Abstract
It has been shown that many real networks are not only “scale-free” (i.e., having a power-law degree distribution), but also contain more complex structures such as “hierarchy” or “self-similarity” that cannot be captured by the preferential attachment random network model. These observations have led to a number of more sophisticated models being proposed in the literature. In this paper we advocate a multivariate analysis perspective based on the notion of MRVs as a unifying framework to study complex structures in networks. We demonstrate the existence of “multivariate heavy tails” in existing network models and real networks, and argue that they better capture the “hierarchical” or “self-similar” structures in these networks.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For example, a multivariate distribution can have a power-law marginal distribution in each variable, but not jointly MRV, i.e., multivariate heavy-tailed.
- 2.
Joint degree-degree pairs are defined for the edges of a network, where each degree belongs to one endpoint of an edge.
References
Ravasz, E., Barabási, A.-L.: Hierarchical organization in complex networks. Phys. Rev. E 67(2), 026112 (2003)
Resnick, S.: Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York (2007)
Dorogovtsev, S.N., Goltsev, A., Mendes, J.F.F.: Pseudofractal scale-free web. Phys. Rev. E 65(6), 066122 (2002)
Chen, L., Comellas, F., Zhang, Z.: Self-similar planar graphs as models for complex networks, arXiv preprint arXiv:0806.1258 (2008)
Song, C., Havlin, S., Makse, H.A.: Origins of fractality in the growth of complex networks. Nat. Phys. 2(4), 275–281 (2006)
Ravasz, E., Somera, A.L., Mongru, D.A., Oltvai, Z.N., Barabási, A.-L.: Hierarchical organization of modularity in metabolic networks. Science 297(5586), 1551–1555 (2002)
Song, C., Havlin, S., Makse, H.A.: Self-similarity of complex networks. Nature 433(7024), 392–395 (2005)
Comellas, F., Fertin, G., Raspaud, A.: Recursive graphs with small-world scale-free properties, arXiv preprint cond-mat/0402033 (2004)
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. EPL (Europhysics Letters) 79(3), 38007 (2007)
Li, L., Alderson, D., Doyle, J.C., Willinger, W.: Towards a theory of scale-free graphs: Definition, properties, and implications. Internet Math. 2(4), 431–523 (2005)
Resnick, S.: On the foundations of multivariate heavy-tail analysis. J. Appl. Probab. 191–212 (2004)
Vázquez, A., Pastor-Satorras, R., Vespignani, A.: Large-scale topological and dynamical properties of the internet. Phys. Rev. E 65(6), 066130 (2002)
Kitsak, M., Havlin, S., Paul, G., Riccaboni, M., Pammolli, F., Stanley, H.E.: Betweenness centrality of fractal and nonfractal scale-free model networks and tests on real networks. Phys. Rev. E 75(5), 056115 (2007)
Acknowledgment
This research was supported in part by DoD ARO MURI Award W911NF-12-1-0385, DTRA grant HDTRA1- 09-1-0050, and NSF grants CNS-10171647, CNS-1117536 and CRI-1305237.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Golnari, G., Zhang, ZL. (2014). Multivariate Heavy Tails in Complex Networks. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_41
Download citation
DOI: https://doi.org/10.1007/978-3-319-12691-3_41
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12690-6
Online ISBN: 978-3-319-12691-3
eBook Packages: Computer ScienceComputer Science (R0)