I was reading the paper on [arixv](https://arxiv.org/abs/1312.5259v1). 


I was confused the equation of nth moment of Poisson distribution.


The detail and partial paper as follow:


...



> 
> For large N, this connection probability takes the form [28]
>  $$\Pi\_T (a,a^\prime) = 1 − exp[−λ(a + a^\prime)]$$, (1)
>  where $\lambda = T/N$ and we have set $m = 1$ to simplify calculations.
>  The topological properties of the integrated network are
>  encoded in the propagator $g\_T (k|a)$, defined as the probability
>  that a node with activity $a$ has integrated degree $k$ at time $T$ , and
>  whose generating function $\hat g\_T (z|a) = \sum\_k g\_T (k|a)z^k$ satisfies
>  the general equation [27,28]
>  $$ln \hat g\_T (z|a) = N\sum\_{a^\prime} F(a^\prime) ln[1 − (1 − z)\Pi\_T(a,a^\prime)]$$. (2)
>  From the propagator, the degree distribution of the integrated
>  network at time $T$ is trivially given by
>  $$P\_T(k) =\sum\_a F(a)g\_T (k|a)$$. (3)
>  In the limit of small $\lambda$ or $N\gg T$ with constant $T$, which we
>  assume from now on, the connection probability Eq. (1) can
>  be approximated as
>  $$\Pi\_T(a,a^\prime)\simeq \lambda(a + a^\prime)$$. (4)
>  In this same limit $\lambda\rightarrow 0$, Eq. (2) can be solved, leading to
>  a propagator with the form of a Poisson distribution with
>  mean $T (a + \langle a\rangle)$. From it, we obtain an asymptotic degree
>  distribution [28]
>  $$P\_T(k)\simeq \frac{1}{T}F(\frac{k}{T}-\langle a \rangle)$$. (5)
> 
> 
> 


...


Here says **Poisson distribution with mean $T (a + \langle a\rangle)$**.


...



> 
> The position of the percolation threshold can be simply
>  obtained by considering that $u = 1$ is always a solution
>  of Eq. (8), corresponding to the lack of giant component.
>  A physical solution $u < 1$, corresponding to a macroscopic
>  giant component, can only take place whenever $G\_1^\prime(1) > 1$,
>  which leads to the Molloy-Reed criterion [32]:
>  $$\frac{\langle k^2\rangle\_T}{\langle k\rangle\_T}> 2$$, (9)
>  where $\langle k^n\rangle\_T=\sum\_k k^n P\_T(k)$ is the $n$th moment of the degree
>  distribution at time $T$ .
>  In the case of the activity-driven model, these moments
>  can be computed noticing, from Eq. (3), that $\langle k^n\rangle\_T = \sum\_a F(a)\sum\_k k^n g\_T(k|a)$. Since the propagator has the form of
>  a Poisson distribution, the moments of the degree distribution
>  simply read as
>  $$\langle k^n\rangle\_T =\sum\_{m=1}^n\genfrac{\{}{\}}{0pt}{}{n}{m}T^m\kappa\_m$$, (10)
>  where $\genfrac{\{}{\}}{0pt}{}{n}{m}$ are the Stirling numbers of the second kind [33] and
>  $$\kappa\_m =\sum\_a F(a)(a+\langle a\rangle)^m=\sum\_{i=0}^m \binom mi \langle a^i\rangle \langle a\rangle^{m-i}$$. (11)
> 
> 
> 




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Well, I find that 


[Moments of the Poisson distribution](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Moments_of_the_Poisson_distribution)


If X is a random variable with a Poisson distribution with expected value $\lambda$, then its $n$th moment is


$$E(X^n)=\sum\_k\genfrac{\{}{\}}{0pt}{}{n}{k}\lambda^k$$




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So, Does $T\kappa$ is the expected value of Poisson distribution in the paper? 


Why $\kappa$ is not $(\sum\_a F(a)(a+\langle a\rangle))^m$ or $(a + \langle a\rangle)^m$? I am confused with $\kappa\_m$. 


Should I supply more information to my question? Did the title suite for my question? 



an application of nth moment of Poisson distribution with stirling number
It is the *conditional* distribution $g(k|a)$ that has a Poisson distribution, with mean $\lambda(T,a)=T(a+\langle a\rangle)$. So the *unconditional* average
$$\langle k^n\rangle = \sum\_a F(a)\langle k^n|a\rangle,$$
with 
$$\langle k^n|a\rangle\equiv\sum\_k k^n g(k|a)=\sum\_m\genfrac{\{}{\}}{0pt}{}{n}{m}\lambda^m$$ 
is given by
$$\langle k^n\rangle= \sum\_a F(a)\sum\_m\genfrac{\{}{\}}{0pt}{}{n}{m}\bigl[T(a+\langle a\rangle)\bigr]^m.$$
This is the equation from the paper, with 
$$\kappa\_m=\sum\_a F(a)(a+\langle a\rangle)^m=T^{-m}\sum\_a F(a)\lambda^m.$$


This is the relation between the expected value $\lambda$ of the Poisson distribution and $\kappa\_m$ requested in the OP.


