By observing and analyzing the economic Network E.U. we can state:

1) There are two strong Hubs ( Germany and France) in the European Union Network caused by the preferential economic connection adding new Members.

2) While the Growth of the national Grand Product of the Hubs appears to be continuous, the other members are characterized by a significant high displacement in their NGP Development.

3) The Welfare Differences between the Members are in the E.U. relatively high. The corresponding Distribution is not of Gauss Character.

4) Economic Statistics do not indicate still an economic Cohesion Trend within the EU.

In this case, the walker performs jumps only on nearest neighbor states. Rigidity and neighbor hopping provide a geometrical constraint factor $G\left(x,y;t\right)$, which is time-independent and homogenous in chemical space in the form

$G\left(x,y;t\right]=g\left(x-y\right)$ with $g\left(x\right)=\frac{1}{2}\ast \left(\delta \left(x-a\right)+\delta \left(x+a\right)\right)$ (16)

On scales larger than the intermonomer space $\alpha$ and times larger than the typical waiting time $\tau$, the process is asymptotically governed by the diffusion limit $\tau ,\alpha \to 0$ with $D={\alpha }^2/2\tau$ = Diffusion Coefficient of the system. Inserting (16) into (15), after algebraic manipulations, and assuming $D=1$, we get the following formula for the master equation:

${\partial }_{t}p={\text{e}}^{-\frac{\beta V}{2}}\Delta {\text{e}}^{\frac{\beta V}{2}}*p-p{\text{e}}^{\frac{\beta V}{2}}\Delta {\text{e}}^{-\frac{\beta V}{2}}={ℒ}_p$ (17)

with Δ = Laplacian, $V=V\left(x\right)-V\left(y\right)$, and $p=p\left(x,t\right)$.

By expanding the Laplacians in (17) one gets the final Form

${\text{e}}^{-\frac{\beta V}{2}}\Delta {\text{e}}^{\frac{\beta V}{2}}\ast p-p{\text{e}}^{\frac{\beta V}{2}}\Delta {\text{e}}^{-\frac{\beta V}{2}}\propto -\left(\nabla F+\Delta \right)p$ with $F\left(x\right)=-\beta \ast \frac{\text{d}V\left(x\right)}{\text{d}x}$ (18)

The above Operator is an ordinary Fokker-Planck Operator, observing deterministic gradient dynamics in the potential V subjected to Gaussian white noise. Thus, on scales larger than the inter-monomer spacing, the process can be modeled by the associated Langevin Ansatz

$\text{d}X\left(t\right)=\beta \ast \frac{\text{d}V\left(x\right)}{\text{d}x}\text{d}t+\text{d}W\left(t\right)$ (19)

In which $\text{d}W\left(t\right)$ is the differential of the Wiener process. In the Limit, the Jump process becomes a diffusion process with continuous sample paths.

Observing the most interesting and relevant cases in our study of rapidly folding polymers it is meaningful to interpret the master equation by averaging the disorder as the geometrical constraint factor $G\left(x,y;t\right)$ isa complex stochastic process itself. We study the expression ${\partial }_t\Vert p\Vert =\Vert {ℒ}_m\left(t\right)p\Vert$, where $\Vert .\Vert$ denotes the disorder average. The disorder average $\Vert G\left(x,y;t\right)\Vert$ is just the probability, of finding monomers x and y at a distance from each other less than “ $\alpha$” in Euclidean space. That means $\Vert G\left(x,y;t\right)\Vert =P\left(\left|R\left(x,t\right)-R\left(y,t\right)\right|\right)\le \alpha$. If the conformational dynamics are equilibrated the above problem is time-independent. This leads to the assumption that the probability is a function “ $\rho$”, which depends on the chemical distance $\left|x-y\right|$. I.e. $P\left(\left|R\left(x,t\right)-R\left(y,t\right)\right|\right)\le \alpha \propto \rho \left(\left|x-y\right|\right)$. Thus the mean-field approximation yields a temporally homogenous master equation with time-independent rates

$w\left(x|y\right)=\frac{1}{\rho }{\text{e}}^{-\beta \left[V\left(x\right)-V\left(y\right)\right]/2}\ast \rho \left(\left|x-y\right|\right)$ (20)

Detailed Balance is on the above function fulfilled, and can be assumed that it decreases with increased distance.

