Scaling expansions:

universal tool for classification of complex systems

Jan Korbel, Rudolf Hanel and Stefan Thurner

Complex systems cover an enormous variety of systems (physics, chemistry, biology, sociology, economy...)
- Different systems have similar statistics
- No united classification
However, statistical complex systems near the thermodynamic limit ($N \rightarrow \infty $) can be characterized by asymptotics of its sample space $W(N)$
Asymptotic behavior can be described by scaling expansion with coefficients corresponding to scaling exponents
Scaling exponents completely determine universality classes

Asymptotic behavior of $W(N)$ for $N \rightarrow \infty$ can be described by the Poincaré asymptotic expansion

$$ f(W(N)) = \sum_{j=0}^n c_j \phi_{j}(N) + \mathcal{O}(\phi_n(N))$$

where $f(W(N)) = \mathcal{O}(\phi_0(N)) $ and $\phi_{j+1}(N) = \mathcal{O}(\phi_{j}(N)) $

Question: What set of functions $ \phi_j(N) $ describes scaling of $W(N)$?

Do the coefficients $ c_j $ have any connection to scaling?

Answer: Yes!

We can define an expansion, where the coefficients

$c_j$ are the scaling exponents of $W(N)$

Notation: $f^{(n)}(x) = \underbrace{f(f(\dots(f(x))\dots))}_{n \ \mathrm{times}} $

Let us choose $\phi_j(N) = \log^{(j+1)}(N)$, Then for some $l$ we can write $$W^{(l)}(N) \equiv \log^{(l+1)}(W(x)) = \sum_{j=0}^n c_j^{(l)} \log^{(j+1)}(N) + \mathcal{O} (\phi_n (N))$$

The coefficients $c_j^{(l)}$ are the scaling exponents of $W$!

We define the set of rescalings $r_\lambda^{(n)}(x) := \exp^{(n)}(\lambda \log^{(n)}(x) $ )

So $r_\lambda^{(0)}(x) = \lambda x$, $r_\lambda^{(1)}(x) = x^\lambda$, $r_\lambda^{(2)}(x) = e^{\log(x)^\lambda} $, ...

Properties: $r_\lambda^{(n)} (r_{\lambda'}^{(n)}(x)) = r_{\lambda \lambda'}^{(n)}(x) $, $r_1^{(n)}(x) = x$

Leading order scaling: $ \lim_{N \rightarrow \infty}\frac{W^{(l)}(r_\lambda^{(0)}(N))}{W^{(l)}(N)} = \lambda^{\bf c_0^{(l)}} $

First correction scaling: $ \lim_{N \rightarrow \infty} \frac{W^{(l)} (r_\lambda^{(1)}(N))}{W^{(l)}(N)} \left(\frac{r_\lambda^{(1)}(N)}{N}\right)^{-c_0^{(l)}} = \lambda^{\bf c_1^{(l)}}$

General rescaling: $ \lim_{N \rightarrow \infty} \frac{W^{(l)}(r_\lambda^{(k)}(N))}{W^{(l)}(N)} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)}(r_\lambda^{(k)}(N))}{\log^{(j)} N}\right)^{-c_j^{(l)}} = \lambda^{\bf c_k^{(l)}}$

Scaling expansion: $ W(N) \sim \exp^{(l)}\left(\prod_{j=0}^n (\log^{(j)}(N))^{c_j^{(l)}} \right)$

J.K., R.H., S.T. New J. Phys. 20 093007

Examples

Random walk

Two possibilities:

step to the left/step to the right

For $N$ steps we have

$W(N) = 2^N$

possible configurations (paths)

Scaling exponents:
- $l=1$
- $c_0^{(1)}=1$
- $c_k^{(1)}=0$ for $ k=1,2,\dots$

Exponential growth:

Aging random walk

R.H., S.T. EPL 96 50003, Entropy 15 5324.

Sub-exponential growth (stretched exponential):

RW with correlations
After 1 step,

2 steps in the same direction,

3 steps in the same direction....

Asymptotically we get

$W(N) \approx 2^{\sqrt{N}/2} \sim 2^{N^{1/2}}$

Scaling exponents:
- $l=1$
- $c_0^{(1)}=1/2$
- $c_k^{(1)}=0$ for $ k=1,2,\dots$

Riemann random walk

Sub-exponential growth:

another version of correlated RW
The walker can make decision only for steps which are prime numbers

Density of prime numbers:

$\pi(N) \sim N/\log N$

$W(N) = 2^{\pi(N)} \sim 2^{N/\log N}$
Scaling exponents:
- $l=1$
- $c_0^{(1)}=1$
- $c_1^{(1)}=-1$
- $c_k^{(1)}=0$ for $ k=2,3,\dots$

Magnetic coins

H. Jensen et al. J. Phys. A: Math. Theor. 51 375002

Super-exponential growth:

$N$ coins - head or tail
Coins are magnetic - any two can stick together (1 state)
$ W(N) \approx N^{N/2} e^{2 \sqrt{N}} \sim e^{N \log N}$
Scaling exponents:
- $l=1$
- $c_0^{(1)}=1$
- $c_1^{(1)}=1$
- $c_k^{(1)}=0$ for $ k=2,3,\dots$

Random networks

Super-exponential growth (compressed exponential):

Undirected network with $N$ nodes has $\binom{N}{2}$ possible links
Number of possible networks $ W(N) = 2^{\binom{N}{2}} \sim 2^{N^2}$
Scaling exponents:
- $l=1$
- $c_0^{(1)}=2$
- $c_k^{(1)}=0$ for $ k=2,3,\dots$

Random walk cascades

Super-exponential growth (double-exponential):

Generalization of RW
The walker can go to the left, to the right or split
After each split, there are two independent walkers
$W(N) = 2^{2^N}-1 \sim 2^{2^N}$
Scaling exponents:
- $l=2$
- $c_0^{(2)}=1$
- $c_k^{(2)}=0$ for $ k=1,2,\dots$

Applications