Stochastic Hybrid Dynamic Multicultural Social Networks

  • Kristina Buzzell Hilton Department of Mathematics and Statistics, University of South Florida, Tampa, FL, United States;
  • G. S. Ladde Department of Mathematics and Statistics, University of South Florida, Tampa, FL, United States;
Keywords: Multi-agent Network, Cohesiveness, Lyapunov Second Method, Invariant Sets


In this work, we investigate the cohesive properties of a stochastic hybrid dynamic multi-cultural network under random environmental perturbations.  By considering a multi-agent dynamic network, we model a social structure and find conditions under which cohesion and coexistence is maintained using Lyapunov’s Second Method and the comparison method.  In this paper, we present a prototype illustration that exhibits the significance of the framework and approach.  Moreover, the explicit sufficient conditions in terms of system parameters are given to exhibit when the network is cohesive both locally and globally.  The sufficient conditions are algebraically simple, easy to verify, and robust.  Further, we decompose the cultural state domain into invariant sets and consider the behavior of members within each set. We also analyze the degree of conservativeness of the estimates using Euler-Maruyama type numerical approximation schemes based on the given illustration.

Author Biographies

Kristina Buzzell Hilton, Department of Mathematics and Statistics, University of South Florida, Tampa, FL, United States;

Department of Mathematics and Statistics

University of South Florida

G. S. Ladde, Department of Mathematics and Statistics, University of South Florida, Tampa, FL, United States;

Department of Mathematics and Statistics


University of South Florida


(1) Chandra, J. and G.S. Ladde, Collective behavior of multi-agent network dynamic systems under internal and external perturbations. Nonlinear Analysis: Real World Applications, 2010. 11(3): p. 1330-1344.

(2) Hilton, K.B. and G.S. Ladde, Deterministic Multicultural Dynamic Networks: Seeking a Balance between Attractive and Repulsive Forces. International Journal of Communications, Network and System Sciences, 2016. 9(12): p. 582-602.

(3) Hilton, K.B. and G.S. Ladde, Stochastic Multicultural Networks. Dynamical Systems and Applications, 2017. In Press.

(4) Anabtawi, M.J., S. Sathannanthan, and G.S. Ladde, Convergence and stability analysis of large-scale parabolic systems under Markovian structural perturbations. International Journal of Applied Mathematics, 2000. 2(1): p. 57-85.

(5) Ladde, G.S. and B.A. Lawrence, Stability and convergence of large-scale stochastic approximation. International Journal of Systems Science, 1995. 26(3): p. 595-618.

(6) Ladde, G.S. and D.D. Siljak, Connective stability of large-scale stochastic systems. International Journal of Systems Science, 1975. 6(8): p. 713-721.

(7) Siljak, D.D., Large-scale Dynamic Systems: Stability and Structure1978, New York, NY: Elsevier North-Holland.

(8) Acemoglu, D., et al., Opinion fluctuations and disagreement in social networks. Mathematics of Operations Research, 2013. 38(1): p. 1-27.

(9) DeGroot, M.H., Reaching a consensus. Journal of the American Statistical Association, 1974. 69(345): p. 118-121.

(10) Friedkin, N.E. Complex objects in the polytopes of the linear state-space. arXiv preprint arXiv:1401.5339, 2004. January 2014.

(11) Ma, H., Literature survey of stability of dynamical multi-agent systems with applications in rural-urban migration. American Journal of Engineering and Technology Research, 2013. 13(1): p. 131-140.

(12) Cao, Y., et al., An overview of recent progress in the study of distributed multi-agen coordination. IEEE Transactions on Industrial Informatics, 2013. 9(1): p. 427-438.

(13) Hu, H.-x., et al., Group consensus in multi-agent systems with hybrid protocol. Journal of the Franklin Institute, 2013. 350: p. 575-597.

(14) Huang, M. and J.H. Manton, Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior. SIAM Journal on Control and Optimization, 2009. 48(1): p. 131-161.

(15) Zhu, Y.-K., X.-P. Guan, and X.-Y. Luo, Finite-time consensus for multi-agent systems via nonlinear control protocols. International Journal of Automation and Computing, 2013. 10(5): p. 455-462.

(16) Axelrod, R., The Complexity of Cooperation: Agent-based Models of Competition1997, Princeton, NJ: Princeton University Press.

(17) Ladde, G.S., Hybrid Dynamical Inequalities and Applications. Dynamical Systems and Applications, 2005. 14: p. 481-514.

(18) Ladde, G.S. and V. Lakshmikantham, Random Differential Inequalities1980, New York, NY: Academic Press.

(19) Ladde, A.G. and G.S. Ladde, An Introduction to Differential Equations: Stochastic Modeling. Vol. 2. 2013, Hackensak, NJ: World Scientific.

(20) Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review, 2001. 43(3): p. 525-546.

(21) Higham, D.J. and P.E. Kloeden, Maple and Matlab for stochastic differential equations in finance, in Programming Languages and Systems in Computational Economics and Finance2002, Springer. p. 233-269.

(22) Kloeden, P.E. and E. Platen, Numerical Solution of Stochastic Differential Equations1992, New York, NY: Springer-Verlag.