What does 'critical' mean?
Experimental observations
via stochastic sampling
versus intrinsic system properties.

Claudius Gros, Dimitrije Markovic

Institut für theoretische Physik
Goethe-Universität Frankfurt a.M.


generating powerlaws: established (?) routes

attractors in critical systems

infinite time-scale separation in sandpiles

× [C. Gros, Complex and Adaptive Dynamical Systems, Springer 2008/10/13/15]

$\ \color{darkRed}\Longrightarrow\ $ mapping to critical branching $(p=1/2)$ in infinite dimesion
$\ \color{darkRed}\Longrightarrow\ $ powerlaws


rich gets richer

Internet 2011: in-degree distribution
large-scale crawl, 32 Million most important sites, 2011
× [Gros, Kaczor, Markovic, EPJB 2012,
Neuropsychological constraints to human data production on a global scale]

- data availabe for further analysis -

generating powerlaws via
preferential attachment

[Barabasi, Albert, ...]
$$ p_k\ \sim\ \frac{1}{k^\gamma},\qquad \gamma = 1+\frac{1}{1-r/2} $$ $\ \color{darkRed}\Longrightarrow\ $ mostly internal growth

continuously active networks

[Markovic, Gros, Powerlaws and Self-Organized Criticality in Theory and Nature, Physics Reports 2014]

[Gazzaley & Rosedale]


absorbing phase transition:

external drivings couple to order parameter


external fields coupling to order parameter

destroy the phase transition

two viewpoints

humans maximizing Shannon information

maximal entropy distributions

with constraint $\displaystyle\quad \langle x\rangle=\mathrm{const.} \qquad\quad\color{darkBlue}{p(x)\ \propto\ \mathrm{e}^{-\lambda x} }$

Weber-Fechner law

$$\scriptstyle \begin{array}{llcll} \color{darkGreen}{\mathrm{music:}} & \mathrm{tone\ pitch} &\propto& \log(\mathrm{frequency}) & \color{darkGreen}{\mathrm{(octave)}} \\ \color{darkGreen}{\mathrm{photometry:}} & \mathrm{brightness} &\propto& \log(\mathrm{intensity}) & \color{darkGreen}{\mathrm{(lumen)}} \\ \color{darkGreen}{\mathrm{acoustics:}} & \mathrm{sound\ level} &\propto& \log(\mathrm{intensity}) & \color{darkGreen}{\mathrm{(decibel)}} \\ \end{array} $$
$$ \color{darkBlue}{x\ \to\ \log(x)} \qquad \color{darkRed}{\Rightarrow} \qquad \color{darkBlue}{p(x)\ \to\ \mathrm{e}^{-\lambda \log(x)} \ \propto\ \frac{1}{x^\gamma} } $$ $\ \color{darkRed}\Longrightarrow\ $ scale invariant distributions

publicly available data files in the internet

[Gros, Kaczor, Markovic, EPJB 2012]

distribution of 2-dimensional data files

[Gros, Kaczor, Markovic, EPJB 2012]

$\ \color{darkGreen}\Longrightarrow\ $ log-normal

maximal entropy distributions for 2D data

$$\color{darkBlue}{ p(x,y)\ \ \propto\ \ \mathrm{e}^{-\,\lambda_1\, xy\, -\,\lambda_2\, x^2\,-\,\lambda_3\, y^2} \ \ =\ \ \mathrm{e}^{-\,\lambda[\alpha\color{darkGreen}{(x+y)}\,+\, \beta\color{darkGreen}{(x-y)}]^2} }$$ logarithmic discounting $$\quad \quad \color{darkGreen}{ x+y \ \ \to\ \ \log(xy) \qquad\qquad x-y \ \ \to\ \ \log(x/y) }$$
× [Gros, Kaczor, Markovic, EPJB 2012]
$$\color{darkBlue}{ p(x,y)\ \ \propto\ \ \mathrm{e}^{-\,\lambda''\, \log^2(xy)\, -\,\lambda'\, \log(xy)} }$$

rank count vs. count probability

[Zipf '49, Cancho & Sole `03]$\qquad\quad$

40.000 most frequent (english) words

$$\color{orange}{\left(\frac{1}{R}\right)^\gamma}$$ $$\color{red}{\left(\frac{1}{C}\right)^{1+1/\gamma}}$$

wrap up - generating powerlaws

some stablished routes

... next: observing powerlaws

NK / Kauffman networks


[Luque & Sole, ‘00]

$$ \begin{array}{rcl} N & : & \mathrm{number\ of\ sites} \\ K & : & \mathrm{number\ of\ controlling\ elements} \\ \Sigma &= & (\sigma_1,\dots,\sigma_N), \qquad\quad \sigma_i=\pm1 \end{array} $$

