Complejos simpliciales heterogéneos en crecimiento

Complex systems are commonly modeled using graphs, where nodes represent entities and edges represent pairwise interactions. Yet, many real-world systems exhibit higher-order interactions that involve multiple entities simultaneously and cannot be adequately captured by pairwise links. Simplicial complexes offer a mathematical framework for modeling such higher-order structures, where interactions among sets of nodes are represented as simplices-such as vertices (0-simplices), edges (1-simplices), and triangles (2-simplices). In practical applications, the sizes of these higher-order interactions can vary significantly. To reflect this heterogeneity, we introduce a growing simplicial complex model in which the dimensions of newly added simplices are sampled from a predefined probability distribution. Theoretical analysis reveals that the generalized degree of faces in this model follows a power-law distribution, with exponents that can be adjusted by varying the simplex-dimension distribution. Numerical simulations support these theoretical predictions and illustrate the model's capacity to generate simplicial complexes with customizable structural properties. Overall, this model offers a versatile theoretical framework for studying heterogeneous higher-order structures and their emergent properties in higher-order systems.