Abstract
While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be obtained by considering further neighborhoods. The current work considers the concept of virtual hierarchies established around each node and the respectively defined hierarchical node degree and clustering coefficient (introduced in cond-mat/0408076), complemented by new hierarchical measurements, in order to obtain a powerful set of topological features of complex networks. The interpretation of such measurements is discussed, including an analytical study of the hierarchical node degree for random networks, and the potential of the suggested measurements for the characterization of complex networks is illustrated with respect to simulations of random, scale-free and regular network models as well as real data (airports, proteins and word associations). The enhanced characterization of the connectivity provided by the set of hierarchical measurements also allows the use of agglomerative clustering methods in order to obtain taxonomies of relationships between nodes in a network, a possibility which is also illustrated in the current article.
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References
M. E. J. Newman, SIAM Rev. 45:167–256 (2003).
A.-L. Barabási and R. Albert, Science 509 (1999).
R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74:47–97 (2002).
P. J. Flory, J. Am. Chem. Soc. 63:3083, 3091, 3096 (1941).
A. Rapoport, Bull. Math. Bioph. 19:257–277 (1957).
P. Erdös and A. Rényi, Publ. Math. 6:290–297 (1959).
D. J. Watts and S. H. Strogatz, Nature 393: 440–442 (1998).
D. J. Watts, Small Worlds, Princeton Studies in Complexity (Princeton University Press, 1999).
R. Albert, H. Jeong, and A. L. Barabási, Nature 401:130, cond-mat/9907038 (1999).
L. daF. Costa, Phys. Rev. Lett. 93:098702 (2004).
L. daF. Costa and L. E. C. da Rocha, Eur. Phys. J. B 50:237–242, cond-mat/0408076 (2006).
E. Ravasz and A. L. Barabási, Phys. Rev. E 67:026112, cond-mat/0206130 (2003).
E. Ravasz, A. L. Somera, A. Mongru, Z. N. Oltvai, and A. L. Barabási, Science 297:1551–1555 cond-mat/0209244 (2002),.
G. Caldarelli, R. Pastor-Satorras, and A. Vespignani, cond-mat/0212026 (2003).
A. L. Barabási, Z. Deszo, E. Ravasz, S.-H. Yook, and Z. Oltvai, Sitges Proceedings on Complex Networks (2004).
A. Trusina, S. Maslov, P. Minnhagen, and K. Sneppen, Phys. Rev. Lett. 92:178702 (2004), cond-mat/0308339 (2003).
M. Boss, H. Elsinger, M. Summer, and S. Thurner, Santa Fe Institute Working Paper No. 03-10-054, Accepted for Quant. Finance, cond-mat/0309582 (2003).
M. Barthélemy, A. Barrat, R. Pastor-Satorras, and A. Vespignani, cond-mat/0311501 (2003).
H. Zhou, Phys. Rev. E 67:061901 (2003).
A. Vázquez, Phys. Rev. E 67:056104 (2003).
M. Steyvers and J. B. Tenenbaum, To appear in Cognitive Science, cond-mat/0212026 (2003).
V. Gold’shtein, G. A. Koganov, and G.I. Surdutovich, cond-mat/0409298 (2004).
R. Cohen, S. Havlin, S. Mokryn, D. Dolev, T. Kalisky, and Y. Shavitt, cond-mat/0305582(2003).
M. E. Newman, cond-mat/0111070 (2001).
M. E. J. Newman and D. J. Watts, Phys. Rev. E 60:7332–7342 (1999).
S. Wuchty and E. Almaas, BMC Evol. Biol. 5:24 (2005).
E. R. Dougherty and R. A. Lotufo, Hands-on Morphological Image Processing (SPIE Press, 2003).
L. Vincent, Signal Proc. 16:365–388 (1989).
H. J. A. M. Heijmans, P. Nacken, A. Toet, and L. Vincent, J. Vis. Comm. and Image Repr. 3:24–38 (1992).
L. daF. Costa, q.bio.MN/0405028 (2004).
L. daF. Costa, q.bio.TO/0412042 (2004).
L. daF. Costa, cond-mat/0405022 (2004).
J. P. Bagrow and E. M. Bollt, cond-mat/0412482 (2004).
L. d. Costa, and R. M. Cesar, Shape Analysis and Classification: Theory and Practice, 1st ed. (CRC Press, Inc., 2000)
R. Duda, P. Hart, and D. Stork, Pattern Classification, 2nd ed. (John Wiley, New York, 2001).
A. C. Zorach and R. E. Ulanowicz, Complexity, 8(3):68–76 (2003).
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms (The MIT Press, 2002).
B. Bollobás, Random Graphs (Academic Press, London, 1985).
G. R. Kiss, C. Armstrong, R. Milroy, and J. Piper, An associative thesaurus of English and its computer analysis, in The Computer and Literary Studies. A. J. Aitken, R. W. Bailey, and N. Hamilton-Smith, eds. (University Press, Edinburgh, 1973).
USAir97, Pajek Package for Large Network Analysis http://vlado.fmf.uni-lj.si/pub/networks/pajek /data/gphs.htm.
S. Sun, L. Ling, N. Zhang, G. Li, and R. Chen, Topological structure analysis of the protein-protein interaction network in budding yeast. Nucl. Acids Res. 31(9), 2443–2450 (2003).
L. daF. Costa, Intl. J. Mod. Phys. C 15:371–379 (2003).
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da Fontoura Costa, L., Silva, F.N. Hierarchical Characterization of Complex Networks. J Stat Phys 125, 841–872 (2006). https://doi.org/10.1007/s10955-006-9130-y
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DOI: https://doi.org/10.1007/s10955-006-9130-y