The prevailing theory of dark matter is the Cold Dark Matter (CDM) hypothesis. This hypothesis is favored because it is assumed that the dark matter particles are non-relativistic - i.e. slow moving. Because they are slow moving, they can essentially orbit in and around the original small density fluctuations, making these small density fluctuatuions stable. These small density fluctuations can clump into denser clumps due to three body gravitational interactions. A three body interaction of small clumps can result in one of the clumps being ejected at a higher speed while the other two clumps slow down and become more gravitationally bound.
However, baryonic matter can clump more effectively than CDM since the electromagnetic interractions allow baryonic matter to cool more effectively than the CDM clumps. That is why the prevailing theory is that the DM forms halos that are more distended than the clumps of baryonic matter. Thus a visible galaxy will have a more extended DM halo that extends far beyond the visible stars of the galaxy. The DM halo will also be more spherical than the flattened galactic disk.
It is true that most DM models assume the DM particles do have weak interactions, but these interactions aren't required by the CDM model. However, these weak interactions are required if any of the dark matter detection experiments are to be successful.
[Note: After more research, I discovered that my original answer was wrong. I now think this answer is correct. Sorry about that.]