Let \(G=(V,E)\) be a graph of order \(N\), and let \(d(i,j)\) be the geodesic distance from vertex \(i\) to vertex \(j\) in \(G\). The "structure statistics" of \(G\) are then given by the series \(s_0,\ldots,s_{N-1}\), where \(s_i = \frac{1}{N^2} \sum_{j \in V} \sum_{k \in V} I\left(d(j,k) \le i\right) \) and \(I\) is the standard indicator function. Intuitively, \(s_i\) is the expected fraction of \(G\) which lies within distance i of a randomly chosen vertex. As such, the structure statistics provide an index of global connectivity.
Structure statistics have been of particular importance to biased net theorists, because of the link with Rapoport's original tracing model. They may also be used along with component distributions or connectedness scores as descriptive indices of connectivity at the graph-level.