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

From: Association between tuberculosis in men and social network structure in Kampala, Uganda

Statistic Definition Equation Notation
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 \(i,j\)
\({b}_{s\in \mathrm{1,2},...n}\)
Number of times node is on shortest path between pairs of other nodesa \(\sum\nolimits_{u \ne s \ne v} {\frac{{\sigma_{uv} \left( s \right)}}{{\sigma_{uv} }}}\) \({\sigma }_{uv}\) 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 \(s\)
\({c}_{s\in \mathrm{1,2},...n}\)
Inverse of the average length of shortest path to all other nodesa \(\frac{1}{{\sum\nolimits_{i \ne s} {d_{si} } }}\) \({d}_{si}\) 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 casea \({\text{min}}\left( {d_{st, t \ne s} } \right)\) \(t\) is the set of TB cases
  1. aNetwork distance, closeness, and betweenness were calculated within the giant component because path length is not defined for disconnected graphs