7.1. Temporal networks
83
The snapshot matrix at time t is combined with the matrices from previous
snapshots with a down-weighting given by parameter α. This value can be chosen to
vary the relative impact of historical network structure on the analysis of the current
snapshot. The aggregate network adjacency matrix A t is computed as follows: if
t = 1, A 1 = W 1 , else A t = W t + α ∗ A t−1 .
Each of these aggregate adjacency matrices is then converted to a random walk
matrix by dividing each row by its row sum to convert entries to probabilities, and
producing aggregate random walk matrices R t .
There is also a low-level technical problem to be addressed. A node may be
isolated in one (or more) of the aggregate matrices, either because it genuinely had
no relationships during that time, or because it is present in other layers and so was
inserted as a placeholder. The row sum of its row in the aggregate random walk matrix will be zero, so we put a 1 in the corresponding diagonal position, representing
a self-loop. This ensures that every row sum is 1.
These random walk matrices are then put on the main diagonal of a cn × cn
matrix M rw (for c time periods), and the edges connecting different versions of the
same nodes are weighted based on a probability, β . (We could also use the weighting
schemes we used previously, basing the “vertical” weights on the connectivity of
the nodes in each layer but, in this setting, the node versions we are connecting
vertically are the same node at different times, so we do not expect big changes in its
connectivity.)
Changing the value of β can be used to bind the structures at different time
steps more or less tightly together. For example, if the goal is to understand the
behavior of an outlier, a large value of β tends to align the central structure of the
network and so make it easier to see changes at the periphery.
The cn × cn random walk matrix M rw is defined as:
M rw =
(1 − β )R 1 · · ·
β
c−1 ∗ I
· · ·
β
c−1 ∗ I
. . .
. . .
. . .
. . .
. . .
β
c−1 ∗ I
· · · (1 − β )R t · · ·
β
c−1 ∗ I
. . .
. . .
. . .
. . .
. . .
β
c−1 ∗ I
· · ·
β
c−1 ∗ I
· · · (1 − β )R c
where I is an identity matrix of size n. The entire matrix M rw reflects not only the
structure during each time period, but also the evolution of the social network with
time.
If the edges of the network are undirected, then the required construction for
the embedding is the one we used for typed networks, with each time period corresponding to a color, and with a slightly different weighting for the “vertical” edges.
The choice of α, which is independent of the spectral embedding, reflects the
amount of smoothing in the base data. It may be used, for example, to compensate
for sampling omissions when data is collected over short time intervals or in settings
such as law enforcement where concealment is an issue.
The choice of β determines how much influence the structure of the social
83
The snapshot matrix at time t is combined with the matrices from previous
snapshots with a down-weighting given by parameter α. This value can be chosen to
vary the relative impact of historical network structure on the analysis of the current
snapshot. The aggregate network adjacency matrix A t is computed as follows: if
t = 1, A 1 = W 1 , else A t = W t + α ∗ A t−1 .
Each of these aggregate adjacency matrices is then converted to a random walk
matrix by dividing each row by its row sum to convert entries to probabilities, and
producing aggregate random walk matrices R t .
There is also a low-level technical problem to be addressed. A node may be
isolated in one (or more) of the aggregate matrices, either because it genuinely had
no relationships during that time, or because it is present in other layers and so was
inserted as a placeholder. The row sum of its row in the aggregate random walk matrix will be zero, so we put a 1 in the corresponding diagonal position, representing
a self-loop. This ensures that every row sum is 1.
These random walk matrices are then put on the main diagonal of a cn × cn
matrix M rw (for c time periods), and the edges connecting different versions of the
same nodes are weighted based on a probability, β . (We could also use the weighting
schemes we used previously, basing the “vertical” weights on the connectivity of
the nodes in each layer but, in this setting, the node versions we are connecting
vertically are the same node at different times, so we do not expect big changes in its
connectivity.)
Changing the value of β can be used to bind the structures at different time
steps more or less tightly together. For example, if the goal is to understand the
behavior of an outlier, a large value of β tends to align the central structure of the
network and so make it easier to see changes at the periphery.
The cn × cn random walk matrix M rw is defined as:
M rw =
(1 − β )R 1 · · ·
β
c−1 ∗ I
· · ·
β
c−1 ∗ I
. . .
. . .
. . .
. . .
. . .
β
c−1 ∗ I
· · · (1 − β )R t · · ·
β
c−1 ∗ I
. . .
. . .
. . .
. . .
. . .
β
c−1 ∗ I
· · ·
β
c−1 ∗ I
· · · (1 − β )R c
where I is an identity matrix of size n. The entire matrix M rw reflects not only the
structure during each time period, but also the evolution of the social network with
time.
If the edges of the network are undirected, then the required construction for
the embedding is the one we used for typed networks, with each time period corresponding to a color, and with a slightly different weighting for the “vertical” edges.
The choice of α, which is independent of the spectral embedding, reflects the
amount of smoothing in the base data. It may be used, for example, to compensate
for sampling omissions when data is collected over short time intervals or in settings
such as law enforcement where concealment is an issue.
The choice of β determines how much influence the structure of the social