Abstract
When network and graph theory are used in the study of complex systems, a typically finite set of nodes of the network under consideration is frequently either explicitly or implicitly considered representative of a much larger finite or infinite region or set of objects of interest. The selection procedure, e.g., formation of a subset or some kind of discretization or aggregation, typically results in individual nodes of the studied network representing quite differently sized parts of the domain of interest. This heterogeneity may induce substantial bias and artifacts in derived network statistics. To avoid this bias, we propose an axiomatic scheme based on the idea of node splitting invariance to derive consistently weighted variants of various commonly used statistical network measures. The practical relevance and applicability of our approach is demonstrated for a number of example networks from different fields of research, and is shown to be of fundamental importance in particular in the study of spatially embedded functional networks derived from time series as studied in, e.g., neuroscience and climatology.
Similar content being viewed by others
References
M.E.J. Newman, SIAM Rev. 45, 167 (2003)
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. Hwang, Phys. Rep. 424, 175 (2006)
L. da F Costa, F.A. Rodrigues, G. Travieso, P.R. Villas Boas, Adv. Phys. 56, 167 (2007)
M.E.J. Newman, Networks: An Introduction (Oxford University Press, 2010)
R. Cohen, S. Havlin, Complex Networks: Structure, Robustness and Function (Cambridge University Press, 2010)
E. Kolaczyk, Statistical Analysis of Network Data: Methods and Models, Springer Series in Statistics (Springer, 2009)
S. Achard, R. Salvador, B. Whitcher, J. Suckling, E. Bullmore, J. Neurosci. 26, 63 (2006)
C. Zhou, L. Zemanová, G. Zamora, C. Hilgetag, J. Kurths, Phys. Rev. Lett. 97, 238103 (2006)
P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli, C.J. Honey, V.J. Wedeen, O. Sporns, PLOS Biol. 6, e159 (2008)
A.A. Tsonis, P.J. Roebber, Physica A 333, 497 (2004)
J.F. Donges, Y. Zou, N. Marwan, J. Kurths, Europhys. Lett. 87, 48007 (2009)
R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001)
R. Pastor-Satorras, A. Vázquez, A. Vespignani, Topology, hierarchy, and correlations in Internet graphs, In Complex Networks, edited by E. Ben-Naim and Z. Toroczkai (2004), Vol. 650, pp. 425–440.
Q. Chen, H. Chang, R. Govindan, S. Jamin, S. Shenker, W. Willinger, The origin of power-laws in internet topologies revisited, in IEEE INFOCOM (2002), Vol. 2, pp. 608–617
G. Siganos, M. Faloutsos, P. Faloutsos, C. Faloutsos, IEEE/ACM Trans. Netw. 11, 514 (2003)
A. Capocci, V. Servedio, F. Colaiori, L. Buriol, D. Donato, S. Leonardi, G. Caldarelli, Phys. Rev. E 74, 036116 (2006)
V. Zlatić, H. Štefančić, Europhys. Lett. 93, 58005 (2011)
M. Comola, M. Fafchamps, CEPR Discussion Papers 7406, C.E.P.R. Discussion Papers, 2009
M. Comola, M. Fafchamps, CSAE Working Paper Series 2010-20, Paris-Jourdan Sciences Économiques, 2010
M.A. Serrano, M. Boguñá, Phys. Rev. E 68, 015101 (2003)
N. Marwan, J.F. Donges, Y. Zou, R.V. Donner, J. Kurths, Phys. Lett. A 373, 4246 (2009)
R.V. Donner, Y. Zou, J.F. Donges, N. Marwan, J. Kurths, New. J. Phys. 12, 033025 (2010)
R.V. Donner, Y. Zou, J.F. Donges, N. Marwan, J. Kurths, Phys. Rev. E 81, 015101 (2010)
M.E.J. Newman, Proc. Natl. Acad. Sci. USA 103, 8577 (2006)
A. Barrat, M. Barthelemy, Proc. Natl. Acad. Sci. USA 101, 3747 (2004)
A.A. Tsonis, K.L. Swanson, P.J. Roebber, B. Am. Meteorol. Soc. 87, 585 (2006)
A.A. Tsonis, K.L. Swanson, Phys. Rev. Lett. 100, 228502 (2008)
K. Yamasaki, A. Gozolchiani, S. Havlin, Phys. Rev. Lett. 100, 228501 (2008)
A. Gozolchiani, K. Yamasaki, O. Gazit, S. Havlin, Europhys. Lett. 83, 28005 (2008)
R.V. Donner, T. Sakamoto, N. Tanizuka, in Nonlinear Time Series Analysis in the Geosciences: Applications in Climatology, Geodynamics and Solar-Terrestrial Physics, edited by R.V. Donner, S.M. Barbosa (Springer, 2008), pp. 125–154
J.F. Donges, Y. Zou, N. Marwan, J. Kurths, Eur. Phys. J. Special Top. 174, 157 (2009)
J.A. Henderson, P.A. Robinson, Phys. Rev. Lett. 107, 018102 (2011)
J.F. Donges, H.C.H. Schultz, N. Marwan, Y. Zou, J. Kurths, Eur. Phys. J. B 84, 635 (2011)
G.R. North, T.L. Bell, R.F. Cahalan, Mon. Weather Rev. 110, 699 (1982)
S.H. Lee, P.J. Kim, H. Jeong, Phys. Rev. E 73, 016102 (2006)
R. Rohde, J. Curry, D. Groom, R. Jacobsen, R.A. Muller, S. Perlmutter, A. Rosenfeld, C. Wickham, J. Wurtele, Under Review (2011), pp. 1–39
T. Plewa, T.J. Linde, V.G. Weirs, in Adaptive mesh refinement, theory and applications, Lecture notes in computational science and engineering (Springer, 2005), Vol. 41
C. Dangalchev, Physica A 365, 556 (2006)
K.A. Stephenson, M. Zelen, Soc. Networks 11, 1 (1989)
V. Latora, M. Marchiori, Physica A 314, 109 (2002)
M.E.J. Newman, Phys. Rev. E 64, 016132 (2001)
A. Arenas, A. Cabrales, A. Díaz-Guilera, R. Guimerà, F. Vega-Redondo, in Statistical Mechanics of Complex Networks, Lecture Notes in Physics, edited by R. Pastor-Satorras, M. Rubi, A. Díaz-Guilera (Springer Berlin, Heidelberg, 2003), Vol. 625, pp. 175–194
M.E.J. Newman, Soc. Networks 27, 39 (2005)
S. Bialonski, M.T. Horstmann, K. Lehnertz, Chaos 20, 013134 (2010)
J. Barcelo, Comput. Netw. 45, 333 (2004)
D. Gfeller, P. De Los Rios, Phys. Rev. Lett. 100, 174104 (2008)
M. Fiedler, Czech. Math. J. 23, 298 (1973)
R. Albert, H. Jeong, A.L. Barabási, Nature 406, 378 (2000)
G. Csárdi, T. Nepusz, Inter Journal Complex Systems CX. 18, 1695 (2006), http://igraph.sf.net
S.N. Soffer, A. Vázquez, Phys. Rev. E 71, 057101 (2005)
P. Bonacich, Am. J. Sociol. 92, 1170 (1987)
J.D. Noh, H. Rieger, Phys. Rev. Lett. 92, 118701 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Heitzig, J., Donges, J.F., Zou, Y. et al. Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes. Eur. Phys. J. B 85, 38 (2012). https://doi.org/10.1140/epjb/e2011-20678-7
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2011-20678-7