User:Douglas R. White/MLE for q-exponential degree distributions

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These notes deal with statistical issues arising from the generative feedback network or "Social-circles" network model. Use Editing Math Formulas.

For Mark Handcock

From Cosma Shalizi (back to Degree distributions):

The pmf is proportional to the density for the continuous case, so

p(x) = C (1+x/\sigma)^{-\theta-1} 

(using {\theta-1} instead of {\theta} just so that it looks like the pdf of the continuous case).

  • Now, assume that the range of x goes from k to infinity. To find the proportionality constant C, which depends on the parameters \sigma and \theta, use the fact that probabilities must sum to one:
1 = C \sum_{x=k}^{\infty}{(1+x/\sigma)^{-\theta-1}}
  = C \sum_{x=k}^{\infty}{((\sigma+x)/\sigma)}^{-\theta-1}    
  = C\sum_{x=k}^{\infty}{(\sigma/(\sigma+x))}^{\theta+1} 
  = C\sigma^{\theta+1}\sum_{x=k}^{\infty}{1/(\sigma+x)^{\theta+1}} 
  = C\sigma^{\theta+1}\sum_{y=1}^{\infty}{1/(\sigma+y+k-1)^{\theta+1}}
  = C\sigma^{\theta+1}\zeta(\theta+1,\sigma+k-1) 

where \zeta is the Hurwicz zeta function (which generalizes the Riemann zeta function). So

       p(x) = (\sigma+x)^{-\theta-1} / \zeta(\theta+1,\sigma+k-1)

Setting up the estimating equations is possible here but it involves taking derivatives of the zeta function, which are not noticeably well-behaved, and my experiments suggest that direct numerical optimization is actually more stable.

For the Tsallis network function

We call the "Tsallis network function" the equation that fits the output distributions from the generative feedback network model, equation (4):

p(k) = p_0k^\delta e_q^{-k/\kappa}

where the Tsallis q-exponential function (a different generalization than the wikipedia q-exponential) e^{x}_q with parameter q is defined as

e^{x}_q = (1+(1-q)x)^{1/(1-q)}   	(e_1^x = e^x)

The function for p(k) arises naturally as the solution of the equation d\kappa/dt=a{\kappa}^q, which is {\kappa}=e^{x}_q = [1+(1-q) a t]^{1/(1-q)}. For the case where q=1, {\kappa}=e^{at}.

My questions are: What is the pmf? Can an MLE be programmed in dnet?

Cosma writes: 13:53, 11 July 2007 (UTC): The problem is that the normalizing factor doesn't seem to be easily expressible in terms of any of the usual special functions. Now, in principle "the Hurwicz zeta function" is just an abbreviation for the infinite series which I got for the discrete Pareto II, and we'd get a similar series and could define it to be a special function in its own right. The problem is that if you try to directly evaluate the Hurwicz zeta function by summing the series up to some maximum term, you find that this series converges very slowly. People have worked out rapidly-converging approximations for the zeta function, because of its mathematical importance, but I haven't been able to massage the normalizing factor for this other distribution into anything in terms of the usual special functions, and I don't know enough numerical analysis to find a rapidly-converging expression for it --- but I do know enough to see that it converges slowly by direct summation. Hence, a problem.