Abstract
Scale free graphs have attracted attention as their non-uniform structure that can be used as a model for many social networks including the WWW and the Internet. In this paper, we propose a simple random model for generating scale free k-trees. For any fixed integer k, a k-tree consists of a generalized tree parameterized by k, and is one of the basic notions in the area of graph minors. Our model is quite simple and natural; it first picks a maximal clique of size k + 1 uniformly at random, it then picks k vertices in the clique uniformly at random, and adds a new vertex incident to the k vertices. That is, the model only makes uniform random choices twice per vertex. Then (asymptotically) the distribution of vertex degree in the resultant k-tree follows a power law with exponent 2 + 1/k, the k-tree has a large clustering coefficient, and the diameter is small. Moreover, our experimental results indicate that the resultant k-trees have extremely small diameter, proportional to o(log n), where n is the number of vertices in the k-tree, and the o(1) term is a function of k.
Similar content being viewed by others
References
Barabási A.-L., Albert R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Bonato, A.: A Course on the Web Graph, American Mathematical Society Graduate Studies in Mathematics 2008, vol. 89 (2008)
Bodlaender H.: A tourist guide through treewidth. Acta Cybernet. 11, 1–21 (1993)
Bodlaender H.: A Partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1–45 (1998)
Bollobás, B., Riordan, O.: Mathematical results on scale-free graphs. In: Bornholdt, S., Schuster, H. (eds.) Handbook of Graphs and Networks, Wiley, Berlin (2002)
Chung, F.R.K., Lu, L.: Complex Graphs and Networks, American Mathematical Society, Providence, Rhode Island (2006)
Faloutsos, M., Faloutsos, P., Faloutsos, C.: On Power-law Relationships of the Internet Topology, SIGCOMM, pp. 251–262 (1999)
Gu Z., Zhou T., Wang B., Yan G., Zhu C., Fu Z.: Simplex triangulation induced scale free networks. Dyn. Contin. Discret. Impuls. Syst. B 13, 505–510 (2006)
Mitzenmacher, M.: A brief history of generative models for power law and lognormal distributions, In: Proc. of the 39th Annual Allerton Conf. on Communication, Control, and Computing, pp. 182–191 (2001)
Newman M.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)
Shigezumi,T., Miyoshi, N., Uehara, R., Watanabe, O.: Scale Free Interval Graphs. Theoretical Computer Science, to appear. (A preliminary version was presented at International Conference on Algorithmic Aspects in Information and Management (AAIM 2008), pp. 202–303. Lecture Notes in Computer Science, vol. 5034. Springer, Berlin (2008))
Watts D.J., Strogatz D.H.: Collective dynamics of ‘Small-World’ networks. Nature 393, 440–442 (1998)
Wormald, N.: The differential equation method for random graph processes and greedy algorithms, In: Karoński, M., Prömel, H.J. (eds.) Lectures on Approximation and Randomized Algorithms, pp. 73–155. PWN, Warsaw (1999)
Yule G.: A mathematical theory of evolution based on the conclusions of Dr. J. C. Willis. Philos. Trans. R Soc. Lond. Ser. B 213, 21–87 (1924)
Zhang Z., Rong L., Comellas F.: High dimensional random Apollonian networks. Phys. A Statist. Mech. Appl. 364, 610–618 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cooper, C., Uehara, R. Scale Free Properties of Random k-Trees. Math.Comput.Sci. 3, 489–496 (2010). https://doi.org/10.1007/s11786-010-0041-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-010-0041-6