The Gifting Problem

Consider $N$ families (numbered $0$ through $N-1$). Each year, each family gives gifts to all other families based on the gifts they received in the previous year. Let's say that in some year, family $i$ received gifts with value $a_i$ (from all families in total). Then, for each valid $j$, the value of the gifts given by family $i$ to family $j$ in the next year is $G(i, j) \cdot a_i$, where $G(i