Abstract
The theory of complex networks and of disordered systems is used to study the stability and dynamical properties of a simple model of material flow networks defined on random graphs. In particular we address instabilities that are characteristic of flow networks in economic, ecological and biological systems. Based on results from random matrix theory, we work out the phase diagram of such systems defined on extensively connected random graphs, and study in detail how the choice of control policies and the network structure affects stability. We also present results for more complex topologies of the underlying graph, focussing on finitely connected Erdös-Réyni graphs, Small-World Networks and Barabási-Albert scale-free networks. Results indicate that variability of input-output matrix elements, and random structures of the underlying graph tend to make the system less stable, while fast price dynamics or strong responsiveness to stock accumulation promote stability.
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References
A. Abel, B. Bernanke, Macroeconomics (Addison Wesley, 2003)
C. Plosser, J. Economic Perspectives 3, 51 (1989)
R. Farmer, J.T. Guo, J. Economic Theory 63, 43 (1994)
G. Mankiw, Euro. Econ. Rev. 36, 559 (1992)
R. Albert, A.L. Barabási, Rev. Mod. Phys. 74, 47 (2002)
S.N. Dorogovtsev, J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, New York, 2003)
M.E.J. Newman, A.L. Barabási, D.J. Watts, The Structure and Dynamics of Networks (Princeton University Press, US, 2006)
B. Drossel, A.J. McKane, Handbook of Graphs and Networks (Wiley-VCH, 2003)
U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits (Chapman and Hall, 2006)
S. Bornholdt, H.G. Schuster, Handbook of Graphs and Networks: From the genome to the internet (Wiley, 2006)
D. Helbing, S. Lämmer, U. Witt, T. Brenner, Phys. Rev. E 70, 56118 (2004)
A. Ponzi, A. Yasutomi, K. Kaneko, Physica A 324, 372 (2003)
D. Helbing, S. Lämmer, T. Seidel, P. Seba, T. Platkowski, Phys. Rev. E 70, 066116 (2004)
A. Ponzi, A. Yasutomi, K. Kaneko, J. Econ. Behavior and Organization 61, 729 (2006)
G. Weisbuch, Complexus, pages 217–227 (2005)
G. Weisbuch, S. Battiston, Production networks and failure avalanches, [arXiv:physics/0507101v1] (2005)
M.D. Köenig, S. Battiston, F. Schweitzer, Innovation Networks – New Approaches in Modelling and Analyzing (Springer, 2008), Chap. Modelling Evolving Innovation Networks
M.D. König, S. Battiston, M. Napoletano, F. Schweitzer, Networks and Heterogeneous Media 3, 201 (2008)
M.L. Mehta, Random Matrices, 3rd edn. (Elsevier/ Academic Press, Amsterdam, 2004)
W. Leontief, Input-Output Economics (Oxford University Press, 1986)
M. Spearman, D. Woodruff, W. Hopp, Int. J. Production Research 28, 879 (1990)
M. Morishima, Econometrica 33, 829 (1965)
K. Lancaster, Mathematical Economics (Dover Publications, New York, 1987)
M. Mézard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987)
A. De Martino, M. Marsili, I. Pérez Castillo, J. Stat. Mech. 2004, 04002 (2004)
A. De Martino, private communication (2007)
A.J. Bray, G.J. Rodgers, Phys. Rev. B 16, 11461 (1988)
J.P.L. Hatchett, R. Kühn, J. Phys. A 39, 2231 (2006)
B. Derrida, E. Gardner, A. Zippelius, Europhys. Lett. 4, 167 (1987)
K. Anand, R. Kühn, Phys. Rev. E 75, 016111 (2007)
H.J. Sommers, A. Crisanti, H. Sompolinsky, Y. Stein, Phys. Rev. Lett. 60, 1895 (1988)
G.J. Rodgers, A.J. Bray, Phys. Rev. B 37, 3557 (1988)
G. Biroli, R. Monasson, J. Phys. A 32, L255 (1999)
T. Nagao, T. Tanaka, J. Phys. A 40, 4973 (2007)
S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, A.N. Samukhin, Phys. Rev. E 68, 046109 (2003)
R. Kühn, J. Phys. A 41, 295002 (2008)
T. Rogers, I. Perez Castillo, R. Kühn, K. Takeda, Phys. Rev. E 78, 031116 (2008)
G. Bianconi, e-print arXiv:0804.1744v1 (2008)
G. Semerjian, L.F. Cugliandolo, J. Phys. A: Math. Gen. 35, 4837 (2002)
D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)
M.E.J. Newman, D.J. Watts, Phys. Rev. E 7332 (1999)
A.L. Barabási, R. Albert, Science 286, 509 (1999)
J.J. Hopfield, Proc. Nat. Acad. Sci. USA 79, 2554 (1982)
A.C.C. Coolen, R. Kühn, P. Sollich, Theory of Neural Information Processing Systems (Oxford University Press, 2005)
A. De Martino, M. Marsili, J. Phys. A: Math. Gen. 39, R465 (2006)
R.M. May, Nature 238, 413 (1972)
R.M. May, Stability and Complexity in Model Ecosystems (Princeton University Press, 2001)
I.D. Rozdilsky, L. Stone, Ecology Lett. 4, 397 (2001)
J.A.A. Vincent, G.D. Kokkoris, Ecology Lett. 6, 498 (2003)
B. Wemmenhove, N.S. Skantzos, A.C.C. Coolen, J. Phys. A: Math. Gen. 37, 7653 (2004)
D. Chowdhury, D. Stauffer, Physica A 346, 7653 (2005)
S. Lämmer, private communication (2007)
G. Biroli, R. Monasson, J. Phys. A: Math. Gen. 32, L255 (1999)
O. Khorunzhiy, W. Kirsch, P. Müller, Ann. Appl. Prob. 16, 295 (2006)
A. De Martino, M. Marsili, J. Stat. Mech.: Theory and Experiment 09, L09003 (2005)
S. Diederich, M. Opper, Phys. Rev. A 39, 4333 (1989)
M. Opper, S. Diederich, Phys. Rev. Lett. 69, 1616 (1992)
P.J. Antsaklis, Proc. IEEE, Special Issue on Hybrid Systems: Theory and Applications 88, 879 (2000)
K.B. Efetov, Advances in Physics 32, 53 (1983)
S.F. Edwards, R.C. Jones, J. Phys. A 9, 1595 (1976)
F. Haake, F. Izrailev, N. Lehmann, D. Saher, H.J. Sommers, Z. Phys. B 88, 359 (1992)
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Anand, K., Galla, T. Stability and dynamical properties of material flow systems on random networks. Eur. Phys. J. B 68, 587–600 (2009). https://doi.org/10.1140/epjb/e2009-00106-7
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DOI: https://doi.org/10.1140/epjb/e2009-00106-7