From this one deduces that the inverse of closeness, a length scale associated with each vertex, varies approximately linearly with the logarithm of degree. The relationship is not exact but it captures a correlation between closeness and degree in large number of networks constructed from real data and this success suggests that shortest-path trees can be a useful approximation in network analysis.
This implies that the inverse of closeness, a measure related to each vertex's distance, is roughly proportional to the logarithm of its degree. Although not precise, this relationship reveals a connection between closeness and degree in many networks constructed from real-world data, suggesting that shortest-path trees can be a useful approximation for network analysis.