Higher order degree in simplicial complexes, multi combinatorial Laplacian and applications of TDA to complex networks
Many real networks in social, biological or computer sciences have an inherent structure of a simplicial complex, which reflects the multi interactions among agents (and groups of agents) and constitutes the basics of Topological Data Analysis. Normally, the relevance of an agent in a network of graphs is given in terms of the number of edges incident to it, its degree, and in a simplicial network there are already notions of adjacency and degree for simplices that, as far as we know, are not valid for comparing simplices in different dimensions. We propose new notions of higher order lower, upper and generalised adjacency degrees for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces. New multi parameter boundary and coboundary operators in an oriented simplicial complex are also given and a novel multi combinatorial Laplacian is defined. These operators generalise the known ones and are proved to be an effective tool for calculating the higher order degrees here presented. Thus, this mathematical framework allows us to elucidate the relevance not only of an agent, but of a bunch of them as a simplicial community, and also to study the degree of collaboration between different communities in a simplicial complex. In addition, they are effective and programmable computational techniques. Some potential applications to simplicial Network Science are also proposed.