Decaying effect of IV over DV. How to analyze? I need to prove my hypothesis:
The relative impact of the direct ties on the project outcome decreases as the direct ties network grows. 
So I have IV (network size) and DV (outcome). I have proved by regression that IV is a significant predictor over the DV. However, how can I prove that the effect decays as the IV grows?
 A: Assuming you are modelling the exponential decay, you can use the following model: 
$DV=\theta_1 \exp(-\theta_2 IV) $ which is non-linear
If $DV >0$, you can make the above model linear 
$\log DV=\log \theta_1 - \theta_2 IV $
The significance of $\theta_2$ indicates an exponential decay. 
A: If I understand your problem correctly, this might be a simple-enough case for some sort of non-parametric regression.
Say $Y$ is project outcome and $X$ is number of ties.
Say that the true function is something like a censored logistic -- growing rapidly at low values of $X$, but not growing as higher values of $X$ are reached.
You run a simple linear regression on something like this, and you'll get a positive slope coefficient.  Big enough sample size, and it'll be "statistically significant".  However, since the true function is a censored logistic, you'll be under-estimating the effect of direct ties at low values of $X$ and overestimating it at high values of $X$.  
If you've only got those two variables, implement some sort of local polynomial or Nadaraya-Watson estimator.  If you've got important control variables, you can use the semi-parametric Robinson estimator (run your regression on all your parametric variables, and then fit the $\epsilon$ nonparametrically with a kernel-type regression), or you could you some sort of smoothed spline estimator like what you get in R's mgcv. 
Edit:  you also shouldn't talk of "proving" your hypothesis.  Read up on what it means to reject the null.  You're not proving anything, you're showing a correlation, which only rarely has a causal interpretation.
Edit2:  Number of ties is probably an integer.  if it is a small integer, like <10, consider running your regression with dummy variables for each number of ties.  Intuitively at least, this is the same thing as a spline estimator.
