# Classification of complex systems by their sample-space scaling exponents

Jan Korbel, Rudolf Hanel and Stefan Thurner

Statistical Physics of Complex Systems, 7-11 May 2019, Nordita, Stockholm

## Classification of statistical complex systems

• Many examples of complex system are statistical (stochastic) systems
• Statistical complex systems near the thermodynamic limit $$N \rightarrow \infty$$            can be characterized by asymptotics of its sample space $$W(N)$$                       - space of all possible configurations
• Asymptotic behavior can be described by
• Coefficients of scaling expansion correspond to scaling exponents
• Scaling exponents completely determine universality classes
• We can find corresponding extensive entropy
• generalization of (c,d)-entropy $$^\star$$

$$^\star$$ R.H., S.T. EPL 93 (2011) 20006

## Rescaling the sample space

• How the sample space changes when we rescale its size $$N \mapsto \lambda N$$?
• The ratio behaves like $$\frac{W(\lambda N)}{W(N)} \sim \lambda^{c_0}$$ for $$N \rightarrow \infty$$
• the exponent $$c_0$$ can be extracted by $$\frac{d}{d\lambda}|_{\lambda=1}$$: $$c_0 = \lim_{N \rightarrow \infty} \frac{N W'(N)}{W(N)}$$
• For the leading term we have $$W(N) \sim N^{c_0}$$.
• Is it only possible scaling? We have $$\frac{W(\lambda N)}{W(N)} \frac{N^{c_0}}{(\lambda N)^{c_0}} \sim 1$$
• Let us use the other rescaling $$N \mapsto N^\lambda$$
• The we get that $$\frac{W(N^\lambda)}{W(N)} \frac{N^{c_0}}{N^{\lambda c_0}} \sim \lambda^{c_1}$$
• First correction is $$W(N) \sim N^{c_0} (\log N)^{c_1}$$
• It is the same scaling like for $$(c,d)$$-entropy
• ### ​Can we go further?

• We define the set of rescalings $$r_\lambda^{(n)}(x) := \exp^{(n)}(\lambda \log^{(n)}(x)$$ )
•  $$f^{(n)}(x) = \underbrace{f(f(\dots(f(x))\dots))}_{n \ times}$$
• ​ $$r_\lambda^{(0)}(x) = \lambda x$$,  $$r_\lambda^{(1)}(x) = x^\lambda$$,   $$r_\lambda^{(2)}(x) = e^{\log(x)^\lambda}$$, ...
• They form a group: $$r_\lambda^{(n)} \left(r_{\lambda'}^{(n)}\right) = r_{\lambda \lambda'}^{(n)}$$,    $$\left(r_\lambda^{(n)}\right)^{-1} = r_{1/\lambda}^{(n)}$$,   $$r_1^{(n)}(x) = x$$
• We repeat the procedure:  $$\frac{W(N^\lambda)}{W(N)} \frac{N^{c_0} (\log N)^{c_1} }{N^{\lambda c_0} (\log N^\lambda)^{c_1}} \sim 1$$,
• We take $$N \mapsto r_\lambda^{(2)}(N)$$
• $$\frac{W(r_\lambda^{(2)}(N))}{W(N)} \frac{N^{c_0} (\log N)^{c_1} }{r_\lambda^{(2)}(N)^{c_0} (\log r_\lambda^{(2)}(N))^{c_1}} \sim \lambda^{c_2}$$,
• Second correction is $$W(N) \sim N^{c_0} (\log N)^{c_1} (\log \log N)^{c_2}$$

## Rescaling the sample space III

• General correction is given by $$\frac{W(r_\lambda^{(k)}(N))}{W(N)} \prod_{j=0}^{k-1} \left(\frac{\log^{(j)} N}{\log^{(j)}(r_\lambda^{(k)}(N))}\right)^{c_j} \sim \lambda^{\bf c_k}$$
• Possible issue: what if $$c_0 = +\infty$$? Then $$W(N)$$ grows faster than any $$N^\alpha$$
• ​We replace $$W(N) \mapsto \log W(N)$$
• The leading order scaling is $$\frac{\log W(\lambda N)}{\log W(N)} \sim \lambda^{c_0}$$ for $$N \rightarrow \infty$$
• So we have $$W(N) \sim \exp(N^{c_0})$$
• If this is not enough, we replace $$W(N) \mapsto \log^{(l)} W(N)$$                                        so that we get finite $$c_0$$
• General expansion of $$W(N)$$ is $$W(N) \sim \exp^{(l)} \left(N^{c_0}(\log N)^{c_1} (\log \log N)^{c_2} \dots\right)$$