Coiled Polymers: The above-described folding polymer can be characterized as a coiled polymer, which performs changes by a typical scale $\sigma$. This length scale $\sigma$ represents a natural cutoff, above which long-range transitions are highly unlikely to occur. Proceeding like in the case of the rigid polymer the dynamics in the

diffusion limit $\sigma ,\rho \to 0$ with $D_c=\frac{{\sigma }^2}{2\tau }$ = Diffusion Coefficient of the system,

with the help of Kramers-Moyal expansion, the governed formula reads as

${\partial }_{t}p\left(x,t\right)=D_c\left(-2\frac{\text{d}}{\text{d}x}{\text{e}}^{\frac{\beta V\left(X\right)}{2}}\left[\frac{\text{d}}{\text{d}x}{\text{e}}^{-\frac{\beta V\left(x\right)}{2}}\right]+\frac{{\text{d}}^2}{\text{d}{x}^2}\right)\propto D_c\left(-\frac{\text{d}}{\text{d}x}F\left(x\right)+\frac{{\text{d}}^2}{\text{d}{x}^2}\right)P\left(x,t\right)$ (21)

The disordered averaged master equation leads like in the case of rigid polymers to a Fokker-Planck Ansatz with limited long-range transitions (cutoff). In general,

it holds the relation $\frac{D_c}{D}=\frac{{\sigma }^2}{{\alpha }^2}$. This means that the diffusive character of the dispersion is not changed when long-range transitions of finite variance come into play.

The dynamics change significantly when scale-free transitions occur between monomers. The function $\rho \left(l\right)$ is now in general terms an inverse power law as $\rho \left(l\right)\propto {\left|l\right|}^{-\left(1+\mu \right)}$ for $\left|l\right|\gg \alpha$. The exponent $\mu$ depends on the polymer configuration, and is typically less unity. In this case holds the following relation

$\rho \left(l\right)=\frac{\mu {\alpha }^{\mu }}{2}\ast {\left|l\right|}^{-\left(1+\mu \right)}$ for $\left|l\right|>\alpha ,0<\mu \le 2$; otherwise $\rho \left(l\right)=0$ (22)

If the exponent $\mu$ lies in the area $\left(0,2\right]$ the density $\rho \left(l\right)$ lacks a finite variance. In the case of $\mu \le 1$ especially the expected value diverges. The statistics of jump lengths along a random polymer became scale-free. Inserting the relation (22) into the master equation and performing diverse algebraic manipulations we

get in the limit $\tau ,\alpha \to 0$, with $D=\frac{\pi \mu }{2\Gamma \left(1+\mu \right)\sin \left(\frac{\pi \mu }{2}\right)}\left(\frac{{\alpha }^{\mu }}{\tau }\right)$ the fractional Fokker-Planck Equation for superdiffusion

${\partial }_{t}P={ℒ}_{M}P={\text{e}}^{-\beta V/2}{\nabla }^{\mu /2}P-P{\text{e}}^{\beta V/2}{\Delta }^{\mu /2}{\text{e}}^{-\beta V/2}$ (23)

with $\left({\Delta }^{\frac{\mu }{2}}f\right)\left(x\right)$ = fractional Laplacian = $C_{\mu }\ast {\displaystyle \int \text{d}y}\frac{f\left(y\right)-f\left(x\right)}{{\left|x-y\right|}^{1+\mu }}$, and $C_{\mu }=\Gamma \left(1+\mu \right)\sin \left(\frac{\pi \mu }{2}\right)$. For $\mu =2$, ${ℒ}_{M}P$ gets out in an ordinary Fokker-Planck

operator. In the case of $\beta V=0$, which takes place if $V=0$ or $\beta \to$ ultra high temperature all the exponentials in (23) are unity and the fractional Laplacian degenerates to the formula ${ℒ}_{\mu }={\Delta }^{\mu /2}$ (free superdiffusion). In this case, the dynamics coincides with the FFPE describing additive Levy stable Noise or associated with highly fluctuating temporal behavior in the next chapter. The following Graphic delivers an overview of fractal Dynamics in Space and Time.

