Supplementary material for the work titled Investigating the Origins of Fractality Based on Two Novel Fractal Network Models - E. Zakar-Polyák, M. Nagy, & R. Molontay

The paper can be found at this arxiv link

The codes can be found in the codes folder of the marcessz/fractal-networks repository.

@article{zakarpolyak2022investigating,
  title={Investigating the Origins of Fractality Based on Two Novel Fractal Network Models},
  author={Zakar-Poly\'ak, Enik\H{o} and Nagy, Marcell and Molontay, Roland},
  journal={arXiv preprint arXiv:2210.15505},
  year={2022}
}

The Repulsion Based Fractal (RBF) model is based on the Song-Havlin-Makse model, it also evolves through time, and we rewire edges to create repulsion among nodes. However, here the probability of an edge to be rewired is not fixed, but depends on the degree of its endpoints. In contrast to the Song-Havlin-Makse model, repulsion is always present in RBFM, moreover, with a predefined parameter we can specify the nodes that repel each other (e.g., hubs or small degree nodes).

The Lattice Small-world Transition Model (LSwTM) utilises the fractal nature of grid-like structures, and at the same time works against it with the preferential attachment mechanism.

The boxplots of four network metrics are displayed, which show the distribution of the given metric for different parameter settings of the Repulsion based fractal model. Each box is based on 30 generations of the model with the given parameters. Subfigure (a) is the boxplot of the normalised diameter (i.e. diameter divided by the logarithm of the size), (b) the average degree, (c) the assortativity coefficient and (d) the average clustering coefficient. The parameter settings (x-axis) are shown in increasing order with respect to the size of the resulting networks.

The examined characteristics can be considered stable because the values of the metrics do not differ significantly for the different realisations of the networks. The average clustering coefficient, and also the maximum of the eigenvector centralities may not seem to be as consistent as the other metrics, but the range, in which the values vary is still quite small. Furthermore, the fluctuation decreases for larger networks, i.e., when the model performs more iterations. Overall, we can conclude that the main characteristics of the network model for a given parameter setting do not depend heavily on the exact realisations.

The distribution of four network metrics for different parameter settings of the LSwTM. Each boxplot is based on 30 generations of the model with the given parameters. Subfigure (a) is the boxplot of the normalised diameter (i.e. diameter divided by the logarithm of the size), (b) the average degree, (c) the assortativity coefficient and (d) the average clustering coefficient. The parameter settings (x-axis) are shown in increasing order with respect to the size of the resulting networks.

It can be seen that for a given parameter setting the generated networks have completely similar characteristics in terms of normalised diameter, average degree, assortativity and average clustering coefficient.