# Simulation of Xfin

### YP, NL, MT, CEP, …

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 $$\mu_{i\rightarrow j,t}$$.

## Use $$A_{dc}$$ only

## working on n = 147 , nd = 99 , nc = 48 , rad = 0.5 , tmax = 2


## Use $$A_{dd}$$ and $$A_{dc}$$

## working on n = 147 , nd = 99 , nc = 48 , rad = 0.5 , tmax = 2