106 research outputs found
Symmetry in Critical Random Boolean Network Dynamics
Using Boolean networks as prototypical examples, the role of symmetry in the
dynamics of heterogeneous complex systems is explored. We show that symmetry of
the dynamics, especially in critical states, is a controlling feature that can
be used both to greatly simplify analysis and to characterize different types
of dynamics. Symmetry in Boolean networks is found by determining the frequency
at which the various Boolean output functions occur. There are classes of
functions that consist of Boolean functions that behave similarly. These
classes are orbits of the controlling symmetry group. We find that the symmetry
that controls the critical random Boolean networks is expressed through the
frequency by which output functions are utilized by nodes that remain active on
dynamical attractors. This symmetry preserves canalization, a form of network
robustness. We compare it to a different symmetry known to control the dynamics
of an evolutionary process that allows Boolean networks to organize into a
critical state. Our results demonstrate the usefulness and power of using the
symmetry of the behavior of the nodes to characterize complex network dynamics,
and introduce a novel approach to the analysis of heterogeneous complex
systems
Martingales, the efficient market hypothesis, and spurious stylized facts
The condition for stationary increments, not scaling,
detemines long time pair autocorrelations. An incorrect
assumption of stationary increments generates spurious
stylized facts, fat tails and a Hurst exponent Hs=1/2, when
the increments are nonstationary, as they are in FX markets.
The nonstationarity arises from systematic uneveness in
noise traders’ behavior. Spurious results arise
mathematically from using a log increment with a ‘sliding
window’. We explain why a hard to beat market demands
martingale dynamics , and martingales with nonlinear
variance generate nonstationary increments. The
nonstationarity is exhibited directly for Euro/Dollar FX
data. We observe that the Hurst exponent Hs generated by
the using the sliding window technique on a time series
plays the same role as does Mandelbrot’s Joseph exponent.
Finally, Mandelbrot originally assumed that the ‘badly
behaved second moment of cotton returns is due to fat tails,
but that nonconvergent behavior is instead direct evidence
for nonstationary increments. Summarizing, the evidence for
scaling and fat tails as the basis for econophysics and
financial economics is provided neither by FX markets nor
by cotton price data
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