Abstract
We discuss three seemingly unrelated quantities that have been introduced in different fields of science for complex networks. The three quantities are the resistance distance, the information centrality and the node displacement. We first prove various relations among them. Then we focus on the node displacement, showing its usefulness as an index of node vulnerability. We argue that the node displacement has a better resolution as a measure of node vulnerability than the degree and the information centrality.
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
References
Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Albert, R., Jeong, H., Barabási, A.L.: Error and attack tolerance of complex networks. Nature 406, 378–382 (2000)
Atilgan, A.R., Akan, P., Baysal, C.: Small-world communication of residues and significance for protein dynamics. Biophys. J. 86, 85–91 (2004)
Bahar, I., Atilgan, A.R., Erman, B.: Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential. Fold. Des. 2, 173–181 (1997)
Barthelemy, M., Barrat, A., Vespignani, A.: The role of geography and traffic in the structure of complex networks. Adv. Complex Syst. 10, 5–28 (2007)
Chen, H.Y., Zhang, F.J.: Resistance distance and the normalized Laplacian spectrum. Discrete Appl. Math. 155, 654–661 (2007)
Choi, J., Barnett, G., Chou, B.: Comparing world city networks: a network analysis of Internet backbone and air transport intercity linkages. Glob. Netw. 6, 81–99 (2006)
de Nooy, W., Mrvar, A., Batagelj, V.: Exploratory Social Network Analysis with Pajek. Cambridge University Press, Cambridge (2005)
del Rio, G., Koschutzki, D., Coello, G.: How to identify essential genes from molecular networks? BMC Syst. Biol. (2009). doi:10.1186/1752-0509-3-102
Dobrynin, A.A., Entringer, R., Gutman, I.: Wiener index of trees: Theory and applications. Acta Appl. Math. 66, 211–249 (2001)
Estrada, E.: Virtual identification of essential proteins within the protein interaction network of yeast. Proteomics 6, 35–40 (2006)
Estrada, E.: Universality in protein residue networks. Biophys. J. 98, 890–900 (2010)
Estrada, E., Bodin, O.: Using network centrality measures to manage landscape connectivity. a short path for assessing habitat patch importance. Ecol. Appl. 18, 1810–1825 (2008)
Estrada, E., Hatano, N.: A vibrational approach to node centrality and vulnerability in complex networks. arXiv:0912.4307
Estrada, E., Hatano, N.: Topological atomic displacements, Kirchhoff and Wiener indices of molecules. Chem. Phys. Lett. 486, 166–170 (2010)
Estrada, E., RodrÃguez, L., Gutiérrez, A.: Matrix algebraic manipulations of molecular graphs. 1. graph theoretical invariants based on distances and adjacency matrices. MATCH Commun. Math. Comput. Chem. 35, 145–156 (1997)
Fiedler, M.: Algebraic connectivity of graphs. Czechoslov. Math. J. 23, 298–305 (1973)
Freeman, L.C.: A set of measures of centrality based upon betweenness. Sociometry 40, 35–41 (1977)
Freeman, L.C.: The Development of Social Network Analysis. Empirical Press, Vancouver (2004)
Harary, F. (ed.): Graph Theory and Theoretical Physics. Academic Press, London (1967)
Jeong, H., Mason, S.P., Barabási, A.L., Oltvai, Z.N.: Lethality and centrality in protein networks. Nature 411(6833), 41–42 (2001)
Jordán, F.: Keystone species and food webs. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 364, 1733–1741 (2009)
Kapferer, B.: Strategy and Transaction in an African Factory. Manchester University Press, Manchester (1972)
Kendall, M.: A new measure of rank correlation. Biometrika 30, 81–89 (1938)
Klein, D., Randić, M.: Resistance distance. J. Math. Chem. 12, 81–95 (1993)
Lind, P., González, M., Herrmann, H.: Cycles and clustering in bipartite networks. Phys. Rev. E 72, 056127 (2005)
Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003)
Soheilifard, R., Makarov, D., Rodin, G.: Critical evaluation of simple network models of protein dynamics and their comparison with crystallographic B-factors. Phys. Biol. 5, 1–13 (2008)
Stephenson, K., Zelen, M.: Rethinking centrality: methods and examples. Soc. Netw. 11, 1–37 (1989)
Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001)
Trinajstić, N.: Chemical Graph Theory. CRC Press, Boca Raton (1992)
Wasserman, S., Faust, K.: Social Network Analysis. Cambridge University Press, Cambridge (1994)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998)
Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)
Xiao, W., Gutman, I.: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284–289 (2003)
Yang, Y.J., Zhang, H.P.: Some rules on resistance distance with applications. J. Phys. A, Math. Theor. 41, 445203 (2008)
Yuan, Z., Bailey, T., Teasdale, R.: Prediction of protein B-factor profiles. Proteins 58, 905–912 (2005)
Zhou, B., Trinajstić, N.: On resistance distance and Kirchhoff index. J. Math. Chem. 46, 283–289 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag London Limited
About this chapter
Cite this chapter
Estrada, E., Hatano, N. (2010). Resistance Distance, Information Centrality, Node Vulnerability and Vibrations in Complex Networks. In: Estrada, E., Fox, M., Higham, D., Oppo, GL. (eds) Network Science. Springer, London. https://doi.org/10.1007/978-1-84996-396-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-84996-396-1_2
Publisher Name: Springer, London
Print ISBN: 978-1-84996-395-4
Online ISBN: 978-1-84996-396-1
eBook Packages: Computer ScienceComputer Science (R0)