Numerous network models have been investigated to gain insights into the origins of fractality. In this work, we introduce two novel network models, to better understand the growing mechanism and structural characteristics of fractal networks. The Repulsion Based Fractal Model (RBFM) is built on the well-known Song-Havlin-Makse (SHM) model, but in RBFM repulsion is always present among a specific group of nodes. The model resolves the contradiction between the SHM model and the Hub Attraction Dynamical Growth model, by showing that repulsion is the characteristic that induces fractality. The Lattice Small-world Transition Model (LSwTM) was motivated by the fact that repulsion directly influences the node distances. Through LSwTM we study the fractal-small-world transition. The model illustrates the transition on a fixed number of nodes and edges using a preferential-attachment-based edge rewiring process. It shows that a small average distance works against fractal scaling, and also demonstrates that fractality is not a dichotomous property, continuous transition can be observed between the pure fractal and non-fractal characteristics.

12 pages, 5 figures, to appear in: 978-3-031-17657-9, Pacheco et al (eds.): Complex Networks XIII