N=7 K=3

state perturbed state

converge, if within the same basin of attraction

number of (cyclic) attrators

$$ \begin{array}{rcl} \color{darkOrange}{\mathrm{Kauffman\ '69:}} &\sim& \sqrt{N} \\ &\Rightarrow& \color{darkGreen}{\mathrm{cell\ differentiation}}\\[1.0ex] \color{darkOrange}{\mathrm{Samuelsson\ \&\ Troein\ '03:}} &>& O(N^p) \ \ \mbox{(any p)} \end{array} $$

are these (additional) small attrators relevant?

vertex routing models

transmission vs. routing

transmission routing
vertex $\ \color{darkOrange}{\Rightarrow}\ $ vertex link (incomming) $\ \color{darkOrange}{\Rightarrow}\ $ link (outgoing)
dynamical variables: vertices dynamical variables: links
[Markovic & Gros, New Journal of Physics `09]

routing dynamics

$\quad$ conserved? $\quad$
$\quad$ $\quad$
yes $\quad$ $\quad$ no
$\quad$ $\quad$
$\quad$ memory? $\quad$

cyclic attractors

random walks through configuration space

number of cycles of length $L$

no memory with memory
$$ \frac{N!}{L(N-1)^{L}(N-L)!} $$ $\qquad$ $$ \frac{N((N-1)^{2})!}{L(N-1)^{2L-1}((N-1)^{2}+1-L)!} $$

$$ \begin{array}{rcl} \color{red}{\langle n\rangle} &:& \mathrm{number\ of\ cycles}\\ \color{black}{\langle T\rangle} &:& \mathrm{cumulative\ cycle\ length} \\[1ex] \color{black}{\langle T\rangle}/\color{red}{\langle n\rangle} &:& \mathrm{mean\ cycle\ length} \\ &\sim& N/\log(N) \end{array} $$
× $$ \begin{array}{rclcl} \color{red}{\langle n\rangle} &:& \mathrm{number\ of\ cycles} &\sim& \color{darkOrange}{\log(\Omega)=2\log(N)} \\ \color{black}{\langle T\rangle} &:& \mathrm{cumulative\ cycle\ length} &\sim& \color{darkOrange}{\sqrt{\Omega}=N} \end{array} $$ [Markovic, Schuelein & Gros, Chaos `13]

stochastig sampling of phase space

experimental observation / biologically active networks

$$ \begin{array}{rcl} \mathrm{start\ randomly} &\color{darkRed}{\Rightarrow} & \mathrm{find\ next\ attractor}\\ &\color{darkRed}{\Rightarrow}& \mathrm{evaluate\ attractor\ properties} \end{array} $$

with memory number of cycles
mean cycle length

no memory number of cycles
mean cycle length

[Markovic, Schuelein & Gros, Chaos `13]

cycle length distribution

large and small basins of attractions

a Gedanken-distribution

criticality in statistical mechanics

second order phase transitions

what are observables?

$\quad \left\langle \sum_{states} \hat A(states)\,\mathrm{e}^{-\beta H(states)} \right\rangle $

thermodynamic averaging $\ \color{darkRed}{\hat=} \ $ stochastic sampling

$\hspace{5ex}\color{darkRed}\Longrightarrow\ $ `intrinsic properties' do not matter

classical thermalization and sampling

[Rigol, Dunjko & Olshanii, Nature `08]

thermal states are 'typical' states

only regions in phase space contribute to the thermal state, which have a finite probability to be visited (closely) by chaotic trajectories


regions in phase space which are not sampled by the internal dynamics do not contribute to the thermal state

caveat for dynamical systems: $\ \ $ thermalization at zero temperature?
$ \ \ $ energy not defined?

a given dynamical system may have interesting internal properties,
but in the end what matters is only what can be observed

textbook: complex systems