i)^K$ is the Stirling number of the second kind. 

Calculating the pgf for this distribution is tedious and we 
do not present analytical results regarding this model but only
simulations and estimation. As we shall see in
Section~\ref{ss.Numerical}, in this model 
 {\em the influenced population is smaller than the population of good pioneers.}

\subsection{Numerical examples}
\label{ss.Numerical}
We will present now a few numerical examples of networks and 
diffusion models presented above.

\subsubsection{Simulations}
In all our examples we simulate the enhanced configuration model on
$N=1000$ nodes
 assuming some particular 
node degree $D$ distribution and influence propagation mechanism
modeled by the conditional distribution of the transmitter degree
$D^{(t)}$. More precisely, we sample the individual node degrees and
transmitter degrees  $(D_i,D_i^{(t)})$ 
$i=1\,...\,N$ independently from  the joint distribution of 
 $(D,D^{(t)})$ 
and use these values to construct an instance of our 
enhanced CM by uniform pairwise matching of the half-edges.
We calculate the relative size of the  influenced population and the
set of good pioneers through the exploration of the
influenced components for all nodes.~\footnote{The simulations are run in
{\em python} using the {\em networkx} package.}
In fact, relative sizes of the
populations reached from different pioneers concentrate 
very clearly, as shown on Figure~\ref{f.Hist}, which illustrates the
statement of Claim~\ref{cl1}.  

\begin{figure}[h!]
\begin{center}
\centerline{\includegraphics[width=1\linewidth]{HistPoi2Bern08.pdf}}
\caption{Concentration of the relative sizes of 
populations reached from different pioneers. CM with Poisson degree of mean
$\lambda=2$ and Bernoulli transmissions with $p=0.8$.
 \label{f.Hist}}
\end{center}
\end{figure}


\subsubsection{Estimation}
\label{sss.Estimation}
We adopt also the following ``semi-analytic'' approach:
Using the sample $(D_i,D_i^{(t)})$, $i=1,\ldots,N$ used to
construct the CM, we consider estimators 
\begin{align}
\hat G_D(x)&:=\frac{1}{N}\sum_{i=1}^N x^{D_i}\label{e.hatD}\\
\hat G_D^{(t)}(x)&:=\frac{1}{N}\sum_{i=1}^N x^{D_i^{(t)}}\label{e.hatDt}\\
\hat H(x)&:=\frac{1}{N}\sum_{i=1}^N \Bigl( D_i x^2-(D_i-D_i^{(t)})x-
D_i^{(t)}x^{D_i}\Bigr)\label{e.hatH}\\
\hat{\overline{H}}(x)&:=\sum_{i=1}^N \Bigl(D_i x^2-D_i^{(t)}x^{D_i^{(t)}} - 
(D_i-D_i^{(t)})x^{D_i^{(t)}+1}\Bigr)\,\label{e.hatHbar}
\end{align}
of the functions $G_D(x)$,  $G_{D^{(t)}}(x)$, $H(x)$ and
 $\overline H(x)$, respectively. We calculate estimators $\hat\alpha$
 and $\hat{\overline\alpha}$ of the fraction of the influenced population
 $\alpha$ and of  good pioneers $\overline\alpha$
using Claim~\ref{e.alpha} and the estimated functions $\hat G_D(x)$,
$\hat G_{D^{(t)}}(x)$, $\hat H(x)$ and
 $\hat{\overline H}(x)$. (That is, we  find numerically zeros $\hat\xi$ and
 $\hat{\overline\xi}$ of  $\hat H(x)$  and $\hat{\overline H}(x)$,
 respectively, and plug them into~(\ref{e.alpha}) and~(\ref{e.alphabar}),
with $\hat G_D(x)$ and $\hat G_{D^{(t)}}(x)$ replacing   $G_D(x)$ and
$G_{D^{(t)}}(x)$.)

Note that in the semi-analytic approach we do not need to know/construct 
the realization of the underlying model. This observation is
a basis  of a  {\em campaign evaluation method} that we propose in
Section~\ref{s.Implications}. In fact, in reality one usually 
does not have the complete insight into the network structure
and needs to rely on  statistics collected  from the  initially
contacted pioneers. 

\subsubsection{Analytic evaluation}
Finally, for all models, except the ``coupon-collector'' one of
Section~\ref{ss.CC}, we calculate numerically the values of $\alpha$
and $\overline\alpha$ using the explicit forms of all the involved
functions.
(For the coupon-collector model we obtained the ``true'' values of   $\alpha$
and $\overline\alpha$ from a sample of $(D_i,D_i^{(t)})$ of a
larger size $N$.)

When comparing these analytic solutions to the simulation and
semi-analytic estimates we see that in some cases $N=1000$ is not big
enough to match the theoretical values. One can easily consider 
larger samples, however we decided to stay with $N=1000$ to show how
the quality of the estimation varies over different model assumptions. 
Also, $N=1000$ seems to be near the lower range of the  number of initial
pioneers one needs to contact to produce a reasonable prognosis for the
development of the campaign.  


\subsubsection{Case study}
Figures~\ref{f.PB} and~\ref{f.PLB}
present  Bernoulli influence propagation on the CM with Poisson and Power-Law degree distribution of mean $\mathbb{E}[D]=2,\,4,\,6$.
Bernoulli transmissions imply the set of good pioneers and influenced population of the same size.
The Power-Law degree with $\beta<3$ leads to positive fraction of good pioneers and influenced component for all $p>0$,
while for the Poisson degree distribution one observes the phase transition at $p=1/\lambda$.
That is,  the fractions of good pioneers and the influenced component are strictly positive if and only if $p>1/\lambda$.

Figure~\ref{f.PBPL} shows again the model with Bernoulli transmissions
on CM with Poisson and Power-Law degree distribution, this time
however for $\mathbb{E}[D]\approx 1.35$ for which both models exhibit the
phase transition in $p$.

A general observation is that the Power-Law degree distribution gives smaller
critical values of $p$ for the existence of a positive fraction of influenced
population  and good pioneers, however for these the size of these sets
increase with the transmission probability $p$ more slowly in the Bernoulli model.  
Obviously the values of $\alpha=\overline\alpha$ at $p=1$ correspond to the size of the biggest
connected component of the underlying CM.


Figure~\ref{f.PNPPL} shows  the node percolation (or ``apathetic and
enthusiastic users) on CM 
with Poisson and Power-Law degree distribution of mean 
$\mathbb{E}[D]\approx 2$. Note that the influenced components have the same size as
for Bernoulli transmissions, however good
components are smaller. The critical values of $p$ for the phase
transition are also the same as for Bernoulli transitions.
Note that estimation of the  node percolation model is more
difficult  than the Bernoulli transmissions because of higher variance of the estimators.

Finally, Figure~\ref{f.PCCPL} shows  that 
the coupon collector dynamics (``absentminded users'') on CM
produces  bigger sets of  good pioneers than  the influenced population.




\section{Application to Viral Campaign Evaluation}
\label{s.Implications}
What does the analysis presented up to now suggest in terms of strategy for a firm which is just about to start
a new marketing campaign on an online social network without having any prior information about
the network structure? 

If the fraction of good pioneers in the network is non-negligible, the 
firm has a strictly positive probability of picking a good pioneer even when it picks a pioneer 
uniformly at random from the network. 
Now when is the fraction of good pioneers non-negligible?
Since the firm has no prior information about the network structure and the campaign effectiveness,
the best it can do is to collect information from its pioneers regarding the number of friends that they
have (total degree) and the number of friends they influence in this campaign (transmitter degree), and 
then