Figure 1.
Pictorial representation of the Kruskal decomposition.
The cube on the left is the original 3-way tensor, which is represented as the sum of rank-1 tensors (on the right), each generated as the outer product of three 1-dimensional vectors (thin rectangles). Each of the rank-1 terms on the right corresponds to one component.
Figure 2.
Schematic representation of the factorization result for an undirected temporal network.
The factors A and C are matrices with R columns, each one corresponding to one extracted component. The rows of A correspond to network nodes, and the rows of C to discrete time intervals. The entries of A give the membership weight of nodes to the different components. The entries of C give the activity level of components at different intervals.
Figure 3.
Community-activity structure detection via non-negative tensor factorization.
The original temporal network is represented as a three-way tensor, which is then decomposed by using non-negative tensor factorization. The complexity of the model (number of components ) is tuned by using quality indicators that provide information on the stability, coverage or redundancy of the decomposition.
Figure 4.
NTF decomposition of an empirical temporal network.
Left panel: core consistency curve. For each value of the number of components used for factorization, the core consistency values for the 5 best decompositions are reported (crosses). The solid line is a guide for the eye. A crossover between two regimes is visible for
. Right panel: component-node matrix for
components. Rows correspond to network nodes and columns to components. The matrix is obtained from the factor
by classifying each node as belonging (lighter rectangles) or not belonging (dark blue rectangles) to a given component. The order of the nodes has been rearranged to expose the block structure of the matrix. Colors identify components, and the community structures that can be matched to school classes are annotated with the corresponding class name.
Figure 5.
Distribution of the membership weights.
Sample histogram of the membership weights for one component of the decomposition (one column of factor for
).
Figure 6.
Activity patterns of the extracted components.
Each panel corresponds to one component obtained by non-negative tensor factorization of the school temporal network, with , and provides the activity level of the component as a function of the time of the day. For clarity, the panels only show the activity patterns for the first day of data (see Fig. S
for the second day). Components that can be matched to classes are marked as class. The other three components that correspond to mixed classes exhibit activity patterns that can be understood in terms of gatherings in the social spaces of the school.
Table 1.
Class structure recovered by non-negative tensor factorization as a function of the number of components.
Table 2.
Components vs classes for .
Figure 7.
Activity patterns of components vs co-location in social spaces.
Each panel corresponds to one of the three components of Fig. 6 that cannot be matched to school classes. The activity pattern of each component is compared with the time series of the co-location vector (
) for two choices of
that correspond, respectively, to the cafeteria and the playground, i.e., the social spaces of the school. The horizontal axis is the time of the day, and the vertical axis has been rescaled for each curve so that its maximum is
.
Table 3.
Reference score matrix containing the correct number of students in each class.
Table 4.
Matrix obtained through NTF containing the number of students in each community projected over the different classes.