  Linear layouts are a graph visualization method that can be used to capture
an entry pattern in an adjacency matrix of a given graph. By reordering the
node indices of the original adjacency matrix, linear layouts provide knowledge
of latent graph structures. Conventional linear layout methods commonly aim to
find an optimal reordering solution based on predefined features of a given
matrix and loss function. However, prior knowledge of the appropriate features
to use or structural patterns in a given adjacency matrix is not always
available. In such a case, performing the reordering based on data-driven
feature extraction without assuming a specific structure in an adjacency matrix
is preferable. Recently, a neural-network-based matrix reordering method called
DeepTMR has been proposed to perform this function. However, it is limited to a
two-mode reordering (i.e., the rows and columns are reordered separately) and
it cannot be applied in the one-mode setting (i.e., the same node order is used
for reordering both rows and columns), owing to the characteristics of its
model architecture. In this study, we extend DeepTMR and propose a new one-mode
linear layout method referred to as AutoLL. We developed two types of neural
network models, AutoLL-D and AutoLL-U, for reordering directed and undirected
networks, respectively. To perform one-mode reordering, these AutoLL models
have specific encoder architectures, which extract node features from an
observed adjacency matrix. We conducted both qualitative and quantitative
evaluations of the proposed approach, and the experimental results demonstrate
its effectiveness.
