Mon Apr 15 10:32:23 2013
Let \( N_{i\rightarrow j,t} \) denote the number of time that
donor \( i \) donates charity \( j \) by time \( t \). We consider some evenly
spread-out time binning scheme, say, \( 0=t_0 < t_1 < \ldots < t_L = 1 \)
so that \( t_{k+1} - t_k = t_{\ell +1} - t_{\ell} \) for any \( k \) and \( \ell \).
Then, assume that for each \( \ell \),
\[
\Delta N_{i\rightarrow j,\ell} = N_{i\rightarrow j,t_{\ell+1}} - N_{i\rightarrow j,t_\ell}
\]
is a Poisson random variable with mean \( \mu_{i\rightarrow j,t} \).
Conditioning on the value of latent position \( X_{1,t},\ldots, X_{n,t} \) and \( Y_{1,t},\ldots, Y_{n,t} \in \mathbb R_+^K \), where
\( \mathbb R_+ = [0,\infty) \), we assume that \( \mu_{i\rightarrow j,t} \) is a
function only of \( X_{i,t} \) and \( Y_{j,t} \) and that the random variables
\[
(\Delta N_{i\rightarrow j,\ell}:i,j,\ell)
\]
are mutually independent. More specifically, we assume
\[
\mu_{i\rightarrow j,t} = \langle X_{i,t}, Y_{j,t} \rangle.
\]
We write \[ X(t) = (X_{1,t};\dots;X_{n,t}) \in \mathbb R_+^{n\times K} \text{ and } Y(t) = (Y_{1,t};\ldots;Y_{n,t}) \in \mathbb R_+^{n\times K}, \] where \( X_{i,t} \) and \( Y_{j,t} \) form the \( i \)-th and the \( j \)-th rows of \( X(t) \) and \( Y(t) \) respectively. Also, we write \( X_i(t) \) and \( Y_j(t) \), for the \( i \)-th and the \( j \)-th columns of \( X(t) \) and \( Y(t) \) respectively.
Then, we have, when \[ M(t) = X(t) Y(t)^\top, \] for \( i \neq j \), \( M_{ij}(t) \) equals \( \mu_{i\rightarrow j,t} \).
## working on n = 147 , nd = 99 , nc = 48 , rad = 0.5 , tmax = 2
## working on n = 147 , nd = 99 , nc = 48 , rad = 0.5 , tmax = 2
## working on n = 147 , nd = 99 , nc = 48 , rad = 0.5 , tmax = 2
