  Consider a bipartite network where $N$ consumers choose to buy or not to buy
$M$ different products. This paper considers the properties of the logistic
regression of the $N\times M$ array of i-buys-j purchase decisions,
$\left[Y_{ij}\right]_{1\leq i\leq N,1\leq j\leq M}$, onto known functions of
consumer and product attributes under asymptotic sequences where (i) both $N$
and $M$ grow large and (ii) the average number of products purchased per
consumer is finite in the limit. This latter assumption implies that the
network of purchases is sparse: only a (very) small fraction of all possible
purchases are actually made (concordant with many real-world settings). Under
sparse network asymptotics, the first and last terms in an extended
Hoeffding-type variance decomposition of the score of the logit composite
log-likelihood are of equal order. In contrast, under dense network
asymptotics, the last term is asymptotically negligible. Asymptotic normality
of the logistic regression coefficients is shown using a martingale central
limit theorem (CLT) for triangular arrays. Unlike in the dense case, the
normality result derived here also holds under degeneracy of the network
graphon. Relatedly, when there happens to be no dyadic dependence in the
dataset in hand, it specializes to recently derived results on the behavior of
logistic regression with rare events and iid data. Sparse network asymptotics
may lead to better inference in practice since they suggest variance estimators
which (i) incorporate additional sources of sampling variation and (ii) are
valid under varying degrees of dyadic dependence.
