Average geodesic distance of skeleton networks of Sierpinski tetrahedron. (English) Zbl 1514.05164
Summary: The average distance is concerned in the research of complex networks and is related to Wiener sum which is a topological invariant in chemical graph theory. In this paper, we study the skeleton networks of the Sierpinski tetrahedron, an important self-similar fractal, and obtain their asymptotic formula for average distances. To provide the formula, we develop some technique named finite patterns of integral of geodesic distance on self-similar measure for the Sierpinski tetrahedron.
MSC:
| 05C82 | Small world graphs, complex networks (graph-theoretic aspects) |
| 05C12 | Distance in graphs |
| 28A80 | Fractals |
Keywords:
fractal network; average distance; Sierpinski tetrahedron; self-similar measure; finite patternReferences:
| [1] | Watts, D. J.; Strogatz, S. H., Collective dynamics of ‘small-world��� networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139 |
| [2] | Barabasi, A. L.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512 (1999) · Zbl 1226.05223 |
| [3] | Newman, M. E.J., The structure and function of complex networks, SIAM Rev., 45, 167-256 (2003) · Zbl 1029.68010 |
| [4] | Newman, M. E.J., Networks: An Introduction (2010), Oxford University Press: Oxford University Press Oxford · Zbl 1195.94003 |
| [5] | (Bonchev, D.; Rouvray, D. H., Chemical Graph Theory: Introduction and Fundamentals (1991), Gordon & Breach Science Publishers: Gordon & Breach Science Publishers New York) · Zbl 0746.05063 |
| [6] | Wiener, H., Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69, 17-20 (1947) |
| [7] | Song, C.; Havlin, S.; Makse, H. A., Self-similarity of complex networks, Nature, 433, 392-395 (2005) |
| [8] | Wang, L.; Wang, Q.; Xi, L.; Chen, J.; Wang, S.; Bao, L.; Yu, Z.; Zhao, L., On the fractality of complex networks: covering problem, algorithms and Ahlfors regularity, Sci. Rep., 7, 41385 (2017) |
| [9] | Xi, L.; Wang, L.; Wang, S.; Yu, Z.; Wang, Q., Fractality and scale-free effect of a class of self-similar networks, Physica A, 478, 31-40 (2017) · Zbl 1400.05227 |
| [10] | Wang, S.; Xi, L.; Xu, H., Scale-free and small-world properties of Sierpinski networks, Physica A, 465, 690-700 (2016) · Zbl 1400.05226 |
| [11] | Andrade Jr., J. S.; Herrmann, H. J.; Andrade, R. F.; Da Silva, L. R., Apollonian networks: Simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs, Phys. Rev. Lett., 94, 1, 018702 (2005) |
| [12] | Zhang, Z.; Zhou, S.; Fang, L.; Guan, J.; Zhang, Y., Maximal planar scale-free Sierpinski networks with small-world effect and power law strength-degree correlation, Europhys. Lett., 79, 38007 (2007) |
| [13] | Zhang, Z.; Zhou, S.; Chen, L.; Yin, M.; Guan, J., The exact solution of the mean geodesic distance for Vicsek fractals, J. Phys. A, 41, 485102 (2008) · Zbl 1156.28003 |
| [14] | Hino, M., Geodesic distances and intrinsic distances on some fractal sets, Publ. Res. Inst. Math. Sci., 50, 2, 181-205 (2013) · Zbl 1294.31010 |
| [15] | Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J., 30, 5, 713-747 (1981) · Zbl 0598.28011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
