7.2. Applications of temporal networks
87
Figure 7.2: Zoomed-in version of the Laplacian embedding of the Caviar network
without showing edges.
figures, N1 and N3. The third cluster forms an arm down and to the right. This is
the finance group with its key figure, N87. The fourth cluster forms an arm down
and to the left, and represents a group including an airport security connection and a
foreign supplier with its key figure, N76. From the figure, it is clear that the hashish
importation group is the heart of this criminal confederation.
The usefulness of global integration of local pairwise information can be illustrated by considering the phone calls between N1 and N12. Over the 11 time periods
the number of calls is [0 2 2 0 2 4 0 0 0 0 0]. Given that N1 made hundreds of phone
calls in each period, this data by itself seems to suggest that there is only a weak
relationship between them. However, the other members of their groups did telephone each other a lot, and this combined indirect connectivity between N1 and N12
causes them to be embedded quite close together. Their global similarity reflects the
sum over many different paths of weak local similarity. This illustrates how spectral
techniques discover connections even when those involved might try to conceal them
by using intermediaries.
Police records indicated that the two drug groups had known each other and
had been cooperating well during the earlier part of the investigation. After consignments began to be seized, the conflicts between them increased. The embedded
temporal network allows us to evaluate these changing relationships.
Figure 7.4 shows the change of positions, over time, for the 5 key participants,
still treating their communication as undirected. Because the embedding is a global
one of the entire set of snapshots, the distances are meaningful, both for any single
snapshot and over the entire period. We can estimate the change in the strength of
relationships by considering how distances change. During the first 4 time periods,
the relative positions of all 5 are stable, but then there is a pattern of accelerating
disintegration. The first seizure (estimated at $2.5 million) took place during time
87
Figure 7.2: Zoomed-in version of the Laplacian embedding of the Caviar network
without showing edges.
figures, N1 and N3. The third cluster forms an arm down and to the right. This is
the finance group with its key figure, N87. The fourth cluster forms an arm down
and to the left, and represents a group including an airport security connection and a
foreign supplier with its key figure, N76. From the figure, it is clear that the hashish
importation group is the heart of this criminal confederation.
The usefulness of global integration of local pairwise information can be illustrated by considering the phone calls between N1 and N12. Over the 11 time periods
the number of calls is [0 2 2 0 2 4 0 0 0 0 0]. Given that N1 made hundreds of phone
calls in each period, this data by itself seems to suggest that there is only a weak
relationship between them. However, the other members of their groups did telephone each other a lot, and this combined indirect connectivity between N1 and N12
causes them to be embedded quite close together. Their global similarity reflects the
sum over many different paths of weak local similarity. This illustrates how spectral
techniques discover connections even when those involved might try to conceal them
by using intermediaries.
Police records indicated that the two drug groups had known each other and
had been cooperating well during the earlier part of the investigation. After consignments began to be seized, the conflicts between them increased. The embedded
temporal network allows us to evaluate these changing relationships.
Figure 7.4 shows the change of positions, over time, for the 5 key participants,
still treating their communication as undirected. Because the embedding is a global
one of the entire set of snapshots, the distances are meaningful, both for any single
snapshot and over the entire period. We can estimate the change in the strength of
relationships by considering how distances change. During the first 4 time periods,
the relative positions of all 5 are stable, but then there is a pattern of accelerating
disintegration. The first seizure (estimated at $2.5 million) took place during time