An Original Convolution Model to analyze Graph Network Distribution Features

Modern Graph Theory is a newly emerging field that involves all of those approaches that study graphs differently from Classic Graph Theory. The main difference between Classic and Modern Graph Theory regards the analysis and the use of graph's structures (micro/macro). The former aims to solve tasks hosted on graph nodes, most of the time with no insight into the global graph structure, the latter aims to analyze and discover the most salient features characterizing a whole network of each graph, like degree distributions, hubs, clustering coefficient and network motifs. The activities carried out during the PhD period concerned, after a careful preliminary study on the applications of the Modern Graph Theory, the development of an innovative Convolutional Model to model brain connections at the cellular level capable of combining exponential models and power law models. This new theoretical framework has been introduced in the first instance with an aspatial graph formulation and then proposed a spatial graph model with Convolutive connectivity able to fit the degree distributions of data driven Connectome reconstructions. In order to evaluate the qualities of the Convolutional Model, theoretical graphical models capable of characterizing brain activity were taken into consideration. In the specific case, the model examined characterizes the epileptic activity through a simple Hindmarsh-Rose model system of point neurons and reproduces the functional characteristics observed in the data driven model. Such a model provides insight into the deep impact of micro connectivity in macro-scale brain activity. Other evaluations have been done in different applications, in the field of image cell segmentation with Explainable Artificial Intelligence's neuronal agents in which has been used a methodology that is not only explainable but also resistant to adversarial noise and also in the field of modelling Covid-19 outbreak in gaining insight on vaccines and role of our habits as individuals in the pandemic spread. Therefore, the core of the thesis is to introduce Modern Graph Theory with a new competitive Convolutive Model and then expose some applications to real-world problems like a characterization of Brain networks, simulation and analysis of Brain dynamics with a particular focus on Epilepsy, Immunofluorescence images segmentation with neuronal based agents and modelling of Covid-19 Epidemic spread with a specific interest in human social networks. All this takes continuously into account the whole dialogue between Graph Theory and its applications.