Linear growth of A on B: For every new B, k new As emerge, for some k (could be and often is fractional).

Quadratic growth of A on B: A grows approximately linearly with the number of handshakes in B, so that in the above language, for every new B, k × B new As emerge, one k for every existing B that the new B could interact with.

On this basis one would say that the number of bugs is quadratic in the lines of source code. Each line has some small probability of interacting with some distant line in some non-negligible buggy way.

Exponential growth of A on B: A grows approximately linearly with the number of connections between A and B, so that in the above language, for every new B, k × A new As emerge, one k for every A that the new B could interact with.

So for “knowledge increases exponentially,” the precise technical claim is that it increases exponentially with time, and this means (somewhat dubiously for long scales) that in a given period of time, every every fact you know has a constant probability of generating a new fact which you now also know. This holds when you don't know very much about a new scenario but tends to be tempered out rapidly, the phenomenon of “low hanging fruit” etc. A meme similarly has an approximately exponential region where the number of folks exposed to the meme drives further exposure to the meme, but this dries up as the probability of “already shared it!” rises and curtails that “I must share it with friends” impulse.