Abstract
A two-state, master equation-based decision-making model has been shown to generate phase transitions, to be topologically complex, and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision-making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy.
Highlights
The utility of the fractional calculus is demonstrated by capturing the dynamics of the individual elements of a complex network from the information quantifying that network’s global behavior
The solution to this fractional equation is obtained through a subordination procedure without the necessity of linearizing the underlying dynamics, that is, the solution retains the influence of the nonlinear network dynamics on the individual
In order to formalize the subordination process we introduce the concept of subjective time to distinguish between clock time that determines the activities of the network and operational or subjective time that determines the activities of the individual
Summary
In the ATA coupling case when the total number of elements within the network becomes infinite (N −→ ∞) the fluctuation frequencies collapse into probabilities according to the law of large numbers In physics this replacement goes by the name of the mean field approximation, in which case the transition rates in the master equation (1) and (2) are written as g12(t) = g0 exp [−K {p1(t) − p2(t)}]. For the time being we retain the ATA coupling within the networks and consider the number of elements N to be finite In this latter case we can no longer make the mean field approximation and the dynamic picture stemming from the above master equation is radically changed. If the number of elements is still very large, but finite, we consider the mean-field approximation to be nearly valid and replace the average Eq(14) with the stochastic quantity ξ(t) = Π(t) + f (t). The complete properties of the DMM on an ATA network are explored by Turalska et al [57, 58]
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