TY - JOUR

T1 - Dynamical Response of Networks Under External Perturbations

T2 - Exact Results

AU - Chinellato, David D.

AU - Epstein, Irving R.

AU - Braha, Dan

AU - Bar-Yam, Yaneer

AU - de Aguiar, Marcus A.M.

N1 - Funding Information:
M.A.M.A. and D.D.C. acknowledge financial support from CNPq and FAPESP. I.R.E. was supported by National Science Foundation Grant CHE-1362477.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.

PY - 2015/1/21

Y1 - 2015/1/21

N2 - We give exact statistical distributions for the dynamic response of influence networks subjected to external perturbations. We consider networks whose nodes have two internal states labeled 0 and 1. We let N0 nodes be frozen in state 0, N1 in state 1, and the remaining nodes change by adopting the state of a connected node with a fixed probability per time step. The frozen nodes can be interpreted as external perturbations to the subnetwork of free nodes. Analytically extending N0 and N1 to be smaller than 1 enables modeling the case of weak coupling. We solve the dynamical equations exactly for fully connected networks, obtaining the equilibrium distribution, transition probabilities between any two states and the characteristic time to equilibration. Our exact results are excellent approximations for other topologies, including random, regular lattice, scale-free and small world networks, when the numbers of fixed nodes are adjusted to take account of the effect of topology on coupling to the environment. This model can describe a variety of complex systems, from magnetic spins to social networks to population genetics, and was recently applied as a framework for early warning signals for real-world self-organized economic market crises.

AB - We give exact statistical distributions for the dynamic response of influence networks subjected to external perturbations. We consider networks whose nodes have two internal states labeled 0 and 1. We let N0 nodes be frozen in state 0, N1 in state 1, and the remaining nodes change by adopting the state of a connected node with a fixed probability per time step. The frozen nodes can be interpreted as external perturbations to the subnetwork of free nodes. Analytically extending N0 and N1 to be smaller than 1 enables modeling the case of weak coupling. We solve the dynamical equations exactly for fully connected networks, obtaining the equilibrium distribution, transition probabilities between any two states and the characteristic time to equilibration. Our exact results are excellent approximations for other topologies, including random, regular lattice, scale-free and small world networks, when the numbers of fixed nodes are adjusted to take account of the effect of topology on coupling to the environment. This model can describe a variety of complex systems, from magnetic spins to social networks to population genetics, and was recently applied as a framework for early warning signals for real-world self-organized economic market crises.

KW - External perturbations

KW - Finite systems

KW - Ising model

KW - Networks

KW - Voter model

UR - http://www.scopus.com/inward/record.url?scp=84939941931&partnerID=8YFLogxK

U2 - 10.1007/s10955-015-1189-x

DO - 10.1007/s10955-015-1189-x

M3 - Article

AN - SCOPUS:84939941931

VL - 159

SP - 221

EP - 230

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -