For over a year now, thanks to the hard work of [1]M. Drew Streib and [2]Jason Harris, analyses of the global PGP keyring have been made available on a roughly monthly basis. These analyses include detailed information about how far each individual key in the strongly connected component (SCC) is from other keys; the "ranking" of each key within the SCC; and a list of the highest-ranking keys in the ring, with fun facts about their holders. It also includes summary information about the total number of keys in the SCC and the average mean distance between keys in the SCC. It is these last two tidbits that we will use to draw some conclusions about the "density" of the SCC and its growth patterns over time.
What follows below is a preliminary analysis of the size and density of the SCC. It is based on a number of assumptions, including assumptions about the validity of formulas which were dug up with google and have not been checked for accuracy. Corrections to any of these assumptions can be sent to vorlon@debian.org.
Let us assume the SCC can be modeled by a sphere of uniform density [3]. From [4], we have the formula r = (35/36)A, where r is the radius of the sphere and A is the average distance between any two points within the sphere. Since the volume V of a sphere of radius r is given by the formula V = 4/3 * π r3, we can now calculate the volume of our theoretically spherical SCC as
Applying these formulas to the keyring results at [1] and [2], we arrive at the figures in the following table.
| date | SCC size | number of reachable keys |
average mean distance | volume | density |
|---|---|---|---|---|---|
| 2002-10-06 | 15169 | 51732 | 6.5101 | 1062.06 | 14.283 |
| 2002-09-22 | 14973 | 51404 | 6.5084 | 1061.23 | 14.109 |
| 2002-09-08 | 14797 | 51085 | 6.5561 | 1084.73 | 13.641 |
| 2002-08-25 | 14586 | 50672 | 6.5327 | 1073.16 | 13.591 |
| 2002-08-11 | 14353 | 50199 | 6.5222 | 1067.99 | 13.439 |
| 2002-07-28 | 14159 | 49866 | 6.5105 | 1062.25 | 13.329 |
| 2002-07-14 | 13879 | 49232 | 6.4960 | 1055.17 | 13.153 |
| 2002-06-30 | 13480 | 47811 | 6.5204 | 1067.11 | 12.632 |
| 2002-06-17 | 13350 | 47463 | 6.5325 | 1073.06 | 12.441 |
| 2002-04-11 | 12285 | 44898 | 6.5809 | 1097.09 | 11.198 |
| 2002-03-19 | 11997 | 44372 | 6.6397 | 1126.76 | 10.647 |
| 2002-02-?? | 11792 | 43946 | 6.6304 | 1122.03 | 10.510 |
| 2002-01-?? | 11484 | 43148 | 6.6514 | 1132.72 | 10.138 |
These figures show a number of trends in the growth of the global SCC. First, the rate of accretion of the SCC has remained relatively constant, averaging roughly 114 new keys per week over the past 16 weeks. The highest rate of increase over any two-week period was seen between 2002/06/30 and 2002/07/14, with 399 new keys added to the ring. This also corresponds to one of the two periods of greatest reduction in the volume of the SCC, this concurrent contraction and increase in mass suggesting the effect of an organized keysigning such as the one that took place at DebConf 2 during this time period. The other period of greatest contraction, from 2002-09-08 to 2002-09-22 (showing an overall reduction in volume of 2.1%) also suggests the presence of a mass-keysigning at work; if anyone knows of an event during this time period that would qualify, please do let me know.
Given the substantial impact that unique mass keysigning events appear to have on the rate of growth of the SCC, it is reasonable to think that the period covered by these reports is not long enough to permit drawing further conclusions about the rate of growth over time. However, in general a linear growth rate of the keyring would be a disappointment, suggesting that PGP advocacy is a constant even as the keyring grows. On the other hand, extrapolating back to the creation of PGP we see that the current rate of growth is certainly higher than it has been in the past, since a sustained rate of 114 new keys per week would cause the SCC to double its present size in only 2 1/2 years.
The other significant trend we see is that even as the number of keys in the SCC grows, the overall volume occupied by the SCC remains constant, or even decreases. As a result, the estimated density of the SCC has increased in every period recorded in the above table. This indicates that even as new keys are added to the edges of the SCC, this growth is more than offset by the strengthening of connections between keys, such that there is a net gain in the reachability of all keys in the set, not just those that were present at the beginning of the period. One possible factor at work here is the tendency of active keysigners to be nearer the center of the SCC, or to gravitate there as a result of their activity, resulting in a sort of "in-filling" of the SCC. Other possibilities are that the SCC's geographical coverage is now great enough that new keys are entering the ring with multiple connections, or that many of those already in the SCC are actively seeking signatures from others in the SCC. The data currently available does not give enough information to determine which of these hypotheses might be correct. Whatever the cause, this trend has positive implications for the Web of Trust, which is growing in strength even as it grows in numbers.
[1] http://dtype.org/keyanalyze/
[2] http://keyserver.kjsl.com/~jharris/ka/
[3] http://www.bbsonline.org/Preprints/SHEPARD-SPECIAL/Commentators/.Boroditsky.html
[4] http://www.math.niu.edu/~rusin/known-math/96/spheric