J.K., R.H., S.T. New J. Phys. 20 (2018) 093007

$$W^{(l)}(N) \equiv \log^{(l+1)}(W(x)) = \sum_{j=0}^n c_j^{(l)} \log^{(j+1)}(N) + \mathcal{O} (\phi_n (N))$$

## Scaling Expansion

• Previous formula can be expressed in terms of                                            Poincaré asymptotic expansion
• Coefficients of the expansion are scaling exponents                                          and can be calculated from:

$$c^{(l)}_k = \lim_{N \rightarrow \infty} \log^{(k)}(N) \left( \log^{(k-1)} \left(\dots\left( \log N \left(\frac{N W'(N)}{\prod_{i=0}^l \log^{(i)}(W(N))}-c^{(l)}_0\right)-c^{(l)}_1\right) \dots\right) - c^{(l)}_k\right)$$

## Extensive entropy

• ​We can do the same procedure with entropy $$S(W)$$
• Leading order scaling: $$\frac{S(\lambda W)}{S(W)} \sim \lambda^{d_0}$$
• First correction $$\frac{S(W^\lambda)}{S(W)} \frac{W^{d_0}}{W^{\lambda d_0}} \sim \lambda^{d_1}$$
• ​First two scalings correspond to $$(c,d)$$-entropy                                                                                 for $$c= 1-d_0$$ and $$d = d_1$$
• Scaling expansion of entropy $$S(W) \sim W^{d_0} (\log W)^{d_1} (\log \log W)^{d_2} \dots$$
• Requirement of extensivity  $$S(W(N)) \sim N$$ determines the relation between $$c$$ and $$d$$ :
• $$d_l = 1/c_0$$,              $$d_{l+k} = - c_k/c_0$$     for $$k = 1,2,\dots$$

# EXAMPLES

Process S(W)
Random walk

0

1

0

Aging random walk

0

2

0
Magnetic coins *

0

1

-1
Random network

0

1/2

0

0

0

1

$$\log W$$

$$(\log W)^2$$

$$(\log W)^{1/2}$$

$$\log \log W$$

$$d_0$$

$$d_1$$

$$d_2$$

$$\log W/\log \log W$$

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

$$W(N) = 2^N$$

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

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

$$W(N) = 2^{\binom{N}{2}} \sim 2^{N^2}$$

$$W(N) = 2^{2^N}-1 \sim 2^{2^N}$$

### How does it change for one more scaling exponent?

R.H., S.T. EPL 93 (2011) 20006

### Parameter space of $$(d_0,d_1,d_2)$$-entropy

To fulfill SK axiom 2 (maximality): $$d_l > 0$$, to fulfill SK axiom 3 (expandability): $$d_0 < 1$$

## Perspectives

Fields of possible applications of scaling expansions:

• Non-equilibrium thermodynamics
• Information geometry$$^\star$$
• Critical phase transitions
• Information theory (Shannon-Khinchin axioms$$^\dag$$)
• Statistical inference (Shore-Johnson axioms$$^\ddag$$)
• Super-exponential processes
• Processes with structures
• ...

$$^\star$$ J.K., R.H., S.T. Entropy 21(2) (2019) 112

$$^\dag$$ P. Tempesta, Proc. R. Soc. A 472 (2016) 2195

$$^\ddag$$ P.J., J.K. Phys. Rev. Lett. 122 (2019), 120601

## References

• J.K., R.H., S.T. Classification of complex systems by their sample-space scaling exponents, New J. Phys. 20 (2018) 093007
•  H. Jensen, R. H. Pazuki, G. Pruessner, P. Tempesta, Statistical mechanics of exploding phase spaces: Ontic open systems, J. Phys. A: Math. Theor. 51 375002
• R.H., S.T. A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and distribution functions,  EPL 93 (2011) 20006
• P. Tempesta, Formal groups and Z-entropies, Proc. R. Soc. A 472 (2016) 2195
• P.J., J.K. Maximum Entropy Principle in Statistical Inference: Case for Non-Shannonian Entropies,   Phys. Rev. Lett. 122 (2019), 120601
• J.K., R.H., S.T. Information Geometric Duality of ϕ-Deformed Exponential Families,         Entropy 21(2) (2019) 112

I am excited to discuss any possible application

of scaling expansions

during the welcome reception or later

By Jan Korbel

# Classification of complex system by their sample-space scaling exponents

Classification of complex systems by their sample-space scaling exponents

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