Innovation Diffusion on Higher-Order Networks

Higher-order networks (HON) provide a suitable frame to model connections that involve groups of nodes—representing interacting individuals or other types of agents—of different sizes. They allow us to take into account not only pairwise interactions but also connections binding three or four or any other natural number of nodes together. Motivated by the consideration that the existence of higher-order interactions may impact, among others, the process of diffusion of new products, the spreading of ideas, and the adoption of practices, we propose and study here a version of the celebrated Bass model on top of HON. We define a mean-field equation that contains terms up to the order at which interactions might make a significant contribution. The impact of the paper is twofold. By considering and comparing different maximal orders of interaction and analyzing how they influence certain times that are important in the diffusion process, we show that HON indeed has an impact and yields a greater accuracy in modeling results. The second contribution of the paper, also of interest for future works, consists of a novel procedure we develop for the construction of HON with assigned generalized mean degrees. We also show that the behavior of the take-off time with the size of the orders contribution undergoes a phase transition where the link density of the network and the related higher-order structures act as the characterizing condition for one phase or the other.