Figure 1.
Left: A directed tree. Right: A more realistic example. The edges in blue are the ones which contribute to the value of trust from Bob to Alice, according to Eq. 7.
Figure 2.
Illustration of the paths used to calculate according to Eq. 7.
The vertices are the in-neighbours of
, and the values
are the values of best trust (Eq. 1) from
to
, with vertex
removed from the graph.
Figure 3.
Neighbourhood of vertex with out-neighbours
with direct trust
.
The best trust from to an arbitrary vertex
,
, is given as a function of
and
, according to Eq. 11.
Figure 4.
Schematic representation of the self-consistency for in Eq. 14.
Each term corresponds to the probability of the vertex having a given number of out-neighbours, and the maximum best trust transitivity being equal the desired value.
Figure 5.
Average values of best trust and pervasive trust
as a function of the fraction of edges with absolute trust
.
Top left: Networks with Poisson in- and out-degree distributions, and uniform trust distribution. Top right and bottom right: Poisson distribution, and single-valued trust distribution. Bottom left: Zipf distribution, and single-valued trust distribution. Solid lines correspond to analytical solutions, and symbols to numerical realizations of several networks of different sizes: (red),
(green) and
(blue) nodes. The dashed line shows the average direct trust
.
Table 1.
Summary of statistics for the whole PGP network (above) and the largest strongly connected component (below).
Figure 6.
Number of keys and signatures as a function of time for the strongly connected component of the PGP network, and waiting time distribution between new keys and signatures.
The straight lines are power-laws , with
(top) and
(bottom).
Figure 7.
Several statistical properties of the PGP Network.
Top left: In- and out-degree distributions, and
respectively. The solid line corresponds to a power-law with exponent
. Top right: Average in- and out-degree of the nearest out-neighbours, as a function of the in- and out-degree. Bottom left: Average lustering coefficient as a function of in- and out-degree. Bottom right: Distribution of community sizes, for the unmodified and shuffled versions of the network. The solid lines correspond to power-laws with exponent
(top) and
(bottom).
Figure 8.
Reciprocity statistics of the PGP network.
Left: Average out-degree as a function of the in-degree of the same vertex. Right: Average edge reciprocity, as a function of the in or out-degree of the source vertex.
Table 2.
The eleven keys with the largest number of signatures in the network, their respective in-degree , out-degree
, average in-degree of the nearest out-neighbours
, clustering coefficient
, and date of creation.
Figure 9.
Two example communities of the PGP network, and their in- and out-degree distributions.
The colors on the vertices correspond to the top-level domain (TLD) of the email addresses. Top: Community containing the CACert.org certificate authority. Bottom: Community composed mostly of Austrian email addresses (.at TLD).
Figure 10.
Average best trust and pervasive trust
, as a function of the fraction of edges with absolute trust
, for the PGP network.
The different curves correspond to the different trust distribution scenarios described in the text.
Figure 11.
Average best in-trust and pervasive in-trust
, as a function of the in-degree
and the fraction of edges with absolute trust
, for the PGP network.
The different plots correspond to the different trust distribution scenarios described in the text: (a) Random distribution, (b) authority-centered distribution and (c) community-centered distribution. The plots (d) correspond to a community-centered distribution, done on a shuffled version of network, with the same degree sequence.