Dynamical Response of Networks Under External Perturbations: Exact Results

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 $$N_0$$N0 nodes be frozen in state 0, $$N_1$$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 $$N_0$$N0 and $$N_1$$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.

[1]  M. Mobilia Does a single zealot affect an infinite group of voters? , 2003, Physical review letters.

[2]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[3]  N. Boccara MODELS OF OPINION FORMATION: INFLUENCE OF OPINION LEADERS , 2007, 0704.1790.

[4]  Alexandre Arenas,et al.  Optimal network topologies for local search with congestion , 2002, Physical review letters.

[5]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[6]  Dan Braha,et al.  From Centrality to Temporary Fame: Dynamic Centrality in Complex Networks , 2006, Complex..

[7]  Yamir Moreno,et al.  Dynamics of rumor spreading in complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Yaneer Bar-Yam,et al.  Analytically solvable model of probabilistic network dynamics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[10]  S. Redner,et al.  Voter model on heterogeneous graphs. , 2004, Physical review letters.

[11]  K. Goh,et al.  Robustness of the avalanche dynamics in data-packet transport on scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  R. Glauber Time‐Dependent Statistics of the Ising Model , 1963 .

[13]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[14]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[15]  Daniele Vilone,et al.  Solution of voter model dynamics on annealed small-world networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Guillermo Abramson,et al.  Vector opinion dynamics in a model for social influence , 2003 .

[17]  I. Epstein,et al.  Response of complex networks to stimuli. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[18]  P. A. P. Moran,et al.  Random processes in genetics , 1958, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[20]  S. Redner,et al.  On the role of zealotry in the voter model , 2007 .

[21]  J. Urry Complexity , 2006, Interpreting Art.

[22]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[23]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[24]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[25]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[26]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

[27]  F. C. Santos,et al.  Evolutionary games in self-organizing populations , 2008 .

[28]  Ying-Cheng Lai,et al.  Oscillations of complex networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  T. Liggett Interacting Particle Systems , 1985 .