The hospital socks do not prevent thrombosis; they help to minimize edema.

Thrombosis is caused when blood flow ceases to be laminar, i.e. smooth and fluid, through the blood vessels. This is especially pertinent in the larger veins of the legs (i.e. femoral), where regular muscle movement is the primary means to ensure blood returns towards the heart. When blood does not move properly, such as when a person is immobile for extended periods of time, there is a risk that clots will begin to form (thrombosis) because platelets (a clotting factor) contact vessel walls.

This can happen from sitting too long without moving from time to time, as well as laying in a bed for too long without moving either. Hospital patients are at high risk for thrombosis, as are the elderly who remain in bed for extended periods.

Regarding sleep, most people move when they sleep, so thrombosis is not a problem. There are few people who lay perfectly and absolutely still while sleeping. Maybe the very elderly, but even they wake up enough to move their legs a little bit.

DVT has often been associated with sitting in economy class, but simply sitting in economy is not a factor in thrombis formation. One could be on a private jet, and still be at risk of thrombosis if the passenger remains immobile for too long. The risk of economy class is that, such as when sitting in the window seat, the oppurtunity to move around is limited. But in first class, a person may be immobile by simply sitting too long.

The big risk of thrombosis is embolism: when a person "throws a clot" that then becomes lodged in the lungs, for example.

I am not a doctor (just in training), but I think simply sleeping is not a risk factor. Most people cannot sleep so deeply on airplanes that they would never move once, anyway.