Sul teorema di Eulero-Poincaré per reti di ordine superiore

Topological data analysis is an approach to the analysis of data, performed by means of topological techniques. Our goal in this thesis is to apply algebraic topological tools and the results from Bjorner and Kalai to explore complex network models. In Chapter 1 we review simplicial complexes and simplicial homology, important tools in the study of the topology of spaces. The advantage of simplicial complexes lies in the fact that they allow topological information to be encoded in terms of purely combinatorial data. From this perspective, we describe ways of representing data as complexes, in order to apply the previously introduced techniques. In the second chapter we review the theory of graphs and networks and discuss some network models, from the simpler Erdos-Renyi model to more complex ones. We illustrate how to apply the results of Chapter 1 to the study of such networks, obtaining what is referred to as higher-order networks. In Chapter 3 we summarize the main results from "An extended Euler-Poincaré theorem", by authors Bjorner and Kalai, which provide a numerical characterization of the (f, β) pair of a simplicial complex and illustrate the combinatorics of compatible pairs. Finally, we apply the above results to the analysis of both real networks and complex network models. In particular, we compute and compare the obtained bounds for the f-vectors and β-vectors for such higher-order networks.