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The idea of minimum curvilinearity (MC) is that the hidden geometry ofcomplex networks, in particular when they are sufficiently sparse, clustered,small-world and heterogeneous, can be efficiently navigated using the minimumspanning tree (MST), which is a greedy navigator. The local topologicalinformation drives the global geometrical navigation and the MST can beinterpreted as a growing path that greedily maximizes local similarity betweenthe nodes attached at each step by globally minimizing their overall distancesin the network. This is also valid in absence of the network structure and inpresence of only the nodes geometrically located over the network generativemanifold in a high-dimensional space. We know that random geometric graphs inthe hyperbolic space are an adequate model for realistic complex networks: theexplanation of this connection is that complex networks exhibit hierarchical,tree-like organization, and in turn the hyperbolic geometry is the geometry oftrees. Here we show that, according to a mechanism that we define similarityattachment, the visited node sequence of a network automaton can efficientlyapproximate the nodes' angular coordinates in the hyperbolic disk, thatactually represent an ordering of their similarities. This is a consequence ofthe fact that the MST, during its greedy growing process, at each stepsequentially attaches the node most similar (less distant) to its own cohort.Minimum curvilinear automata (MCA) displays embedding accuracy which seemssuperior to HyperMap-CN and inferior to coalescent embedding, however its linkprediction performance on real networks is without precedent for methods basedon the hyperbolic space. Finally, depending on the data structure used to buildthe MST, the MCA's time complexity can also approach a linear dependence fromthe number of edges.

Minimum curvilinear automata with similarity attachment for network embedding and link prediction in the hyperbolic space