Abstract.
We briefly describe the toolkit used for studying complex systems: nonlinear dynamics, statistical physics, and network theory. We place particular emphasis on network theory--the topic of this special issue--and its importance in augmenting the framework for the quantitative study of complex systems. In order to illustrate the main issues, we briefly review several areas where network theory has led to significant developments in our understanding of complex systems. Specifically, we discuss changes, arising from network theory, in our understanding of (i) the Internet and other communication networks, (ii) the structure of natural ecosystems, (iii) the spread of diseases and information, (iv) the structure of cellular signalling networks, and (v) infrastructure robustness. Finally, we discuss how complexity requires both new tools and an augmentation of the conceptual framework--including an expanded definition of what is meant by a “quantitative prediction.”
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References
J.M. Ottino, AIChE J. 49, 292 (2003)
G.L. Baker, J.P. Gollub, Chaotic Dynamics: An Introduction (Cambridge University Press, Cambridge, UK, 1990)
L. Glass, M.C. MacKey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, NJ, 1988)
J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge University Press, Cambridge, UK, 1989)
S. Wolfram, A New Kind of Science (Champaign, IL, Wolfram Media, 2002)
J.M. Epstein, R.L. Axtell, Growing Artificial Societies: Social Science from the Bottom Up (Cambridge, MA, MIT Press, 1996)
H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1971)
H.E. Stanley, Rev. Mod. Phys. 71, S358-S364 (1999) [Special Issue for the Centennial of the American Physical Society]
C.-K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Chaos 5, 82 (1995)
P.Ch. Ivanov, L.A.N. Amaral, A.L. Goldberger et al. , Nature 399, 461 (1999)
L.A.N. Amaral, P.Ch. Ivanov, N. Aoyagi et al. , Phys. Rev. Lett. 86, 6026 (2001)
A.L. Goldberger, L.A.N. Amaral, J.M. Hausdorff et al. , Proc. Nat. Acad. Sci. USA 99 Supp. 1, 2466 (2002)
U.A. Muller, M.M. Dacorogna, R.B. Olsen et al. , J. Banking Fin. 14, 1189 (1990)
A. Pagan, J. Empirical Finance 3, 15 (1996)
P. Gopikrishnan, V. Plerou, L.A.N. Amaral et al. , Phys. Rev. E 60, 5305 (1999)
V. Plerou, P. Gopikrishnan, L.A.N. Amaral et al. , Phys. Rev. E 60, 6519 (1999)
M.H.R. Stanley, L.A.N. Amaral, S.V. Buldyrev et al. , Nature 379, 804 (1996)
L.A.N. Amaral, S.V. Buldyrev, S. Havlin et al. , J. Phys. I France 7, 621 (1997)
Y. Lee, L.A.N. Amaral, D. Canning et al. , Phys. Rev. Lett. 81, 3275 (1998)
L.A.N. Amaral, S.V. Buldyrev, S. Havlin et al. , Phys. Rev. Lett. 80, 1385 (1998)
V. Plerou, L.A.N. Amaral, P. Gopikrishnan et al. , Nature 400, 433 (1999)
L.A.N. Amaral, P. Gopikrishnan, K. Matia et al. , Scientometrics 51, 9 (2001)
H.E. Stanley, L.A.N. Amaral, S.V. Buldyrev et al. , Proc. Nat. Acad. Sci. USA 99 Supp. 1, 2561 (2002)
R. Jackiw, Phys. Today 25, 23 (1972)
P.J.E. Peebles, The Large-Scale Structure of the Universe (Princeton University Press, Princeton, NJ, 1980)
F. Liljeros, C.R. Edling, L.A.N. Amaral, Microbes Infect. 5, 189 (2003)
J.J. Binney, N.J. Dowrick, A.J. Fisher, M.E.J. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford University Press, Oxford, 1992)
J. von Newmann, Theory of self-reproducing automata, edited by A.W. Burks (University of Illinois Press, Urbana and London, 1966)
S. Ulam, in Proc. Int. Congress Math. 2, 264 (1950)
M. Gardner, Scientific American 223, 120 (1970)
D.J. Watts, Small Worlds: The Dynamics of Networks Between Order and Randomness (Princeton University Press, Princeton, NJ, 1999)
M.E.J. Newman, J. Stat. Phys. 101, 819 (2000)
S.H. Strogatz, Nature 410, 268 (2001)
R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)
