Why is the minimum runway length non-linear in relation to the landing weight? The graph below shows the landing runway length requirements for a Boeing 747-8.


(taken from the Boeing 747-8 Airplane Characteristics for Airport Planning)

Considering the fact that the kinetic energy and impulse are directly proportional to the mass, I would have expected this graph to be linear –­ but clearly that's not the case. What is causing the non-linearity here?
While the kinetic energy is proportional to mass, it is not proportional to velocity. 

The heavier the aircraft is, the higher the landing speed. So the stopping distance increases mainly because of the higher landing speed.

Comparing two landings, first at mass = $m_1$, the second at mass = $m_2 =1.1 \cdot m_1$, the landing speed will be $\sqrt{\frac{m_2}{m_1}}$ higher during the second landing.

The kinetic energy during the landing will scale proportional to the square of the mass.