Abstract: When our topology ancestors were playing with simplicial complexes to devise models of RP^2 or common triangulations of surfaces they could not have envisioned how ubiquitous and large graph structures are nowadays. Starting with reminders on elementary collapses, clique complexes and a touch of NP-completeness for maximal cliques, I will illustrate the relevant business examples we have found in our Data Science practice as elbformat consulting in Hamburg, Germany. Cases will include Structural Website Optimisations, User Funnel Analyses, Process Mining insights and a Recommendation Engine Prototype.