  We study properties of some standard network models when the population is
split into two types and the connection pattern between the types is varied.
The studied models are generalizations of the Erd\H{o}s-R\'{e}nyi graph, the
configuration model and a preferential attachment graph. For the
Erd\H{o}s-R\'{e}nyi graph and the configuration model, the focus is on the
component structure. We derive expressions for the critical parameter,
indicating when there is a giant component in the graph, and study the size of
the largest component by aid of simulations. When the expected degrees in the
graph are fixed and the connections are shifted so that more edges connect
vertices of different types, we find that the critical parameter decreases. The
size of the largest component in the supercritical regime can be both
increasing and decreasing as the connections change, depending on the
combination of types. For the preferential attachment model, we analyze the
degree distributions of the two types and derive explicit expressions for the
degree exponents. The exponents are confirmed by simulations that also
illustrate other properties of the degree structure.