S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)
S. Wasserman, K. Faust, Social Network Analysis (Cambridge University Press, Cambridge, UK, 1994)
A.F.J. van Raan, Nature 347, 626 (1990)
M.E.J. Newman, P. Natl. Acad. Sci. USA 98, 404 (2001)
F. Liljeros, C.R. Edling, L.A.N. Amaral et al. , Nature 411, 907 (2001)
E.M. Jin, M. Girvan, M.E.J. Newman, Phys. Rev. E 64, 046132 (2001)
M. Girvan, M.E.J. Newman, P. Natl. Acad. Sci. USA 99, 7821 (2002)
D.J. Watts, P.S. Dodds, M.E.J. Newman, Science 296, 1302 (2002)
M.E.J. Newman, Ego-centered networks and the ripple effect. Soc. Networks. 25, 83 (2003)
H. Ebel, L.-I. Mielsch, S. Bornholdt, Phys. Rev. E 66, 035103(R) (2002)
R. Guimera, L. Danon, A. Diaz-Guilera, F. Giralt, A. Arenas, Self-similar community structure in organisations, arXiv:cond-mat/0211498
M. Faloutsos, P. Faloutsos, C. Faloutsos, ACM SIGCOMM ‘99, Comput. Commun. Rev. 29, 251 (1999)
R. Albert, H. Jeong, A.-L. Barabási, Nature 401, 130 (1999)
B.A. Huberman, L.A. Adamic, Nature 401, 131 (1999)
S.L. Pimm, J.H. Lawton, J.E. Cohen, Nature 350, 669 (1991)
R.T. Paine, Nature 355, 73 (1992)
J. Camacho, R. Guimerá, L.A.N. Amaral, Phys. Rev. E 65, 030901(R) (2002)
J. Camacho, R. Guimerá, L.A.N. Amaral, Phys. Rev. Lett. 88, 228102 (2002)
J.A. Dunne, R.J. Williams, N.D. Martinez, P. Natl. Acad. Sci. USA 99, 12917 (2002)
C. Koch, G. Laurent, Science 284, 96 (1999)
L.F. Lago-Fernandez, R. Huerta, F. Corbacho, J.A. Siguenza, Phys. Rev. Lett. 84, 2758 (2000)
U. Alon, M.G. Surette, N. Barkai, S. Leibler, Nature 397, 168 (1999)
R. Milo, S. Shen-Orr, S. Itzkovitz et al. , Science 298, 824 (2002)
Z.N. Oltvai, A.-L. Barabási, Science 298, 763 (2002)
H. Jeong, S.P. Mason, A.-L. Barabási, Z.N. Oltvai, Nature 411, 41 (2001)
A.C. Gavin et al. , Nature 415, 141 (2002)
Y. Ho et al. , Nature 415, 180 (2002)
H. Jeong, B. Tombor, R. Albert et al. , Nature 407, 651 (2000)
A. Wagner, D.A. Fell, P. Roy. Soc. London B-Biol. Sci. 268, 1803 (2001)
E. Ravasz, A.L. Somera, D.A. Mongru et al. , Science 297, 1551 (2002)
L. Euler, Solutio problematis ad geometriam situs pertinentis (1736)
The Small World, edited by M. Kochen (Norwood, NJ, Ablex, 1989)
S. Milgram, Psychol. Today 1, 61 (1967)
J. Travers, S. Milgram, Sociometry 32, 425 (1969)
B. Bollobás, Random Graphs (London, Academic Press, 1985)
D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)
M. Barthélemy, L.A.N. Amaral, Phys. Rev. Lett. 82, 3180 (1999)
A. Barrat, M. Weigt, Eur. Phys. J. B 13, 547 (2000)
A.-L. Barabási, R. Albert, Science 286, 509 (1999)
P.O. Seglen, J. Am. Soc. Inf. Sci. 43, 628 (1992)
S. Redner, Eur. Phys. J. B 4, 131 (1998)
R. Cohen, S. Havlin, Phys. Rev. Lett. 90, 058701 (2003)
L.A.N. Amaral, A. Scala, M. Barthélemy, H.E. Stanley, Proc. Nat. Acad. Sci. USA 97, 11149 (2000)
S. Mossa, M. Barthélemy, H.E. Stanley, L.A.N. Amaral, Phys. Rev. Lett. 88, 138701 (2002)
M.E.J. Newman, SIAM Rev. 45, 167 (2003)
S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks: From Biological nets to the Internet and WWW (Oxford University Press, Oxford, 2003)
A. Tanenbaum, Computer Networks, 3rd edn. (Prentice Hall, Englewood Cliffs, 1996)
D. Comer, Internetworking with TCP/IP, Vol. 1, 4th edn. (Prentice Hall, Englewood Cliffs, 2000)
P.W. Anderson, K.J. Arrow, D. Pines, The Economy as an Evolving Complex System, SFI Studies in the Sciences of Complexity (Addison Wesley Longman, 1988)
W. Brian Arthur, S.N. Durlauf, D.A. Lane, The Economy as an Evolving Complex System II, SFI Studies in the Sciences of Complexity (Addison Wesley Longman, 1997)
A. Bunde, J. Kropp, H.J. Schellnhuber, The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes (Springer-Verlag, Berlin, 2002)
Unsolved Problem on Noise 1999, edited by D. Abbott, L.B. Kish, AIP Conf. Proc. 511 (American Institute of Physics, Melville NY, 2000)
