  The idea of minimum curvilinearity (MC) is that the hidden geometry of
complex networks, in particular when they are sufficiently sparse, clustered,
small-world and heterogeneous, can be efficiently navigated using the minimum
spanning tree (MST), which is a greedy navigator. The local topological
information drives the global geometrical navigation and the MST can be
interpreted as a growing path that greedily maximizes local similarity between
the nodes attached at each step by globally minimizing their overall distances
in the network. This is also valid in absence of the network structure and in
presence of only the nodes geometrically located over the network generative
manifold in a high-dimensional space. We know that random geometric graphs in
the hyperbolic space are an adequate model for realistic complex networks: the
explanation of this connection is that complex networks exhibit hierarchical,
tree-like organization, and in turn the hyperbolic geometry is the geometry of
trees. Here we show that, according to a mechanism that we define similarity
attachment, the visited node sequence of a network automaton can efficiently
approximate the nodes' angular coordinates in the hyperbolic disk, that
actually represent an ordering of their similarities. This is a consequence of
the fact that the MST, during its greedy growing process, at each step
sequentially attaches the node most similar (less distant) to its own cohort.
Minimum curvilinear automata (MCA) displays embedding accuracy which seems
superior to HyperMap-CN and inferior to coalescent embedding, however its link
prediction performance on real networks is without precedent for methods based
on the hyperbolic space. Finally, depending on the data structure used to build
the MST, the MCA's time complexity can also approach a linear dependence from
the number of edges.
