Dogs in general, but especially small dogs are not designed to run along at bike speeds for any amount of time. It's bad for their joints; on hard surfaces even more so. If they had the chance to trot only, it would be more safe, but require for you to go very, very slow for a tiny Papillon or other small dog.
Just imagine the very painful injury of getting one of the little paws caught in the spikes of the wheel....that alone should make you wanting to find another, and much safer interaction between you and your friend.
Stay away from taking your small dog along while bike riding.