J. Postel, Transmission Control Protocol. RFC 793 (1981)
W.E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, IEEE/ACM Trans. Network 2, 1 (1994)
V. Paxson, S. Floyd IEEE/ACM Trans. Network 3, 226 (1995)
M. Crovella, A. Bestavros, IEEE/ACM Trans. Network 5, 835 (1997)
I. Csabai, J. Phys. A 27, 417 (1994)
M. Takayasu, H. Takayasu, T. Sato, Physica A 233, 924 (1996)
M. Takayasu et al. , Physica A 277, 248 (2000)
K. Fukuda et al. , Physica A 287, 289 (2000)
K. Fukuda et al. , in Traffic and Granular Flow 2001, edited by Y. Sugiyama et al. (Springer, Heidelberg, 2002)
P. Bernaola-Glaván, P.Ch. Ivanov, L.A.N. Amaral, H.E. Stanley, Phys. Rev. Lett. 87, 168105 (2001)
K. Fukuda, L.A.N. Amaral, H.E. Stanley, Europhys. Lett. 62, 189 (2003)
K.A. Eriksen, I. Simonsen, S. Maslov, K. Sneppen, Phys. Rev. Lett. 90, 148701 (2003)
M. Barthélémy, B. Gondran, E. Guichard, Physica A 319, 633 (2003)
J.-J. Pansiot, D. Grad, ACM Comp. Comm. Rev. 28, 41 (1998)
R. Govindan, A. Reddy, in Proceedings of IEEE INFOCOM’97 (1997), p. 850
H. Chou, A note on power-laws on Internet topology, arXiv:cs.NI/0012019 (2000)
G. Caldarelli, R. Marchetti, L. Pietronero, Europhys. Lett. 52, 386 (2000)
S.-H. Yook, H. Jeong, A.-L. Barabási, Proc. Nat. Ac. Sci. USA 99, 13382 (2002)
The National Laboratory for Applied Network Research (NLANR), sponsored by the National Science Foundation, provides Internet routing related information based on border gateway protocol data (see http://moat.nlanr.net/)
R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001)
A. Vázquez, R. Pastor-Satorras, A. Vespignani Phys. Rev. E 65, 066130 (2002)
J. Camacho, R. Guimerá, D.B. Stouffer, L.A.N. Amaral, Quantitative patterns in the structure of model and empirical food webs, Ecol. Mon. (submitted, 2004)
D. Garlaschelli, G. Caldarelli, L. Pietronero, Nature 423, 165 (2003)
R.M. Anderson, R.M. May, Infectious Diseases in Humans (Oxford University Press, Oxford, 1992)
R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)
R. Pastor-Satorras, A, Vespignani, Phys. Rev. E 65, 036104 (2002)
D.J. Watts, Proc. Nat. Ac. Sci. USA 99, 5766 (2002)
R. Cohen, S. Havlin, D. ben-Avraham, Efficient Immunization Strategies for Computer Networks and Populations, arXiv:cond-mat/0207387
Z. Dezso, A.-L. Barabási, Phys. Rev. E 65, 055103 (2002)
V.M. Egu\’iluz, K. Klemm, Phys. Rev. Lett. 89, 108701 (2002)
A. Vazquez et al. , Phys. Rev. E 67, 046111 (2003)
L.S. Liebovitch, I.B. Schwartz, Phys. Rev. E 68, 017101 (2003)
M.E.J. Newman, S. Forrest, J. Balthrop, Phys. Rev. E 66, 035101 (2002)
B. Vogelstein, D. Lane, A.J. Levine, Nature 408, 307 (2000)
R. Albert, H. Jeong, A.-L. Barábasi, Nature 406, 378 (2000)
A. Broder et al. , Computer Networks 33, 309 (2000)
R. Cohen, K. Erez, D. Ben-Avrham, S. Havlin, Phys. Rev. Lett. 85, 4626 (2000)
D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)
R. Cohen, K. Erez, D. Ben-Avrham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001)
P. Holme, B.J. Kim, Phys. Rev. E 65, 066109 (2002)
P. Holme, Phys. Rev. E 66, 036119 (2002)
T. McMahon, Science 179, 1201 (1973)
G.B. West, J.H. Brown, B.J. Enquist, Science 276, 122 (1997)
J.R. Banavar, A. Maritan, A. Rinaldo, Nature 399, 130 (1999)
L.P. Kadanoff, Physics Today 54, 34 (2001)
J.M. Ottino, Nature 427, 399 (2004)
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Received: 12 November 2003, Published online: 14 May 2004
PACS:
89.75.Fb Structures and organization in complex systems - 89.75.Da Systems obeying scaling laws
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Amaral, L.A.N., Ottino, J.M. Complex networks. Eur. Phys. J. B 38, 147–162 (2004). https://doi.org/10.1140/epjb/e2004-00110-5
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DOI: https://doi.org/10.1140/epjb/e2004-00110-5