Fig. 5.
The relationship between friendship probability and rank. The probability P(r) of a link from a randomly chosen source u to the rth closest node to u, that is, the node v such that ranku(v) = r, in the LiveJournal network, averaged over 10,000 independent source samples. A link from u to one of the nodes Sδ = {v:d(u, v) = δ}, where the people in Sδ are all tied for rank r + 1,..., r +|Sδ|, is counted as a (1/|Sδ|) fraction of a link for each of these ranks. As before, the value of ε represents the background probability of a friendship independent of geography. (A) Data for every 20th rank are shown. Because we do not have fine-grained geographic data, on average the ranks r through r + 1,305 all represent people in the same city; thus we have little data to distinguish among these ranks. (B) The data are averaged into buckets of size 1,306: for each displayed rank r, the average probability of a friendship over ranks {r – 652,..., r + 653} is shown. (C and D) The same data are replotted (unaveraged and averaged, respectively), correcting for the background friendship probability: we plot the rank r versus P(r) – 5.0 × 10–6.