Geometric separability of mesoscale patterns in embedding representation and visualization of multidimensional data and complex networks

Complexity science studies physical phenomena that cannot be explained by the mere analysis of the single units of a system but requires to account for their interactions. A feature of complexity in connected systems is the emergence of mesoscale patterns in a geometric space, such as groupings in bird flocks. These patterns are formed by groups of points that tend to separate from each other, creating mesoscale structures. When multidimensional data or complex networks are embedded in a geometric space, some mesoscale patterns can appear respectively as clusters or communities, and their geometric separability is a feature according to which the performance of an algorithm for network embedding can be evaluated. Here, we introduce a framework for the definition and measure of the geometric separability (linear and nonlinear) of mesoscale patterns by solving the travelling salesman problem (TSP), and we offer experimental evidence on embedding and visualization of multidimensional data or complex networks, which are generated artificially or are derived from real complex systems. For the first time in literature the TSP’s solution is used to define a criterion of nonlinear separability of points in a geometric space, hence redefining the separability problem in terms of the travelling salesman problem is an innovation which impacts both computer science and complexity theory.