Intrinsic Stability, Time Delays and Transformations of Dynamical Networks

  • Leonid Bunimovich
  • Benjamin Webb
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 20)


The dynamics of most real networks are time delayed. These time delays are important as they can often have a destabilizing effect on the network’s dynamics. Here we introduce the notion of intrinsic stability, which is stronger than the standard notion of stability used in the study of network dynamics or more generally in the study of multidimensional systems. The difference between a stable and an intrinsically stable network is that an intrinsically stable network maintains its (intrinsic) stability as nondistributed time-delays are incorporated into or removed from the network. Importantly, determining whether a network is intrinsic stability is fairly easy for concrete systems. Therefore, one can analyze the stability of a time-delayed network by analyzing the stability of the much simpler undelayed version of the network. One can simplify this analysis even further by removing the network’s implicit time-delays, which further reduces the network to a lower-dimensional system which can be used to gain improved stability estimates of the original unreduced network. A network can also be expanded in ways that preserves its intrinsic stability, which suggests how a network with an evolving structure can maintain its stability as it grows. Thus, intrinsic stability is the property that ensures the stability not only of a network but also of its reductions, expansions, and its time-delayed variants. We illustrate these results and techniques on various examples of Cohen–Grossberg neural networks.


Network Stability Time delay 



The work of Leonid Bunimovich is partially supported by the NSF grant DMS-1600568.


  1. 1.
    Alpcan, T., Basar, T.: A globally stable adaptive congestion control scheme for internet-style networks with delay. IEEE/ACM Trans. Netw. 13, 6 (2005)CrossRefGoogle Scholar
  2. 2.
    Bunimovich, L.A., Webb, B.Z.: Isospectral graph transformations, spectral equivalence, and global stability of dynamical networks. Nonlinearity 25, 211–254 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bunimovich, L.A., Webb, B.Z.: Restrictions and stability of time-delayed dynamical networks. Nonlinearity 26, 2131–2156 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bunimovich, L.A., Webb, B.Z.: Isospectral Transformations: A New Approach to Analyzing Multidimensional Systems and Networks. Springer Monographs in Mathematics. Springer, Berlin (2014)CrossRefMATHGoogle Scholar
  5. 5.
    Bunimovich, L.A., Webb, B.Z.: Mechanisms for network growth that preserve spectral and local structure (2016). arXiv:1608.06247 [nlin.AO]Google Scholar
  6. 6.
    Cao, J.: Global asymptotic stability of delayed bi-directional associative memory neural networks. Appl. Math. Comput. 142 (2–3), 333–339 (2003)MathSciNetMATHGoogle Scholar
  7. 7.
    Chena, S., Zhaoa, W., Xub, Y.: New criteria for globally exponential stability of delayed Cohen–Grossberg neural network. Math. Comput. Simul. 79, 1527–1543 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cheng, C.-Y., Lin, K.-H., Shih, C.-W.: Multistability in recurrent neural networks. SIAM J. Appl. Math. 66 (4), 1301–1320 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cohen, M., Grossberg S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst. Man Cybern. SMC-13, 815–821 (1983)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Newman, M.E.J.: Networks an Introduction. Oxford University Press, Oxford (2010)CrossRefMATHGoogle Scholar
  11. 11.
    Tao, L., Ting, W., Shumin, F.: Stability analysis on discrete-time Cohen–Grossberg neural networks with bounded distributed delay. In: Proceedings of the 30th Chinese Control Conference, Yantai, 22–24 July (2011)Google Scholar
  12. 12.
    Wang, L., Dai, G.-Z.: Global stability of virus spreading in complex heterogeneous networks. SIAM J. Appl. Math. 68 (5), 1495–1502 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsBrigham Young UniversityProvoUSA

Personalised recommendations