Overview of Fractal Dynamics in Space, and Time

The Existence of Steady States can be derived setting

${\partial }_{t}P={ℒ}_{M}P={\text{e}}^{-\frac{\beta V}{2}}{\nabla }^{\frac{\mu }{2}}P-P{\text{e}}^{\frac{\beta V}{2}}{\Delta }^{\frac{\mu }{2}}{\text{e}}^{-\frac{\beta V}{2}}=0$. (24)

The solution of the above Equation results in $P_{st}\left(x\right)={\text{e}}^{-\beta V\left(x\right)}$, and does not depend on the Fractality-Order Factor $\mu$. This is not a surprising factum as the evolution of the master equation takes place under Gibbs-Boltzmann thermal conditions. In this case holds the normalization Condition ${\displaystyle \int P_{st}\left(x\right)\text{d}x}=1$ on the Maxwell Spectrum. In the case of rigid chains or transitions that are limited in

scale length $\alpha$, the government reads as ${\text{e}}^{-\frac{\beta V}{2}}{\nabla }^{\frac{1}{2}}P-P{\text{e}}^{\frac{\beta V}{2}}{\Delta }^{\frac{1}{2}}{\text{e}}^{-\frac{\beta V}{2}}={\partial }_{t}P$, which

results like above in

$P_{st}\left(x\right)={\text{e}}^{-\beta V\left(x\right)}$ (25)

Interestingly the operator ${ℒ}_{1/2}P$ can be interpreted as an ordinary Fokker-Planck operator operating in gradient dynamics subjected to Gaussian white noise. It must be denoted that the master equation describing topologically induced Diffusion does have stationary states according to Gibbs-Boltzmann Equilibrium thermodynamics.

This economic Szenario (Constellation) corresponds to a global economic constellation in which diverse economic clusters exist. The inter-cluster economic connections are relatively rare, while each cluster exhibits its self-scale-free behavior. The corresponding steady states are of Gibbs-Boltzmann Character of the Forrm $P_{st}\left(x\right)={\text{e}}^{-\beta V\left(x\right)}$ like in the scale-free case.

Remaining in the superdiffusive area dynamics can occur, while the time-fractional Order becomes more and more lower values (scale-free Memory). The process is superdiffusive in space and time. The operational time $\tau \left(t\right)$ is a stochastic process itself. This is a specific subordination process in the superdiffusive area. The corresponding conditional pdf ${\partial }_{t}P\left(x,t|x_0,t_0\right)$ can be represented by fractional Fokker-Planck Equation for symmetric Levy stable processes. It holds ${\partial }_{t}P\left(x,t|x_0,t_0\right)={\Delta }_x^{\mu /2}P\left(x,t|x_0,t_0\right)$. For non-vanishing Force the correct Equation reads: ${\partial }_{t}P\left(x,t|x_0,t_0\right)=-{\left(-L_{x,FP}\right)}^{\mu /2}P\left(x,t|x_0,t_0\right)$ with

$L_{x,FP}={\nabla }_{x}F\left(x,t\right)+\frac{1}{2}\ast {\Delta }_{x}D\left(x,t\right)$. As the Operator $L_{x,FP}$, and the term

$-{\left(-L_{x,FP}\right)}^{\mu /2}$ commute with each other we can state that superdiffusive subordinated processes evolve under Gibbs-Boltzmann thermal Conditions. Thus if $D\left(x,t\right)=1$, and $F\left(x,t\right)=-\beta \text{d}V\left(x\right)/\text{d}x$ the stationary solution reads $P_{st}={\text{e}}^{-\beta V}$.

Summary: Assuming dynamics that can be analyzed by the topologically induced Diffusion, we can state the existence of Gibbs-Boltzmann Stationary Solutions of the form $P_{st}={\text{e}}^{-\beta V}$. This result can be derived by simulating the Dynamics by the Master Equation in the Superdiffusiv Area. It must be noted that the topologically induced Diffusion exhibits Levy Flights with stationary Solutions as opposed to the generalized Langevin ansatz where the Levy Flight dynamics fail to generate Stationarity.

The transition rate $w\left(x|y\right)=\frac{1}{\rho }{\text{e}}^{-\beta \left[V\left(x\right)-V\left(y\right)\right]/2}\ast \rho \left(\left|x-y\right|\right)$, which has been used in the master equation ansatz to get concrete solutions, can be generalized, as the Potentials contribute in the above equation only by their Difference. The Term $\Delta V=\frac{1}{2}\left[V\left(x\right)-V\left(y\right)\right]$ can be generalized assuming that a contribution of the

potentials by their offset is more appropriate. Inserting in the above relation $\Delta V=cV\left(x\right)-\left(1-c\right)V\left(y\right)$ with $c\in \left[0,1\right]$, we obtain a more general rate in the form

$w\left(x|y\right)=\frac{1}{\rho }{\text{e}}^{-\beta cV\left(x\right)}\rho \left(\left|x-y\right|\right)\ast {\text{e}}^{\beta \left(1-c\right)V\left(y\right)}$ (26)

