Dynamic social network models incorporating stochasticity and delays

Banks, H.; Rehm, Keri; Sutton, Karyn

doi:10.1090/S0033-569X-2010-01201-X

Authors: H. T. Banks, Keri Rehm and Karyn L. Sutton
Journal: Quart. Appl. Math. 68 (2010), 783-802
MSC (2000): Primary 91D30, 91C20, 34F05, 34K50
DOI: https://doi.org/10.1090/S0033-569X-2010-01201-X
Published electronically: September 23, 2010
MathSciNet review: 2761244
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Networks are typically studied via computational models, and often investigations are restricted to the static case. Here we extend the work in Banks, Karr, Nguyen and Samuels (2008), which demonstrated a simple dynamical system framework in which to study social network behavior, to include a discrete delay. This delay represents the time lag that is likely required for an agent to change his/her own characteristics (e.g., opinions, viewpoints or behavior) after interacting with an agent possessing different characteristics. Thus this modification adds significantly to the relevance of the model in many potential applications. We have shown that the delays can be incorporated into a stochastic differential equations (SDE) framework in an efficient and computationally tractable way. Through numerical studies, we see novel outcomes when stochasticity, delay, or both are considered, demonstrating the need to include these features should they be present in the network application.

References

A. V. Balakrishnan, Active control of airfoils in unsteady aerodynamics, Appl. Math. Optim. 4 (1977/78), no. 2, 171–195. MR 468596, DOI https://doi.org/10.1007/BF01442138
H. T. Banks, Control of functional differential equations with function space boundary conditions, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 1–16. MR 0389319
H. T. Banks, Parameter identification techniques for physiological control systems, Mathematical aspects of physiology (Proc. Summer Sem., Univ. Utah, Salt Lake City, Utah, 1980) Lectures in Appl. Math., vol. 19, Amer. Math. Soc., Providence, R.I., 1981, pp. 361–383. MR 623301

H. T. Banks, Identification of nonlinear delay systems using spline methods, in Nonlinear Phenomena in Mathematical Sciences (V. Lakshmikantham, ed.), Academic Press, New York, 1982, 47–55.

H. T. Banks, D. M. Bortz, and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. Biosci. 183 (2003), no. 1, 63–91. MR 1965457, DOI https://doi.org/10.1016/S0025-5564%2802%2900218-3

H. T. Banks and J. A. Burns, Hereditary control problems: numerical methods based on averaging approximations, SIAM J. Control Optim. 16 (1978), no. 2, 169–208. MR 483428, DOI https://doi.org/10.1137/0316013

H. T. Banks, J. A. Burns, and E. M. Cliff, Parameter estimation and identification for systems with delays, SIAM J. Control Optim. 19 (1981), no. 6, 791–828. MR 634954, DOI https://doi.org/10.1137/0319051

H. T. Banks, A. F. Karr, H. K. Nguyen, and J. R. Samuels Jr., Sensitivity to noise variance in a social network dynamics model, Quart. Appl. Math. 66 (2008), no. 2, 233–247. MR 2416772, DOI https://doi.org/10.1090/S0033-569X-08-01124-0

H. T. Banks and F. Kappel, Spline approximations for functional differential equations, J. Differential Equations 34 (1979), no. 3, 496–522. MR 555324, DOI https://doi.org/10.1016/0022-0396%2879%2990033-0

H. T. Banks, K. L. Rehm and K. L. Sutton, Conversion of a dynamic social network stochastic differential equation model to Fokker-Planck model, Technical Report CRSC-TR09-10, Center for Research in Scientific Computation, North Carolina State University, April, 2009.

Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745

Steve Blythe, Xuerong Mao, and Xiaoxin Liao, Stability of stochastic delay neural networks, J. Franklin Inst. 338 (2001), no. 4, 481–495. MR 1833972, DOI https://doi.org/10.1016/S0016-0032%2801%2900016-3

E. Boukas and Z. Liu, Deterministic and Stochastic Time Delay Systems, Springer-Verlag, New York NY, 2002.

P. J. Carrington, J. Scott and S. Wasserman, Models and Methods in Social Network Analysis, Cambridge University Press, New York, 2005.

Rodney D. Driver, Some harmless delays, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 103–119. MR 0385277

L. È. Èl′sgol′c, Introduction to the theory of differential equations with deviating arguments, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966. Translated from the Russian by Robert J. McLaughlin. MR 0192154

G. R. Grimmett and D. R. Stirzaker, Probability and random processes, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1992. MR 1199812

A. Halanay, Differential equations: Stability, oscillations, time lags, Academic Press, New York-London, 1966. MR 0216103

Jack K. Hale, Functional differential equations, Springer-Verlag New York, New York-Heidelberg, 1971. Applied Mathematical Sciences, Vol. 3. MR 0466837

Y. Jiang and Y. Liu, Stochastic Network Calculus, Springer-Verlag, New York NY, 2008.

N. Minorsky, Experiments with activated tanks, Trans. ASME, 69 (1941), 735–747.

N. Minorsky, Self-excited oscillations in dynamical systems possessing retarded actions, J. Appl. Mechanics, 9 (1942), A65–A71.

Nicolas Minorsky, Nonlinear oscillations, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962. MR 0137891

K. Park and W. Willinger, Self-similar Network Traffic and Performance Evaluation, John Wiley & Sons, Inc., New York NY, 2000.

S. Wasserman and K. Faust, Social Network Analysis: Methods and Applications, Cambridge University Press, New York, 1994.

S. Wasserman and J. Galaskiewicz, Advances in Social Network Analysis: Research in the Social and Behavioral Sciences, Sage Publications, Thousand Oaks, CA, 1994.