identified contact between \(i,j\)

Betweenness,

\({b}_{s\in \mathrm{1,2},...n}\)

Number of times node is on shortest path between pairs of other nodes^a

\(\sum\nolimits_{u \ne s \ne v} {\frac{{\sigma_{uv} \left( s \right)}}{{\sigma_{uv} }}}\)

\({\sigma }_{uv}\) is the total number of shortest paths from node \(u\) to \(v\) and \({\sigma }_{uv}\left(s\right)\) is the number of those paths that pass through \(s\)

Closeness,

\({c}_{s\in \mathrm{1,2},...n}\)

Inverse of the average length of shortest path to all other nodes^a

\(\frac{1}{{\sum\nolimits_{i \ne s} {d_{si} } }}\)

\({d}_{si}\) is the network distance between nodes \(s\) and \(i\)

Distance to TB case,

\({y}_{s\in \mathrm{1,2},...n}\)

Network distance to a TB case^a

\({\text{min}}\left( {d_{st, t \ne s} } \right)\)

\(t\) is the set of TB cases