  Professional networks -- the social networks among people in a given line of
work -- can serve as a conduit for job prospects and other opportunities. Here
we propose a model for the formation of such networks and the transfer of
opportunities within them. In our theoretical model, individuals strategically
connect with others to maximize the probability that they receive opportunities
from them. We explore how professional networks balance connectivity, where
connections facilitate opportunity transfers to those who did not get them from
outside sources, and congestion, where some individuals receive too many
opportunities from their connections and waste some of them.
  We show that strategic individuals are over-connected at equilibrium relative
to a social optimum, leading to a price of anarchy for which we derive nearly
tight asymptotic bounds. We also show that, at equilibrium, individuals form
connections to those who provide similar benefit to them as they provide to
others. Thus, our model provides a microfoundation in professional networking
contexts for the fundamental sociological principle of homophily, that
"similarity breeds connection," which in our setting is realized as a form of
status homophily based on alignment in individual benefit. We further explore
how, even if individuals are a priori equally likely to receive opportunities
from outside sources, equilibria can be unequal, and we provide nearly tight
bounds on how unequal they can be. Finally, we explore the ability for online
platforms to intervene to improve social welfare and show that natural
heuristics may result in adverse effects at equilibrium. Our simple model
allows for a surprisingly rich analysis of coordination problems in
professional networks and suggests many directions for further exploration.
