SPectral graph theory And Random walK (SPARK) toolbox for static and dynamic characterization of (di)graphs: a tutorial

Spectral graph theory and its applications constitute an important step forward modern network theory. Over the last decades the spectral graph theory has been gaining an increasing consensus that fostered the development of innovative tools enabling network theory to model a variety of different scenarios while answering questions of increasing complexity. Nevertheless, a comprehensive understanding of spectral graph theory’s principles requires a solid technical background which often prevent its diffusion through the scientific community. To overcome such an issue, we developed and released an open-source MATLAB toolbox - SPectral graph theory And Random walK (SPARK) toolbox - that combines spectral graph theory and random walk concepts to provide both static and dynamic characterization of digraphs. Following the description of the theoretical principles grounding the toolbox, we presented SPARK folder structure and the list of available indices and measures. Thereafter, we tested SPARK in three different scenarios: two-toy examples on synthetic networks and a real-data scenario relying on EEG data recorded from stroke patients in resting state condition. Results on synthetic data showed that indices extracted using SPARK toolbox allow to correctly characterize the topology of a bi-compartmental network. Furthermore, they could also be used to find the "optimal" vertex set partition (i.e. the one that minimizes the number of between-cluster links) for the underlying network and compare it to a given a-priori partition. Finally, the application to real EEG-based networks provides a practical case study where the SPARK toolbox has been used to describe networks’ alterations in stroke patients and put them in relation with the motor impairment.