Recursive simplicial networks: Degree characteristics and Zagreb index analysis

Simplicial networks model group-level interactions rather than traditional pairwise connections, enabling the capture of higher-order dependencies in complex systems such as brain function, epidemic spread, and social collaboration. In this study, we develop an iterative simplicial network in which higher-order relations are represented through the recursive addition of clique structures at each growth step. We derive the structural properties induced by the network’s degree configuration. A recursive framework is then introduced for computing Zagreb indices and their variants, supported by closed-form approximations that allow efficient characterization of large networks. Our simulations show that the model exhibits both scale-free and small-world properties. We further assess the accuracy of the approximation methods through extensive computational analysis. Overall, the results highlight the practical relevance of simplicial networks for capturing complex higher-order interactions and their potential in network-based modeling of real-world systems.