Parenclitic and Synolytic Networks Revisited

Abstract

Parenclitic networks provide a powerful and relatively new way to coerce multidimensional data into a graph form, enabling the application of graph theory to evaluate features. Different algorithms have been published for constructing parenclitic networks, leading to the question—which algorithm should be chosen? Initially, it was suggested to calculate the weight of an edge between two nodes of the network as a deviation from a linear regression, calculated for a dependence of one of these features on the other. This method works well, but not when features do not have a linear relationship. To overcome this, it was suggested to calculate edge weights as the distance from the area of most probable values by using a kernel density estimation. In these two approaches only one class (typically controls or healthy population) is used to construct a model. To take account of a second class, we have introduced synolytic networks, using a boundary between two classes on the feature-feature plane to estimate the weight of the edge between these features. Common to all these approaches is that topological indices can be used to evaluate the structure represented by the graphs. To compare these network approaches alongside more traditional machine-learning algorithms, we performed a substantial analysis using both synthetic data with a priori known structure and publicly available datasets used for the benchmarking of ML-algorithms. Such a comparison has shown that the main advantage of parenclitic and synolytic networks is their resistance to over-fitting (occurring when the number of features is greater than the number of subjects) compared to other ML approaches. Secondly, the capability to visualise data in a structured form, even when this structure is not a priori available allows for visual inspection and the application of well-established graph theory to their interpretation/application, eliminating the “black-box” nature of other ML approaches.