5.3. Applications of directed networks
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families to the Medici group since the Medici in node is close to the oligarch families
and the Medici out node is close to the Medici group.
We can use this data for edge prediction by computing and comparing the
distances between the out version of a node and the in version of the other and the
converse. For example, the Strozzi family were one of the most powerful of the
oligarchs. The distance from Medici to Strozzi is 0.0458 and from Strozzi to Medici
is 0.0413 so we would (weakly) predict that such an edge, if created, would point
from Strozzi to Medici. At the time of this dataset, Medici was still relatively weak,
so this seems plausible. A more asymmetric example is the relationship between
the Tornabuoni family and the Guasconi family. The distance from Tornabuoni to
Guasconi is 0.2186 while the converse distance is 0.2412, so we would predict the
direction of such an edge to be from Tornabuoni to Guasconi.
While the figures could provide more detailed useful information if they could
be rotated and zoomed into in real time, there is also useful information to be gleaned
from measures computed about the embedding. The in-out normalized edge length
and the average neighborhood edge length are given in Table 5.1. Recall that the
in-out lengths reveal the amount of asymmetric flow through each (original) node,
while the normalized edge length reveals the amount of local distortion associated
with each (original) edge.
The Medici-aligned families tend to have long normalized edges since they
are weakly connected. However, the fact that Medici has long normalized edges but
high degrees makes it special. The average neighborhood edge length indicates that
Medici is the key broker between the two blocs in the embedding. The in-out edge
associated with Medici has the highest normalized value of all normalized added
edges, indicating their importance to flow of influence, and the direction indicates
that the oligarch families can influence the Medici-aligned families via the Medici
family, but influence does not flow strongly in the reverse direction. Again, this is
consistent with historians’ views of this period.
Macaque brain connections
The third real-world dataset we use is the visuotactile brain areas and connection
network model of the macaque monkey. The model was defined by Felleman & Van
Essen [32] as well as Hilgetag et al. [39] and used by N´ egyessy et al. [67]. The
model consists of 45 areas and 463 directed connections.
Figure 5.6 shows the embeddings of the network of the main cortical areas in
two dimensions. Both embeddings reveal similar structure for this network, and there
is a clear separation between visual and sensorimotor areas in both embeddings. The
area VIP occupies a central position in both embeddings, and areas LIP, 7a, and 46
of the visual cortex are also close to the center. This is similar to the findings of
N´ egyessy et al. [67]. However, in Figure 5.6(a) (Chung’s embedding), the perirhinal
cortex areas, 35 and 36, are far from the visual cortical areas, even though they have
connections to both groups. In Figure 5.6(b) (our new embedding), areas 35 and 36
are not only closer to the visual cortical areas, but it becomes obvious that they tend
to transmit information from visual cortical areas to sensorimotor areas.
55
families to the Medici group since the Medici in node is close to the oligarch families
and the Medici out node is close to the Medici group.
We can use this data for edge prediction by computing and comparing the
distances between the out version of a node and the in version of the other and the
converse. For example, the Strozzi family were one of the most powerful of the
oligarchs. The distance from Medici to Strozzi is 0.0458 and from Strozzi to Medici
is 0.0413 so we would (weakly) predict that such an edge, if created, would point
from Strozzi to Medici. At the time of this dataset, Medici was still relatively weak,
so this seems plausible. A more asymmetric example is the relationship between
the Tornabuoni family and the Guasconi family. The distance from Tornabuoni to
Guasconi is 0.2186 while the converse distance is 0.2412, so we would predict the
direction of such an edge to be from Tornabuoni to Guasconi.
While the figures could provide more detailed useful information if they could
be rotated and zoomed into in real time, there is also useful information to be gleaned
from measures computed about the embedding. The in-out normalized edge length
and the average neighborhood edge length are given in Table 5.1. Recall that the
in-out lengths reveal the amount of asymmetric flow through each (original) node,
while the normalized edge length reveals the amount of local distortion associated
with each (original) edge.
The Medici-aligned families tend to have long normalized edges since they
are weakly connected. However, the fact that Medici has long normalized edges but
high degrees makes it special. The average neighborhood edge length indicates that
Medici is the key broker between the two blocs in the embedding. The in-out edge
associated with Medici has the highest normalized value of all normalized added
edges, indicating their importance to flow of influence, and the direction indicates
that the oligarch families can influence the Medici-aligned families via the Medici
family, but influence does not flow strongly in the reverse direction. Again, this is
consistent with historians’ views of this period.
Macaque brain connections
The third real-world dataset we use is the visuotactile brain areas and connection
network model of the macaque monkey. The model was defined by Felleman & Van
Essen [32] as well as Hilgetag et al. [39] and used by N´ egyessy et al. [67]. The
model consists of 45 areas and 463 directed connections.
Figure 5.6 shows the embeddings of the network of the main cortical areas in
two dimensions. Both embeddings reveal similar structure for this network, and there
is a clear separation between visual and sensorimotor areas in both embeddings. The
area VIP occupies a central position in both embeddings, and areas LIP, 7a, and 46
of the visual cortex are also close to the center. This is similar to the findings of
N´ egyessy et al. [67]. However, in Figure 5.6(a) (Chung’s embedding), the perirhinal
cortex areas, 35 and 36, are far from the visual cortical areas, even though they have
connections to both groups. In Figure 5.6(b) (our new embedding), areas 35 and 36
are not only closer to the visual cortical areas, but it becomes obvious that they tend
to transmit information from visual cortical areas to sensorimotor areas.