Let's have a deeper look at these growing random network models, and in particular we're going to start talking just about Mean Field Approxiamtions, a useful technique for these kinds of models.
Continuous time approximation will allow us to just solve a differential equation to figure out what the nodes expected degrees should be over time. Let's go back and think again about simple Erdos-Renyi variation where now each node is born, forms m links at random to the existing ones. But we'll just smooth this out and do a continuous time approximation.
starting condition => $d_i(i) = m$
new links gained per unit time => $dd_i(t)/dt = m/t$
$d_i(t) = m + m log(t/i)$
More high and low degree nodes than predicted at random
Related to other settings (wealth, city size, word usage...)
Rich get richer - more links you have, the easier it is to get links, so more wealth you get, the easier it is to get more wealth. The bigger your city is, the easier it is to get more population. These kinds of things where you get a multiplicative growth together with new objects being born over time, so new articles, new cities, in this case new nodes. Those things being grown born over time are going to gain proportionally to how large they already are and we'll end up with power loss