Figure 1.

Basic principles and examples of ElPiGraph usage. (A) Schematic workflow of the ElPiGraph method. Left, construction of the elastic graph starts by defining the initial graph topology and embedding it into the data space. The graph structure is fit to the data, using minimization of the mean square error regularized by the elastic energy. The elastic energy includes a term reflecting the overall stretching of the graph (symbolically shown as contractive red springs) and a term reflecting the overall bending of the graph branches and the harmonicity of branching points (shown as repulsive green springs). Middle, ElPiGraph explores a large region of the structural space by exhaustively applying a set of graph rewriting rules (graph grammars) and selecting, at each step, the structure leading to the minimum overall energy of the graph embedding. (B) Example of elastic matrix for a simple graph and transforming this matrix into several parts used for optimization (see details in the Methods). (C) Left and middle, illustration of the robust local workflow of ElPiGraph, which makes it possible to deal with the presence of noise. In the global version, the graph structure is fit to all data points at the same time, while in the local version, the structure is fit to the points in the local graph neighborhood, which expands as the graph grows. Right, an illustration of principal graph ensemble approach: 100 elastic principle graphs are superimposed, each constructed on a fraction of data points randomly sampled at each execution. (D) Application of a set of graph grammar rules to generate a set of tree topologies with up to seven nodes.