Abstract
This chapter examines the relationship between the dynamics of large networks and of their smaller factor networks (factors) obtained through the factorization of the network’s digraph representation. We specifically examine dynamics of networks which have Z-matrix state matrices. We perform a Cartesian product decomposition on its network structure producing factors which also have Z-matrix dynamics. A factorization lemma is presented that represents the trajectories of the large network in terms of the factors’ trajectories. An interval matrix lemma provides families of network dynamics whose trajectories are bounded by the interval bounds’ factors’ trajectories.
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
R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,” Proc. IEEE, vol. 95, no. 1, pp. 215–233, Jan. 2007.
H. G. Tanner, G. J. Pappas, and V. Kumar, “Leader-to-formation stability,” IEEE Transactions on Robotics and Automation, vol. 20, no. 3, pp. 443–455, 2004.
M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks. Princeton: Princeton University Press, 2010.
R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge University Press, 1990.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Academic Press, 1979.
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. Hwang, “Complex networks: structure and dynamics,” Physics Reports, vol. 424, no. 4–5, pp. 175–308, 2006.
N. Ganguly, A. Deutsch, and A. Mukherjee, Dynamics on and of Complex Networks: Applications to Biology, Computer Science, and the Social Sciences. Boston: Birkhauser, 2009.
L. Kocarev and G. Vattay, Complex Dynamics in Communication Networks. Berlin: Springer, 2005.
A. Nguyen and M. Mesbahi, “A factorization lemma for the agreement dynamics,” in 46th IEEE Conference on Decision and Control, no. 1, 2007, pp. 288–293.
W. Imrich and S. Klavzar, Product Graphs: Structure and Recognition. New York: Wiley, 2000.
G. Sabidussi, “Graph multiplication,” Mathematische Zeitschrift, vol. 72, pp. 446–457, 1960.
V. G. Vizing, “The Cartesian product of graphs,” Vycisl. Sistemy, pp. 30–43, 1963.
J. Feigenbaum, “Directed graphs have unique Cartesian factorizations that can be found in polynomial time,” Discrete Applied Mathematics, vol. 15, pp. 105–110, 1986.
M. Mansour, “Robust stability of interval matrices,” in Proc. 28th IEEE Conference on Decision and Control, 1989, pp. 46–51.
J. Rohn, “Forty necessary and sufficient conditions for regularity of interval matrices: a survey,” Electronic Journal Of Linear Algebra, vol. 18, pp. 500–512, 2009.
A. Chapman and M. Mesbahi, “Advection on graphs,” in Proc. 50th IEEE Conference on Decision and Control, 2011, pp. 1461–1466.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chapman, A. (2015). Cartesian Products of Z-matrix Networks: Factorization and Interval Analysis. In: Semi-Autonomous Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-15010-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-15010-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15009-3
Online ISBN: 978-3-319-15010-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)