Fractal Dimensions of Generalized Sierpiński Networks

Quantifying the self-similarity of complex networks via discrete fractal dimensions is a fundamental problem in network science. In this paper, we introduce a generalized framework for Sierpiński network families by allowing arbitrary intercopy edge connection patterns, and derive two associated families: touching-type and skeletal-type, formed by contracting inter-copy edges and intra-copy edges, respectively. By integrating the theory of self-similar touching networks with analysis of geodesic distances within these structures, we prove that all three families of generalized Sierpiński networks share the same discrete Hausdorff and mass dimensions.