In the case of economic preferential attachment where Hubs occur by adding new members to the network, we can assume that the entire potential increases by

a constant $V_c$. That means $c\ne \frac{1}{2}$ and $c<0.5$. in a physical sense, the influence

of the source location increases, while an increase in the rate takes place. This is however a general problem in networks with hubs, while the latter have a catalytic and accelerated influence on the further dynamics of the network. In this context ( $c<0.5$) if the entire potential increases the algebraic structure of the rate remains the same but the waiting time constant $\tau$ changes effectively. That means

$V\to V+V_C\Rightarrow \tau =\frac{\tau }{{\text{e}}^{\beta V_c\left(2c-1\right)}}$ (27)

Equation (27) indicates that high temperatures and abrupt changes of $V_c$ could transfer the dynamics from the superdiffusive area to that of ambivalent processes. It should be emphasized here that ambivalent processes exhibit different Steady State solutions. However, we do not handle this topic notwithstanding that could represent dynamics that are probable within the general Globalization evolution. It is important to say that we observe specific economic Constellations in the global network, which can be effectively approximated by the topologically induced Diffusion.

The coiled Polymer Dynamics could be characterized as the most probable real scenario affecting the evolution of the general Globalization process. This assumption relies on the actual economic relations and trends worldwide. It consists of diverse economic clusters with intercluster connections characterized by a typical scale (cutoff), while the dynamics in every single cluster are scale-free dependent naturally on the specific conditions and agreements needed to establish these as discrete (i.e. European Union). In the case of ordinary Diffusion in the rate

$w\left(x|y\right)=\frac{1}{\rho }{\text{e}}^{-\beta cV\left(x\right)}\rho \left(\left|x-y\right|\right)\ast {\text{e}}^{\beta \left(1-c\right)V\left(y\right)}$, the factor $\rho \left(\left|x-y\right|\right)$ must be replaced

by $\delta \left(x-y\right){\Delta }_y$. The master Equation consequently results in

${\partial }_{t}p={\text{e}}^{-\beta cV}\Delta e^{1-c}{\text{e}}^{\beta \left(1-c\right)V}p-p{\text{e}}^{\beta \left(1-c\right)V}\Delta {\text{e}}^{-c\beta V}$ (28)

Important Note: The generalization of the probability rate $w\left(x|y\right)$ does have only in the case of $c=1/2$ a quantum-mechanical Equivalent and can not be recast in the Fokker-Planck Ansatz

${\partial }_{t}p=\left\{-\nabla F+0.5\Delta D\right\}$ (29)

like In the case of $\rho \left(l\right)=\frac{\mu \alpha }{2}\ast {\left|l\right|}^{-\left(1+\mu \right)}$, which induces scale-free long-range transitions. Concerning Coiled Polymer Dynamics and varying the Value of $c\in \left[0,1\right]$

we could use Equation (29) to gain results that are most probably in real Dynamics. The more the value c tends to 1 the more the Influence of the target location is higher. In the opposite case a low value of c in the area $0\le c\le 0.5$ indicates a higher influence of the source location ( Brockmann, 2003 ). Drift and diffusion coefficients can be represented by the potential $V\left(x\right)$ as

$F\left(x\right)=-c\beta \frac{\text{d}V}{\text{d}x}\ast D\left(x\right)$, $D\left(x\right)=2{\text{e}}^{-\beta \left(2c-1\right)V\left(x\right)}$ (30)

Setting $c=0.5$ we obtain the usual diffusion equation in a gradient force field. In the case of $c=0$, which corresponds to a situation where the transitional rates are determined by the potential at the source location, we obtain an equation without drift term. The environment inhomogeneity is incorporated in the diffusion coefficient. We get the following Identities assuming $c=0$: $F=0$,

$D=2{\text{e}}^{\beta V}$, ${\partial }_{t}p=\frac{1}{2}\Delta Dp$ (31)

From the Relation in (31) we can assert that the dynamics in this case are caused only by the Laplacian of the Diffusion coefficient. In the extremal case of $c=1$ (Target location determines the transition rate) we gain the identity ${\partial }_{t}p={\text{e}}^{-\beta V}p-p\Delta {\text{e}}^{-\beta V}$. In this case, only the Laplacian of the Term ${\text{e}}^{-\beta V}$ occurs in the master equation, while the probability density is respected with the prefactor ${\text{e}}^{-\beta V}$.