Combining correlations I'm not sure whether this is a question that can be analyzed statistically, but I will present it and be very interested in any feedback, whether it's an answer, or an explanation why the question can't be meaningfully asked.
Suppose there some kind of phenomenon, P1, such that standard normal deviates which are under the influence of P1 always, if there are no other influences, have a fixed correlation C1 between each pair of deviates.
Now suppose there is another phenomenon, P2, such that standard normal deviates under the influence of P2 always, if there are no other influences, have fixed correlation C2 between each pair.
Now suppose there are some pairs of deviates that fall under the influence of both P1 and P2. Is there a way to calculate the correlation, by somehow combining C1 and C2, that would result between those deviates?
If so, is there a way to extend it to handle weights for the strength of those influences, and having more than two of them?
To me, it seems intuitively sensible that there could be such phenomena, and that they could be combined to end up with specific results.
And it seems like simply computing the arithmetic average of the correlations would probably be the answer. But I'm not sure if there would be more to it than that.
It is OK if one has to make some extra assumptions for this to be calculable. Just specifying what the assumptions are would be useful for understanding.
 A: Writing as an answer rather than a comment for more space.
I think the issue I'm having is how P1 is "influencing" the correlation. I don't think I've ever seen that type of description before. You might need to add a substantial amount of detail to your post if I'm completely misunderstanding you. But, starting from the top,
(Pearson) Correlation is defined as
$$ \rho_{X,Y} = \frac{E[(X-\mu_X)(Y-\mu_Y)]}{\sigma_X\sigma_Y} = \frac{E[XY] - E[X]E[Y]}{\sigma_X\sigma_Y}$$
(Recall that if $X$ and $Y$ are independent then they are uncorrelated, but the converse is not true in general. In the specific case when $X$ and $Y$ are jointly normally distributed they are uncorrelated iff they are independent.)
When talking about multiple random variables we can discuss the correlation matrix. For example with three random variables, $X_1, X_2, X_3$, the correlation matrix would be
$$
\begin{bmatrix}
1 & \rho_{X_1,X_2} & \rho_{X_1,X_3} \\
\rho_{X_2,X_1} & 1 & \rho_{X_2,X_3} \\
\rho_{X_3,X_1} & \rho_{X_3,X_2} & 1
\end{bmatrix}
$$
Because correlation is symmetric, so is this matrix. If $X_1 = X_2 = X_3$ then all the entries will be 1. If $X_1, X_2, X_3$ are all pairwise independent then all of the off-diagonal terms will be 0. Many combinations are possible, but we cannot fill in just any values we like and always get a well defined correlation matrix. There is a requirement that the matrix be positive semi-definite. For example, this is a valid correlation matrix
$$
\begin{bmatrix}
1 & 0.707 & 0.5 \\
0.707 & 1 & 0.707 \\
0.5 & 0.707 & 1
\end{bmatrix}
$$
So is this
$$
\begin{bmatrix}
1 & 0.9 & 0.9 \\
0.9 & 1 & 0.9 \\
0.9 & 0.9 & 1
\end{bmatrix}
$$
This is not
$$
\begin{bmatrix}
1 & 0.9 & 0 \\
0.9 & 1 & 0.9 \\
0 & 0.9 & 1
\end{bmatrix}
$$
It's perfectly possible for $\rho_{X_1,X_2} = 0.9$, or $\rho_{X_1,X_3} = 0$, or $\rho_{X_2,X_3} = 0.9$. But we can't have all three simultaneously. So there is a sort of "global" restriction on the correlations that can't be detected looking only at pairs one at a time.
Now, I haven't made any mention of other phenomenons "influencing" the $X$'s. There's been no external force affecting $X_1,X_2,X_3$ that I could add or remove. I can't think of a way do to so without creating new random variables. For example, I could consider adding new random values $Y_1,Y_2,Y_3$ to the $X$'s and end up with new variables $Z_1,Z_2,Z_3$. These new variables would have correlations that are related to the original correlations, but also depend on the $Y$'s. There isn't anything general we could say about those new correlations without knowing how the $X$'s and $Y$'s are related to each other.
So, could you please clarify how these phenomenons influencing the random variables fit into the above definitions or point us to a reference discussing the type of relation you have in mind?
