REVIEW article

Front. Appl. Math. Stat.
Sec. Mathematical Biology
doi: 10.3389/fams.2022.947053

A unified Framework for analyzing complex systems: Juxtaposing the (Kernel) PCA method and Graph Theory

 Andreas A. Ioannides1*, Constantinos Kourouyiannis1, Christodoulos Karittevlis1, 2,  Lichan Liu1, Ioannis Michos2, Michalis Papadopoulos2, Evangelos Papaefthymiou2, Orestis Pavlou2,  Vicky Papadopoulou Lesta2 and Andreas Efstathiou2
  • 1Laboratory for Human Brain Dynamics, AAI Scientific Cultural Services Limited, Cyprus
  • 2European University Cyprus, Cyprus
Provisionally accepted:
The final, formatted version of the article will be published soon.

In this article, we present a unified framework for the analysis and characterization of a complex system and demonstrate its application in two diverse fields: neuroscience and astrophysics. The framework brings together techniques from graph theory, applied mathematics and dimensionality reduction through Principal Component Analysis, separating linear PCA and its extensions. The implementation of the framework maps an abstract multidimensional set of data into reduced representations, which enables the extraction of the most important properties (features) characterizing the complex data. The reduced representations can be sign-posted by known examples to provide meaningful descriptions of the results that can spur explanations of phenomena and support or negate proposed mechanisms in each application. In this work, we focus on the clustering aspects highlighting relatively fixed stable properties of the system under study. We include examples where clustering leads to semantic maps and representations of dynamic processes within the same display.

Although the framework is composed of existing theories and methods, its usefulness is exactly that it brings together seemingly different approaches, into a common framework, revealing their differences/commonalities, advantages/disadvantages and suitability for a given application. The framework provides a number of different computational paths and techniques to choose from, as it concerns the dimension reduction method to apply, the clustering approaches to be used, as well as the representations (embeddings) of the data in the reduced space. Although here it is applied to just two scientific domains, neuroscience and astrophysics, it can potentially be applied in several other sciences, since it is not based on any specific domain knowledge.

Keywords: Principal Component Analysis, Kernel PCA, graphs, graph theory, graph clustering, Graph spectral clustering, Manifold Learning, Semantic maps, Sleep staging, Early Brain Responses, Galaxy evolution

Received: 18 May 2022; Accepted: 31 Aug 2022.

Copyright: © 2022 Ioannides, Kourouyiannis, Karittevlis, Liu, Michos, Papadopoulos, Papaefthymiou, Pavlou, Papadopoulou Lesta and Efstathiou. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Andreas A. Ioannides, Laboratory for Human Brain Dynamics, AAI Scientific Cultural Services Limited, Nicosia, 1065, Nicosia, Cyprus