  We analyze dynamic random network models where younger vertices connect to
older ones with probabilities proportional to their degrees as well as a
propensity kernel governed by their attribute types. Using stochastic
approximation techniques we show that, in the large network limit, such
networks converge in the local weak sense to randomly stopped multitype
branching processes whose explicit description allows for the derivation of
asymptotics for a wide class of network functionals. These asymptotics imply
that while degree distribution tail exponents depend on the attribute type
(already derived by Jordan (2013)), Page-rank centrality scores have the
\emph{same} tail exponent across attributes. Moreover, the mean behavior of the
limiting Page-rank score distribution can be explicitly described and shown to
depend on the attribute type. The limit results also give explicit formulae for
the performance of various network sampling mechanisms. One surprising
consequence is the efficacy of Page-rank and walk based network sampling
schemes for directed networks in the setting of rare minorities. The results
also allow one to evaluate the impact of various proposed mechanisms to
increase degree centrality of minority attributes in the network, and to
quantify the bias in inferring about the network from an observed sample.
Further, we formalize the notion of resolvability of such models where, owing
to propagation of chaos type phenomenon in the evolution dynamics for such
models, one can set up a correspondence to models driven by continuous time
branching process dynamics.
