Complexity and Synchronization

In this article, the authors study a fully connected network (cluster) of interacting two-state units as a model of cooperative decision making. Each unit in isolation generates a Poisson process with rate g. The authors show that when the number of nodes is finite, the decision-making process becomes intermittent. The decision-time distribution density is characterized by inverse power-law behavior with index μ=1.5 and is exponentially truncated. The authors find that the condition of perfect consensus is recovered by means of a fat tail that becomes more and more extended with increasing numbers of nodes N. The intermittent dynamics of the global variable are described by the motion of a particle in a double well potential. The particle spends a portion of the total time τs at the top of the potential barrier. Using theoretical and numerical arguments it is proved that τs ∝ (1/g)1n(const X N). The second portion of its time, τk, is spent by the particle at the bottom of the potential well and it is given by τk=(1/g)exp(const X N). The authors show that the time τk is responsible for the Kramers fat tail. This generates a stronger ergodicity breakdown than that generated by the inverse power law without truncation. The authors establish that the condition of partial consensus can be transmitted from one cluster to another provided that both networks are in a cooperative condition. No significant information transmission is possible if one of the two networks is not yet self-organized. The authors find that partitioning a large network into a set of smaller interacting clusters has the effect of converting the fat Kramers tail into an inverse power law with μ=1.5.