Estimating Tsallis q for degree distributions

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[edit] Hurwicz function

http://en.wikipedia.org/wiki/Hurwitz_zeta_function goes with new estimation methods

[edit] In general

Figure from Thurner and Tsallis 2005:200 click for reference
Figure from Thurner and Tsallis 2005:200 click for reference
e_q^x \equiv [1+(1-q)x]^{1/(1-q)} \ \mathrm{(equation\ 2} if (1 + (1 - q)x) > 0); otherwise e_q^x=0, where (e_1^x = e^x) and

e_{Qc}^x \equiv [1+(1-{Qc})x]^{1/(1-{Qc})} \ \mathrm{(equation\ 2} if (1 + (1 - Qc)x) > 0); otherwise e_{Qc}^x=0, where Qc is for a cumulative distribution.

While Nonextensive aspects of self-organized scale-free gas-like networks Stefan Thurner and Constantino Tsallis give:

for k \ge 2, x = e_{Qc}^{-(k-2)/\kappa}, where Qc is for a cumulative distribution.


And if for k \ge, x = e_{Qc}^{-(k-1)/\kappa}. then for K (cumulative degree distribution for k) we might use the estimator

The upper part of these distributions tends to follow the semi-q-log straight line characteristic of the q-exponential, with steeper slopes in the tails. 

P( \ge k) is defined as (K=cum freq for each k) / cum freq for all k. Then the expectation is that:

Z(k) = Ln_q P( \ge k) = (P( \ge k)^{(1-qc)}-1) / (1-qc) \approx -x/\kappa

Z(k) = ln_q P( \ge k)

with lnQc(k) adjusted for k \ge 1 or k \ge 2

Qc = 1 / (2 − q),q < 2

q = 2 − 1 / Qc

[edit] Method for Generative Feedback Model ("social circles") simulated networks

We begin with the model of WKTFW2006:3 (equation 4) for degree distributions with the probability density function:

p(k)=p_0\kappa^{\delta}e_q^{-k/\kappa}\ \mathrm{(equation\ 1)}

Then, since

e_q^x \equiv [1+(1-q)x]^{1/(1-q)} \ \mathrm{(equation\ 2)} if (1 + (1 - q)x) > 0); otherwise e_q^x=0, where (e_1^x = e^x),


p(x)=p_0\kappa^{\delta}e_q^{-x/\kappa} \equiv p_0\kappa^{\delta}[1-(1-q)x/\kappa]^{1/(1-q)} \ \mathrm{(equation\ 3)}

An alternative distribution is given in TKT2007(5-6, equations 13-14):

p(k) = \frac{e_q^{-\beta(k-1)}}{dk'e_q^{-\beta(k'-1)}} = \beta(2-q)e_q^{-\beta(k-1)}

Image:ZqLnqMethod.jpg

[edit] Cumulative CDF

Tsallis q distribution project Tambayong, Clauset, Shalizi, White

[edit] Links

http://en.wikipedia.org/wiki/Probability_distribution#Discrete_distributions

[edit] References

WKTFW2006 White, Douglas R., Nataša Kejžar, Constantino Tsallis, Doyne Farmer, and Scott White. Physical Review E, 016119 11pp. http://arxiv.org/abs/cond-mat/0508028 http://tinyurl.com/ylpbn3 http://en.wikipedia.org/wiki/Social-circles_network_model

TKT2007 Unified model for network dynamics exhibiting nonextensive statistics Stefan Thurner, Fragiskos Kyriakopoulos, Constantino Tsallis published as Phys. Rev. E 76, 036111 (2007) (8 pages)

STMS2005 Preferential attachment growth model and nonextensive statistical mechanics Europhys. Lett., 70 (1), p. 70 (2005) D. J. B. Soares, C. Tsallis, A. M. Mariz, L. R. da Silva.

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