Growing simplicial complex with face dimension selection and preferential attachment

When simplicial complexes are used to represent higher-order systems, information regarding when and how interactions happen may be lost. In this paper, we propose the concept of temporal simplicial complexes, in which simplices with timestamps (or temporal simplices) are used to represent interactions, and faces with weights are used to represent relations. Then, we propose a growing model with two rules, face dimension selection (FDS), and preferential attachment. By properly setting the probability parameter vector q in the FDS rule, one can balance network diameter expansion and network centrality, thus attaining more flexibility in the growing process. Our theoretical analysis and simulations that followed show the generalized degree of faces of any dimension follows a power-law distribution, with a scaling component controlled by q. Our work provides a flexible growing model and can be used to study higher-order systems with temporal properties.